The effect of the Reynolds number on the Reynolds stresses and vorticity in a turbulent far-wake

Using two orthogonal arrays of 16 X-wires, eight in the (x,y)-plane and eight in the (x,z)-plane, the effect of the Reynolds number in a turbulent plane far-wake has been investigated for two values of Re_θ (based on the free stream velocity and the momentum thickness), i.e. 1350 and 4600. It is observed that as the Reynolds number increases the magnitudes of the measured Reynolds stresses increase, as does the size of two-point vorticity correlation iso-contours. Discernible differences are also observed in probability density function, spectra and three-dimensional topologies. The Reynolds number dependence seems to vanish when Re_θ5000.     

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.     

Hole diameter effect on flow characteristics of wake behind porous fences having the same porosity

The hole diameter effect on the flow characteristics of wake behind porous fences has been investigated experimentally in a circulating water channel having a test section of 300^w×200^h×1200^l (mm). Three porous fences having different hole diameters of d=1.4,2.1,2.8 mm were tested in this study, but they have the same =38.5% geometric porosity. One thousand instantaneous velocity fields for each fence were measured consecutively by the hybrid PTV system employing a high-speed CCD camera. Free stream velocity was fixed at 10 cm/sec and the corresponding Reynolds number based on the fence height was Re=2,985. Consequently, the fence with the smallest hole diameter d=1.4 mm (d_1.4) decreases the streamwise velocity component and increases the vertical velocity component. Among the three hole diameters tested in this study, the d_1.4 fence has the largest turbulence intensity in the shear layer developed from the fence top. Regardless of the hole diameter, however, all three fences having the same porosity reduce the reduction of turbulent intensity in the lower region below the fence height (y/H<1).     

Three-dimensional numerical study and field synergy principle analysis of wavy fin heat exchangers with elliptic tubes

Three dimensional numerical studies were performed for laminar heat transfer and fluid flow characteristics of wavy fin heat exchangers with elliptic/circular tubes by body-fitted coordinates system. The simulation results of circular tube were compared with the experiment data, then circular and elliptic (e = b/a = 0.6) arrangements with the same minimum flow cross-sectional area were compared. A max relative heat transfer gain of up to 30% is observed in the elliptic arrangement, and corresponding friction factor only increased by about 10%. The effects of five factors on wavy fin and elliptic tube heat exchangers were examined: Reynolds number (based on the smaller ellipse axis, 500  4000), eccentricity (b/a, 0.6  1.0), fin pitch (F_p/2b, 0.05  0.4), fin thickness (F_t/2b, 0.006  0.04) and tube spanwise pitch (S₁/2b, 1.0  2.0). The results show that with the increasing of Reynolds number and fin thickness, decreasing of the eccentricity and spanwise tube pitch, the heat transfer of the finned tube bank are enhanced with some penalty in pressure drop. There is an optimum fin pitch (F_p/2b = 0.1) for heat transfer, but friction factor always decreases with increase of fin pitch. And when F_p/2b is larger than 0.25, it has little effects on heat transfer and pressure drop. The results were also analyzed from the view point of field synergy principle. It was found that the effects of the five factors on the heat transfer performance can be well described by the field synergy principle.     

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 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.     

Impingement heat transfer at a circular cylinder due to a submerged slot jet of water

An experimental investigation was carried out on the heat transfer due to a submerged slot jet of water impinging on a circular cylinder in crossflow. The cylinder diameter and the slot width are of the same order of magnitude, specifically D_s = 2.0 and 3.0 mm and D_c = 2.5 and 3.0 mm. The experimental apparatus allowed variation of the slot width, the cylinder diameter, and the distance from nozxle exit to heater. Conditions of impingement from the bottom (ascending flow) were taken into consideration as well as impingement from above (descending flow). The Nusselt number was determined as a function of Reynolds and Prandtl numbers in the range 1.5 × 10³ < Re < 2.0 × 10⁴, 2.7 < Pr < 7.0, and 1.5 ≤ z/D_s ≤ 10. The experimental data were correlated with a simple equation that fits 90% of the data with a precision of 20%.     

