Vortical structures in a turbulent far-wake: effect of Reynolds number

The effect of the Reynolds number on vortical structures in a turbulent far-wake has been investigated for Re_d (based on the free stream velocity and the cylinder diameter) =2800 and 9750. Velocity data were obtained using two orthogonal arrays of 16 X-wires, eight in the (x,y)-plane and eight in the (x,z)-plane. Structures were detected in both planes using a technique based on vorticity concentration and circulation. Conditional streamlines and contours of vorticity based on spanwise structures, i.e. detections in the (x,y)-plane, reveal that the streamwise size of spanwise structures increases as Re_d increases. The interrelationship is investigated between detections simultaneously identified in the two planes. Transverse structures, i.e. detections in the (x,z)-plane, correspond, with a relatively high probability, to spanwise structures, in conformity with a distortion in the (y,z)-plane of spanwise structures. Those that correspond, with relatively high probability, to the saddle between consecutive spanwise structures are interpreted in terms of ribs, whose signatures are detectable in instantaneous data. The probability is also high for transverse structures to occur between the focus of a spanwise structure and its associated saddle when Re_d=9750, but not when Re_d=2800. This is consistent with an increased vortex pairing frequency at the higher Re_d, as observed in instantaneous sectional streamlines.     

Pressure spectra in homogeneous low Reynolds number turbulent shear flow subjected to weak rotation

In this paper, pressure spectra have been derived from the authors’ model (Eur. J. Mech., B/Fluids 12 (1) (1993) 31–42) developed by means of rapid distortion theory (RDT) of homogeneous low Reynolds number turbulent shear flow subjected to weak rotation. The combined effects of uniform shear dU₁/dx₂ and weak rotation Ω₃ on the evolution of pressure spectra have been examined in terms of the rotation number 2Ω₃/(dU₁/dx₂). It is found that the system rotation exhibits the opposite effect on the pressure field as compared with the influence of rotation on the velocity fluctuations.     

Further consideration of the shape-factor relationship for turbulent boundary layers

The lag-entrainment predictive scheme developed by Green et al. has been modified to include the pressure-gradient parameter Π₁. In the original model suggested by Green et al. the mass-flow shape factor H₁ is related to the common shape factor H, H₁ = f(H). In the present model H₁ is related to H, Reynolds number based on the local momentum thickness θ, and Π₁; thus H₁ = f(H, R_eθ, Π₁). The modified formula for H₁, is introduced into the original lag-entrainment integral model. Calculations are made to examine the present model for the predictions of the development of boundary layers approaching separation studied experimentally by the authors. Slightly improved predictions are obtained using the model developed by El Telbany et al. However, the present model proved to give an improved representation of the development of wall shear stress in cases the two-equation turbulence model proved to be unsuccessful.     

The near wake structure of a square cylinder

The near wake structure of a square cross section cylinder in flow perpendicular to its length was investigated experimentally over a Reynolds number (based on cylinder width) range of 6700–43,000. The wake structure and the characteristics of the instability wave, scaling on θ at separation, were strongly dependent on the incidence angle () of the freestream velocity. The nondimensional frequency (St_θ) of the instability wave varied within the range predicted for laminar instability frequencies for flat plate wakes, jets and shear layers. For = 22.5°, the freestream velocity was accelerated over the side walls and the deflection of the streamlines (from both sides of the cylinder) towards the center line was higher compared to the streamlines for = 0°. This caused the vortices from both sides of the cylinder to merge by x/d 2, giving the mean velocity distribution typical of a wake profile. For = 0°, the vortices shed from both sides of the cylinder did not merge until x/d 4.5. The separation boundary layer for all cases was either transitional or turbulent, yet the results showed good qualitative, and for some cases even quantitative, agreement with linearized stability results for small amplitude disturbances waves in laminar separation layers.     

