Mode transition and oscillation suppression in supersonic cavity flow

Supersonic flows past two-dimensional cavities with/without control are investigated by the direct numerical simulation(DNS). For an uncontrolled cavity, as the thickness of the boundary layer declines, transition of the dominant mode from the steady mode to the Rossiter Ⅱ mode and then to the Rossiter Ⅲ mode is observed due to the change of vortex-corner interactions. Meanwhile, a low frequency mode appears. However, the wake mode observed in a subsonic cavity flow is absent in the current simulation.The oscillation frequencies obtained from a global dynamic mode decomposition(DMD)approach are consistent with the local power spectral density(PSD) analysis. The dominant mode transition is clearly shown by the dynamic modes obtained from the DMD. A passive control technique of substituting the cavity trailing edge with a quarter-circle is studied. As the effective cavity length increases, the dominant mode transition from the Rossiter Ⅱ mode to the Rossiter Ⅲ mode occurs. With the control, the pressure oscillations are reduced significantly. The interaction of the shear layer and the recirculation zone is greatly weakened, combined with weaker shear layer instability, responsible for the suppression of pressure oscillations. Moreover, active control using steady subsonic mass injection upstream of a cavity leading edge can stabilize the flow.     

Imaging of the self-excited oscillation of flow past a cavity during generation of a flow tone

Flow through a pipeline-cavity system can give rise to pronounced flow tones, even when the inflow boundary layer is fully turbulent. Such tones arise from the coupling between the inherent instability of the shear flow past the cavity and a resonant acoustic mode of the system. A technique of high-image-density particle image velocimetry is employed in conjunction with a special test section, which allows effective laser illumination and digital acquisition of patterns of particle images. This approach leads to patterns of velocity, vorticity, streamline topology and hydrodynamic contributions to the acoustic power integral. Comparison of global, instantaneous images with time- and phase-averaged representations provides insight into the small-scale and large-scale concentrations of vorticity, and their consequences on the topological features of streamline patterns, as well as the streamwise and transverse projections of the hydrodynamic contribution to the acoustic power integral. Furthermore, these global approaches allow the definition of effective wavelengths and phase speeds of the vortical structures, which can lead to guidance for physical models of the dimensionless frequency of oscillation.     

Influence of flow path modification on oscillation of cavity shear layer

This study investigates the influence on the oscillating characteristics of a cavity shear layer by introducing either a sloped bottom or a flow path modifier at the bottom of the cavity. All the experiments are performed in a recirculating water channel. The laser Doppler velocimetry system and the laser sheet technique are employed to perform the quantitative velocity measurements and the qualitative flow visualization, respectively. The Reynolds number, based on the momentum thickness at the upstream edge of the cavity, is kept at about Re _{θ 0}=194 ± 3.4. It is found that, in addition to the feedback effect, the upstream moving part of the recirculating flow inside the cavity also plays an important role in changing the oscillating characteristics of the unstable shear layer. As the bottom of the cavity is either negatively or positively sloped, the oscillating characteristics of the cavity shear layer are modified to different extents. Significant reduction of the oscillating amplitude within the cavity is found while the bottom slope increases up to d/L=± 2/5. As the bottom slope further increases up to d/L=± 1/2, the self-excited oscillation is completely suppressed. In addition, the ability to suppress the self-excited oscillation by the negative bottom slopes is superior to that in the case of a positive bottom slope. Depending upon the fence locations, the upstream moving part of the recirculating flow will perturb the unstable shear layer at different x/L locations, leading to different oscillating amplitudes. The ability to promote the enlarged oscillating amplitude of the unstable shear layer is better for a fence inclined at a positive angle than for one at a negative angle. Received: 31 May 2000/Accepted: 11 January 2001     

Analysis of two-dimensional cavity flow by finite elements

The variational principle in terms of stream function ψ for free surface gravity flow is discussed by the formulation of first-order variation in a variable domain. Because of different transversal conditions adopted, there are four forms of variational principle in terms of ψ.A n air-gilled cavity flow with given discharge and total energy is then analysed by finite element method. At the end of the cavity, the free stream line is tangent to a short fictitious plate of given length, which joins the fixed boundary at on angle to be determined. The condition that the free stream line should be tangent to the fixed boundary at the point of separation makes the solution unique.Finally curves giving the cavity length as a function of the Fraude number, cavity pressure and channel bottom slope are presented.     