Large-scale structure of a leading-edge separation bubble with local forcing

The Karhunen-Loeve (K-L) expansion is used to extract coherent structures from a leading-edge separation bubble with local forcing. A leading-edge separation bubble is simulated using the discrete vortex method, where a time-dependent source forcing is perturbed near the separation point. Based on the wealth of numerical data, the K-L procedure is applied in a range of the forcing amplitude (A₀ = 0, 0.5, 1.0 and 1.5) and forcing frequency (0 f_FH/U_∞ 0.3). Application of K-L procedure reveals that the eigenstructures are changed noticeably by local forcings. In an effort to investigate the mechanism of decreasing reattachment length (x_R), dynamic behaviors of the expansion coefficients and contributions of the eigenfunctions are scrutinized. As the forcing amplitude increases, large-scale vortex structures are formed near the separation point. Furthermore, the flow becomes more organized, which results in the reduction of x_R. Two distinctive regimes are classified: the regime of decreaing x_R and the regime of increasing x_R. The K-L global entropy indicates that x_R is closely linked to the organization of the flow structure.     

The wake behaviour behind a heated horizontal cylinder

The behaviour of vortex structures shed from a heated cylinder is experimentally investigated by means of 2-D particle tracking velocimetry. Within this investigation the Re_D number was chosen to be 73. The Ri_D number, the dimensionless number which presents the relative importance of the induced heat, varies between 0 and 1. The experiments were carried out in a large towing tank where the disturbances caused by boundary layers could be minimised. The results show that for small Ri_D numbers the induced heat results in a deflection of the vortex street in negative y-direction. Within the vortex street a linking of two subsequently shed vortices occurs where the vortex shed from the lower half of the cylinder rotates around the vortex shed from the upper half. These phenomena are assumed to be caused by a strength difference between the vortices shed from the upper half of the cylinder and the lower half. For Ri_D=1 the effect of the induced heat and buoyancy becomes even more pronounced resulting in a more upwards directed vortex street.     

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.     

Uniformly valid approximations for non-linear oscillations with small damping

A uniformly valid zeroth-order approximation is obtained for the general equation y + εH(y)y + M(y)y = 0, where ε is a small parameter. The notion of multiple scaling is utilized to set up a systematic approximation scheme. Examples are given for simple polynomials for H(y) and M(y), which lead to results involving elliptic integrals. Further restrictions allow progress to be made in terms of gamma functions.     

Vortex shedding behind rings and discs

The wake structure of discs and bluff rings has been investigated experimentally in a wind tunnel. The rings have an inner diameter d_i, and an outer diameter d_o and are classified according to the parameter (d_o + d_i)/(d_o − d_i) = d/w. the ratio of mean diameter to ring width. As d/w → ∞ the flow approaches that around a two dimensional bluff body whereas as d/w tends to unity the body approaches a solid disc. A distinct change in the vortex shedding pattern is found around d/w = 5. Below this critical value velocity fluctuations in the wake have a weak periodic component which is 180° out of phase across a diameter of the body. Above d/w = 5. regular and coherent axisymmetric vortex ring shedding is observed with shedding occurring alternately from the inner and outer circumferences of the bluff body. Flow visualization and conditional averaging of hot-wire data are used to investigate the vortex structure.     