The effect of Reynolds number on a turbulent far-wake

The turbulent far-waked generated by a circular cylinder is investigated for two values (1350 and 4600) of the Reynolds number Re _θ (based on the free stream velocity and the momentum thickness). Two arrays of sixteen X-wires, eight in the (x,?y)-plane and eight in the (x,?z)-plane, are used to capture the main features of the large-scale motion in two orthogonal planes. Both the magnitude of the measured Reynolds stresses and the size of the two-point velocity and vorticity correlation contours increase with Reynolds number. The probability density function and spectra of the velocity signals also exhibit differences with Re _θ. A comparison of centerline turbulence intensities with those in the literature suggests that the Reynolds number dependence may disappear for Re _θ?5000.     

Experimental study of the Kolmogorov refined similarity variable

Recently Thoroddsen and Van Atta (1992, Phys. Fluids A4, 2592) showed that Kolmogorov's refined similarity hypothesis (1962, J. Fluid Mech. 13, 77; 82) is supported by experimental data from a wind-tunnel study of a cylinder wake, at Re_λ of 550. We show here that the probability density of the Kolmogorov similarity variable approaches a Gaussian distribution. The data also suggest that it may obey an even more demanding conditional similarity, which leads to important conclusions regarding the scaling exponents.     

Experimental study of turbulence in isothermal and stably stratified homogeneous shear flows: Mean flow measurements

The evolution of freestream turbulence under the combined action of linear shear and stable linear temperature profile is investigated. The experiment is carried out in a small, open circuit, low-speed test cell that uses air as working fluid. The temperature gradient formed at the entrance to the test section by means of an array of 24 horizontal, differentially heated elements is varied to get a maximum Brunt-Vaisala frequency N_o[=({g/T_m}{∂T/∂y})^1/2] of 3.1⁻¹. Linear velocity profiles are produced using screens of variable mesh size. The Reynolds number Re_M based on centre-line velocity and mesh size is varied from 80 to 175. Isothermal studies are carried out in four different experiments with varying velocity gradients. The effect of inlet turbulence level on growth of turbulence is studied in these flows by keeping the shear parameter Sh (=(x/u)(∂u/∂y)) constant. The range of shear parameters considered is 2.5–7.0. Shear and stratification combined produce a maximum gradient Richardson number Ri_g (= N_o²/(∂u/∂y)²) of 0.0145. Results have been presented in terms of evolution of variance of velocity fluctuations, Reynolds shear stress and temperature fluctuations. Measurements show the following: In isothermal flows the growth rate of turbulence quantities depends on both shear parameter and inlet turbulence level. There are distinct stages in the evolution of the flow and that can be identified by the power-law exponent of growth of turbulence. Shear is seen to promote the growth of turbulence and accelerate it towards a fully developed equilibrium state. Stratification initially suppresses the growth of turbulence and, hence, enhances the degree of underdevelopment. Under these conditions shear becomes active and subsequently enhances the growth rate of turbulence quantities.     

Boundary-layer relaxation after a separated region

With reference particularly to the work of Peter Bradshaw and his associates, some remarks are made about the recovery of previously distorted shear flows. It is emphasized that such recovery is usually extremely slow, and this is further illustrated by new measurements of the velocity field and turbulence structure in the relaxing flow downstream of a separated region. Data have been obtained for downstream distances (x) up to about 20 times the length of the separated region (x_r), or about 75 times the flow thickness at reattachment. This is a significantly more extensive region than has been previously studied, and the data are more comprehensive than any previously available.

It is shown that the recovery is even slower than previously surmized. Furthermore, the measurements demonstrate that the turbulence stresses eventually fall below standard boundary-layer values (at the same Reynolds number), although around reattachment they are very much higher, having values more akin to those in plane mixing layers. This undershoot is apparently a new finding and is argued to be a result of the influence of the outer part of the flow on the growing inner region. The usual log-law only begins to appear beyond x/x_r = 2.5. It effectively “sees” a turbulent outer region that recovers even more slowly than itself, and the response of the inner region therefore has similarities to the response of an ordinary boundary layer to free-stream turbulence.

It is concluded that even current second-order (i.e., Reynolds stress) models may not capture the exquisitely slow decay of the strong, large eddy motions in the outer part of the flow and the subtleties of their influence on the inner region.     