Effect of an axial throughflow on buoyancy-induced flow in a rotating cavity

In this paper large-eddy simulation is used to study buoyancy-induced flow in a rotating cavity with an axial throughflow of cooling air. This configuration is relevant in the context of secondary air systems of modern gas turbines, where cooling air is used to extract heat from compressor disks. Although global flow features of these flows are well understood, other aspects such as flow statistics, especially in terms of the disk and shroud boundary layers, have not been studied. Here, previous work for a sealed rotating cavity is extended to investigate the effect of an axial throughflow on flow statistics and heat transfer. Time- and circumferentially-averaged results reveal that the thickness of the boundary layers forming near the upstream and downstream disks is consistent with that of a laminar Ekman layer, although it is shown that the boundary layer thickness distribution along the radial direction presents greater variations than in the sealed cavity case. Instantaneous profiles of the radial and azimuthal velocities near the disks show good qualitative agreement with an Ekman-type analytical solution, especially in terms of the boundary layer thickness. The shroud heat transfer is shown to be governed by the local centrifugal acceleration and by a core temperature, which has a weak dependence on the value of the axial Reynolds number. Spectral analyses of time signals obtained at selected locations indicate that, even though the disk boundary layers behave as unsteady laminar Ekman layers, the flow inside the cavity is turbulent and highly intermittent. In comparison with a sealed cavity, cases with an axial throughflow are characterised by a broader range of frequencies, which arise from the interaction between the laminar jet and the buoyant flow inside the cavity.     

Laminar mixed convection on a horizontal plate of finite length in a channel of finite width

The steady two-dimensional laminar mixed-convection flow past a horizontal plate of finite length is analysed for large Péclet numbers, small Prandtl numbers and weak buoyancy effects. The plate is placed in a channel of finite width, with the plane walls of the channel being parallel to the plate. The temperature of the plate is assumed to be constant. The hydrostatic pressure difference across the wake behind the plate is compensated by a perturbation of the inviscid channel flow. This outer flow perturbation affects the temperature distribution in the thermal boundary layer at the plate and the heat transfer rate, respectively. Solutions in closed form are given. The forces acting on the plate due to the potential flow perturbation are also determined.     

The evolution of a finite rate periodic oscillation

Finite rate oscillations of a gas in a closed tube arise when the amplitude of the applied periodic piston velocity is small while its acceleration is unrestricted. The asymptotic form of the periodic motion for large acceleration is given. The evolution to the final periodic motion from the initial state of rest is constructed for a finite rate oscillation. Exact results for a piecewise linear piston velocity are used to illustrate the solutions.     

Aerosol aspiration through a slot of finite width

Aerosol aspiration has been calculated by Levin for a sink simulating a fairly narrow slot or a thin tube [1].In the present study we determine the aspiration coefficient for aerosol particles for a two-dimensional nozzle of finite width. For strongly inertial aerosol particles, when kk_*, where k is the Stokes number and k_* is its critical value, the approximate equations proposed in [2] are applicable.For kk_*, when there is no inertial deposition of aerosol particles on the nozzle wall, we use the small parameter method. The aspiration coefficient is calculated with accuracy to k². Some numerical data and a comparison with sink flow are presented.     

矩形空腔内Stokes流的状态空间有限元法

基于Hellinger-Reissner二类变分原理,从平面Stokes流问题的平衡方程、连续性要求和边界条件出发,得到相应的Hamilton函数,建立Hamilton正则方程后,采用分离变量法对场变量进行离散求解:在x方向采用有限元插值,在y方向采用状态空间法给出控制坐标方向的解析解。计算过程中的指数矩阵均采用精细积分法求解,使得本文算法具有高效率、高精度、对步长不敏感的优点。通过对侧边自由液面边界条件的单板驱动矩形空腔Stokes流问题的求解,得到与文献相同的结果,从而验证了本文方法的有效性。本文旨在将弹性力学状态空间有限元法的思想引入到低雷诺数流体力学中,为Hamilton体系下研究复杂边界Stokes流问题提供新的途径。     

Pressure oscillation on a plane upstream of an obstacle in a supersonic flow

The levels and spectra of pressure oscillation on a plane upstream of a vertical cylinder and a step in an M=3 supersonic flow are measured in the presence of a turbulent boundary layer. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 69–74, January–February, 1998.     

Influence of external turbulence on flow in a rectangular cavity

This paper presents the results of an experimental investigation of the influence of external turbulence on a flow in a rectangular cavity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 162–166, March–April, 1986.     

Effect of water density inversion on the plane-parallel flow and heat transfer in a channel of constant width

The results of a numerical study of the effect of cold-water density inversion (Prandtl number Pr=11.59) on the flow and heat transfer in a horizontal plane-parallel channel with isothermal top and bottom walls are presented. The calculations were performed for the Grashof number Gr=3·10⁴, the Reynolds number Re=10, and the channel segment length-to-height ratiol/d=40. The wall temperature was so varied that the temperature difference between the top and bottom walls remained constant. Surgut. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 72–78, January–February, 2000.     