Asymptotic soliton train solutions of Kaup–Boussinesq equations

Asymptotic soliton trains arising from a ‘large and smooth’ enough initial pulse are investigated by the use of the quasiclassical quantization method for the case of Kaup–Boussinesq shallow water equations. The parameter varying along the soliton train is determined by the Bohr–Sommerfeld quantization rule which generalizes the usual rule to the case of ‘two potentials’ h₀(x) and u₀(x) representing initial distributions of height and velocity, respectively. The influence of the initial velocity u₀(x) on the asymptotic stage of the evolution is determined. Excellent agreement of numerical solutions of the Kaup–Boussinesq equations with predictions of the asymptotic theory is found.     

The instability of rhombic cell flows

Instability of two-dimensional periodic flows with rhombic cell structure represented by the stream function Ψ=cos kx+cosy is investigated. Stability characteristics are obtained for the Reynolds number R=1, 2, 3 and 4 and the ratio of the diagonals of the cell . Variation of the critical Reynolds number R_c with k is obtained, and the square cell flow (k=1) is found to be most stable (R_c=√2). It is found that R_c → 1 as k → 0, which leads to a finite gap between this limiting R_c and R_c=√2 for K=0 (Ψ=cos y).     

Numerical and experimental study of turbulent impinging twin-jet flow

The two dimensional impinging circular twin-jet flow with no-cross flow is studied numerically and experimentally. The theoretical predications are carried out through numerical procedure based on finite volume method to solve the governing mass, momentum, turbulent kinetic energy and turbulent kinetic energy dissipation rate. The parameters studied were jet Reynolds number (9.5 × 10⁴  Re  22.4 × 10⁴), nozzle to plate spacing (3  h/d  12), nozzle to nozzle centerline spacing (l/d = 3, 5 and 8) and jet angle (0°  θ  20°). It is concluded that the stagnation primary point moves away in the radial main flow direction by increasing the jet angle. This shift becomes stronger by increasing the nozzle to nozzle centerline spacing (l/d). A secondary stagnation point is set up between two jets. The value of pressure at this point decreases by decreasing Reynolds number and/or increasing the jet angle.

The sub atmospheric region occurs on the impingement plate. It increases strongly by increasing Reynolds number and decreases as the jet angle and/or a nozzle to plate spacing increases. The spreading of jet decreases by increasing nozzle to plate spacing. The intensity of re-circulation zone between two jets decreases by increasing of h/d and jet angle. The increase of turbulence kinetic energy occurs within high gradient velocity.     

Rotation invariant constitutive relation for Reynolds stress structure parameter

A new Reynolds stress constitutive formula is constructed using the firstorder statistics of turbulent fluctuations instead of the mean strain rate. It includes zero empirical coefficients. The formula is validated with the direct numerical simulation(DNS) data of turbulent channel flow at Reτ =180. The Reynolds stresses given by the proposed formula agree very well with the DNS results. The good agreement persists even after the multi-angle rotation of the coordinate system, indicating the rotation invariance of the formula. The autocorrelation of the fluctuating velocity rather than the mean strain rate is close to the essence of the Reynolds stress.     

Effect of tube diameter on Preston tube calibration curves for the measurement of wall shear stress

The effect of tube diameter (d) on Preston tube calibration curves for the measurement of wall shear stress (τ_w) in a zero pressure gradient turbulent boundary layer has been investigated. Five different outside diameter tubes of 1.46, 1.82, 3.23, 4.76 and 5.54 mm, corresponding to (d/δ) of 0.022, 0.027, 0.048, 0.071 and 0.082 were used to measure τ_w at Reynolds numbers based on momentum thickness (R_θ) of 2800–4100. The calibration curves of Patel (V.C. Patel, J. Fluid Mech. 23 (part I) (1965) 185–208) and Bechert (D.W. Bechert, AIAA J. 34 (1) (1995) 205–206) are both dependent on the tube diameter. The maximum difference in the τ_w measurements from the different tubes using Patel's calibration is about 8%, while Bechert's calibration gives a maximum difference of approximately 18%.     

圆盘近尾迹流场相关谱分析

对Re=22 000 时径厚比D/h=5 圆盘近尾迹开展大涡模拟数值研究. 通过对x/D=1, 2 和8 处脉动速度进行快速傅里叶变换（fast Fourier transformation, FFT）,发现3 个特征频率：斯特劳哈尔数St₂=0.123 为自然脱落频率,与文献结果相符;较小频率St₁=0.035,与回流区伸缩和剪切层附近涡旋脱落点的旋转相关;高频率St₃=1.3～1.7 则与剪切层湍流结构相关. 通过分析截面r/D=2.8 圆周上两点间流向速度相关系数、相干谱和相位谱,发现相关系数受涡旋脱落影响出现以30°或45°为周期的正负波动,表明轴面上涡旋脱落点具有随机性.     

Reynolds方程在纹理表面动压润滑计算中的有效性评价

从理论上阐述了纹理表面动压润滑计算中决定Reynolds方程有效性的两个关键因素为油膜厚度与纹理特征长度的比值h/L和缩减的雷诺数re;只有当h/L和re同时趋近于零时Reynolds方程才能够获得准确的结果,并由此在h/L-Re平面上标注了Reynolds方程的适用范围.继而以二维矩形沟槽为实例,采用数值方法计算了h/L和re对Reynolds方程误差的影响规律;分析了Reynolds方程在不同条件下的失效机制;分析了矩形沟槽纹理表面Reynolds方程有效性的评价标准:当缩减的雷诺数re小于0.20,并且h/L小于0.015时能够保证Reynolds方程的误差在10%以下.     

Stress state of compound polygonal plate

The constructions made of bars and plates with holes, openings and bulges of various forms are widely used in modern industry. By loading these structural elements with different efforts, there appears concentration (accumulation) of stress whose values sometimes exceeds the admissible one. The durability of the given element is defined according to the quantity of these stresses. Since the failure of details and construction itself begins from the place where the stress concentration has the greatest value.

Therefore the exact determination of stress distribution in details (bars, plates, beams) is of great scientific and practical interest and is one of the important problems of the solid fracture.

Compound details (when the nucleus of different material is soldered to the hole) are often used to decrease the stress concentration.

In the present paper, we study a stress–strain state of polygonal plate weakened by a central elliptic hole with two linear cracks info which a rigid nucleus (elliptic cylinder with two linear bulges) of different material was put in (soldered) without preload.

The problem is solved by a complex variable functions theory stated in papers [Theory of Elasticity, Higher School, Moscow, 1976, p. 276; Plane Problem of Elasticity Theory of Plates with Holes, Cuts and Inclusions, Publishing House Highest School, Kiev, 1975, p. 228; Bidimensional Problem of Elasticity Theory, Stroyizdat, Moscow, 1991, p. 352; Science, Moscow (1996) 708; MSB AH USSR OTH 9 (1948) 1371].

Kolosov–Mushkelishvili complex potential (z) and ψ(z) satisfying the definite boundary conditions are sought in the form of sums of functional series.

After making several strict mathematical transformations, the problem is reduced to the solution of a system of linear algebraic equations with respect to the coefficients of expansions of functions (z) and ψ(z).

Determining the values of (z) and ψ(z), we can find the stress components σ_r, σ_θ and τ_rθ at any point of cross-section of the plate and nucleus on the basis of the known formulae. The obtained solution is illustrated by numerical example.

Changing the parameters A₁, m₁, e, A₂, and m₂ we can get the various contour plates.

For example, if we assume m₁=0, A₁=r, then the internal contour of L₁ becomes the circle of radius r with two rectilinear cracks (for the nucleus––a rectilinear bulges).

Further, if we assume a small semi-axis of the ellipse b₁ to be equal to zero (b₁=0), then a linear crack becomes the internal contour of L₁ (and the nucleus becomes the linear rigid inclusion made of other material). For m₂=0; A₂=R, the external contour L₂ turns into the circle of radius R.

The obtained method of solution may be applied and in other similar problems of elasticity theory; tension of compound polygonal plate, torsion and bending of compound prismatic beams, etc.     