Biomagnetic fluid flow in a driven cavity

In this study, the fundamental problem of the biomagnetic fluid flow in a lid driven cavity under the influence of a steady localized magnetic field is studied. The mathematical model used for the formulation of the problem is consistent with the principles of Ferrohydrodynamics (FHD) and Magnetohydrodynamics (MHD). The biomagnetic fluid is considered as a homogeneous Newtonian fluid and is treated as an electrically conducting magnetic fluid which also exhibits magnetization. A known biomagnetic fluid which exhibits such magnetic properties is blood. For the numerical solution of the problem, which is described by a coupled, non linear system of PDEs, with appropriate boundary conditions, the SIMPLE algorithm is used. The solution is obtained by the development of a numerical methodology using finite volumes on a staggered, properly stretched, grid. Results concerning the velocity indicate that the presence of the magnetic field influences considerably the flow field.     

Kolmogorov scales in a driven cavity flow

Turbulent flow in a three-dimensional driven cavity has been simulated directly by solving the Navier–Stokes equations. The results at Re=3200 and 10 000 compare well with the experimental data. Viscous dissipation rate has been calculated without making the assumption of isotropy. Near the top moving wall, the instantaneous dissipation rate is very high and also has high amplitude. Its frequency increases but amplitude decreases as one moves away from the wall and it becomes intermittent in the vortex core. The high Reynolds number assumption that dissipation is mainly due to the fluctuating velocity components is seen to be true in the present case except near the wall. The Kolmogorov length scale attains higher values in the core of the primary vortex due to low dissipation rate there. A value of 0.01 times the size of the cubic cavity is a good representative value at Re=10 000. Even though the present (84×84×84) grid cannot resolve this scale very well, it can resolve all the scales dynamically significant in the flow as seen from the velocity and dissipation spectra.     

Effect of finite depth of stratified flow on turbulence formation in the wake of a cylinder

The experimental results of investigating the effect of the finite depth of a linearly stratified flow channel on turbulence formation associated with the horizontal motion of a cylinder are presented. The limits of the interval of internal Froude numbers and dimensionless channel depths (with respect to the cylinder diameter) corresponding to local instabilites in the disturbed flow density distribution, leading to the formation of turbulence, are found. The dynamics of formation of the turbulent zones and their evolution are investigated. Unsteady periodic regimes are found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 160–163, September–October, 1990.The authors are grateful to A. T. Onufriev for his interest in their work.     

Surface cracks in a plate of finite width under extension or bending

In this paper the problem of a finite plate containing collinear surface cracks is considered. The problem is solved by using the line spring model with plane elasticity and Reissner's plate theory. The main purpose of the study is to investigate the effect of interaction between two cracks or between cracks and stress-free plate boundaries on the stress intensity factors and to provide extensive numerical results which may be useful in applications. First, some sample results are obtained and are compared with the existing finite element results. Then the problem is solved for a single (internal) crack, two collinear cracks and two corner cracks for wide range of relative dimensions. Particularly in corner cracks the agreement with the finite element solution is surprisingly very good. The results are obtained for semielliptic and rectangular crack profiles which may, in practice, correspond to two limiting cases of the actual profile of a subcritically growing surface crack.     

Plane-parallel flow around a gas cavity

The stationary motion of a gas cavity in an ideal incompressible fluid is studied taking account of surface tension by using a variational equation. Approximate analytical dependences of the dimensionless parameters on the degree of cavity deformation are obtained. It is shown that the variational equation admits of an exact analytical solution. The stability of motion corresponding to the exact solution is proved relative to arbitrary perturbations in the cavity shape. A solution is given for the problem of stationary motion of an elliptical cavity in a gravity viscous fluid and the stability problem is investigated. Dependences are found for the velocity of cavity rise, the Reynolds number, and the Froude number as a function of the cavity size.     

Effect of cavity aspect ratio on flow and heat transfer characteristics in pipes: a numerical study

Using the standard k–ε turbulence model, a two-dimensional turbulent pipe flow was simulated with and without square cavities. Effect of cavity aspect ratio on flow and heat transfer characteristics was investigated. Uncertainty was approximated through experimental validation and grid independence. The simulation revealed circulation inside the cavities. Cavity boundaries were shown to contribute significantly toward turbulence production. Cavity presence was shown to enhance overall heat transfer through the wall, while increasing pressure drop significantly across the pipe. It was predicted that cavities with higher aspect ratio enhance heat transfer more while increasing pressure drop.