The effect of Reynolds number on boundary layer turbulence

A new facility for studying high Reynolds number incompressible turbulent boundary layer flows has been constructed. It consists of a moderately sized wind tunnel, completely enclosed by a pressure vessel, which can raise the ambient air pressure in and around the wind tunnel to 8 atmospheres. This results in a Reynolds number range of about 20:1, while maintaining incompressible flow. Results are presented for the zero pressure gradient flat plate boundary layer over a momentum thickness Reynolds number range 1500–15?000. Scaling issues for high Reynolds number non-equilibrium boundary layers are discussed, with data comparing the three-dimensional turbulent boundary layer flow over a swept bump at Reynolds numbers of 3800 and 8600. It is found that successful prediction of these types of flows must include length scales which do not scale on Reynolds number, but are inherent to the geometry of the flow.     

Inertial instability of the Stewartson -layer

Inertial stability of a vertical shear layer (Stewartson E^1/4-layer) on the sidewall of a cylindrical tank with respect to stationary axisymmetric perturbations is inverstigated by means of a linear theory. The stability is determined by two non-dimensional parameters, the Rossby number Ro = U/2ΩL and Ekman number E = v/ΩH², where U and L = (E/4)1/4H are the characteristic velocity and width of the shear layer, respectively, Ω the angular velocity of the basic rotation, v the kinematic viscosity and H the depth of the tank.

For a given Ekman number, the flow is more unstable for larger values of the Rossby number. For E = 10⁻⁴, which is a typical value of the Ekman number realized in rotating tank experiments, the critical Rossby number Ro_c for instability and the critical axial wavenumber m_c non-dimensionalized by L⁻¹ are found to be 1.3670 and 8.97, respectively. The value of Ro_c increases and that of m_c decreases with increasing E.     

Coherent fine scale eddies in turbulence transition of spatially-developing mixing layer

To investigate the relationship between characteristics of the coherent fine scale eddy and a laminar–turbulent transition, a direct numerical simulation (DNS) of a spatially-developing turbulent mixing layer with Re_ω,0 = 700 was conducted. On the onset of the transition, strong coherent fine scale eddies appears in the mixing layer. The most expected value of maximum azimuthal velocity of the eddy is 2.0 times Kolmogorov velocity (u_k), and decreases to 1.2u_k, which is an asymptotic value in the fully-developed state, through the transition. The energy dissipation rate around the eddy is twice as high compared with that in the fully-developed state. However, the most expected diameter and eigenvalues ratio of strain rate acting on the coherent fine scale eddy are maintained to be 8 times Kolmogorov length (η) and :β:γ = −5:1:4 in the transition process. In addition to Kelvin–Helmholtz rollers, rib structures do not disappear in the transition process and are composed of lots of coherent fine scale eddies in the fully-developed state instead of a single eddy observed in early stage of the transition or in laminar flow.     

Flow in a cylinder driven by rotating split-disk endwalls

Flow of an incompressible viscous fluid contained in a cylindrical vessel (radius R, height H) is considered. Each of the cylinder endwalls is split into two parts which rotate steadily about the central axis with different rotation rates: the inner disk (r < r₁) rotating at Ω₁, and the outer annulus (r₁ < r < R) rotating at Ω₂. Numerical solutions to the axisymmetric Navier-Stokes equations are secured for small system Ekman numbers E ( v/(ΩH²)). In the linear regime, when the Rossby number Ro

Full-size image (1K)

, the numerical results are shown to be compatible with the theoretical prediction as well as the available experimental measurements. Emphasis is placed on the results in the nonlinear regime in which Ro is finite. Details of the structures of azimuthai and meridional flows are presented by the numerical results. For a fixed Ekman number, the gross features of the flow remain qualitatively unchanged as Ro increases. The meridional flows are characterized by two circulation cells. The shear layer is a region of intense axial flow toward the endwall and of vanishing radial velocity. The thicknesses of the shear layer near r = r₁ and the Ekman layer on the endwall scale with E and E , respectively. The numerical results are consistent with these scalings.