Reconstruction of a quasi-instantaneous image of coherent structures from hotwire signals obtained by a multi-point simultaneous measurement system

The turbulent velocity components (u, v) at 11 points in a reciprocating oscillatory turbulent flow have been measured simultaneously by a set of eleven X-type hotwire probes located in a plane perpendicular to the mean flow. Using a conditional sampling technique and a new method of data analysis for the inverse estimation of flow fields called the “virtual plate/load and MASCON model”, a quasi-instantaneous three-dimensional image of coherent structures of turbulence was first reconstructed directly from the experimental velocity data. The quasi-instantaneous image was expressed in terms of the velocity components u, v, w and the vorticity components ω_x ω_y, ω_z and we found that the large-scale coherent structure was composed of a pair of counter-rotating fluid motions with asymmetry which was quite different from that of the ensemble-averaged one. Flow patterns induced by the large-scale structure have been clarified by perspective representations visualized by computer simulations that produce timelines and streaklines of fluid particle traces. Results showed that the new experimental method was applicable for investigating the three-dimensional feature of coherent structures including asymmetry.     

Simulation of vortex shedding behind a bluff body flame stabilizer using a hybrid U-RANS/PDF method

The present study aims at the investigation of the effects of turbulence-chemistry interaction on combustion instabilities using a probability density function(PDF) method.The instantaneous quantities in the flow field were decomposed into the Favre-averaged variables and the stochastic fluctuations,which were calculated by unsteady Reynolds averaged Navier-Stokes(U-RANS) equations and the PDF model,respectively.A joint fluctuating velocityfrequency-composition PDF was used.The governing equations are solved by a consistent hybrid finite volume/MonteCarlo algorithm on triangular unstructured meshes.A nonreacting flow behind a triangular-shaped bluff body flame stabilizer in a rectilinear combustor was simulated by the present method.The results demonstrate the capability of the present method to capture the large-scale coherent structures.The triple decomposition was performed,by dividing the coherent Favre-averaged velocity into time-averaged value and periodical coherent part,to analyze the coherent and incoherent contributions to Reynolds stresses.A simple modification to the coefficients in the turbulent frequency model will help to improve the simulation results.Unsteady flow fields were depicted by streamlines and vorticity contours.Moreover,the association between turbulence production and vorticity saddle points is illustrated.     

Experimental investigation of various turbulator inserts in gas-heated channels

This experimental research was focused on the investigation of the heat transfer augmentation by various turbulator inserts in gas-heated channels. The work was conducted directly in a convective part of a two fire-tube boiler. The flue ducts were positioned vertically and horizontally for various design applications. Twisted-tape insert (with the twist ratio y=4.12), the straight-tape insert, and the combined turbulator insert (the internal twisted tape with the twist ratio of 180° y=2.16 and an external tape, which spirally winded on an internal tape, with longitudinal pitch H_360°=110 mm and the relative height of a tape (rib) e/D₀=0.098;0.2) were investigated. The working fluids were the combustion products of light oil fuel and wood pellets. In addition, the experiments were conducted in the two fire-tube boiler without any inserts. Despite of relatively large data scattering obtained in these experiments some qualitative and quantitative conclusions were drawn.     

Convection regimes and heat transfer characteristics along a continuously moving heated vertical plate

The steady laminar flow and thermal characteristics of a continuously moving vertical sheet of extruded material are studied close to and far downstream from the extrusion slot. The velocity and temperature variations, obtained by a finite volume method, are used to map out the entire forced, mixed and natural convection regimes. The effects of the Prandtl number (Pr) and the buoyancy force parameter (B) on the friction and heat transfer coefficients are investigated. Comparisons with experimental measurements and solutions by others in the pure forced and pure natural convection regions are made. In the mixed convection region, the results are compared with available finite-difference solutions of the boundary layer equations showing excellent agreement. The region close to the extrusion slot is characterized as a non-similar forced-convection dominated region in which Nu_xRe_x^−1/2 drops sharply with increasing Richardson number (Ri_x). This is followed by a self-similar forced-convection dominated region in which Nu_xRe_x^−1/2 levels off with increasing Ri_x until the buoyancy effect sets in. The existence and extent of the latter region depend upon the value of B. A non-similar mixed convection region where increasing buoyancy effect enhances the heat transfer rate follows. Finally, this region is followed downstream by a self-similar natural-convection dominated region in which Nu_xRe_x^−1/2 approaches the pure natural convection asymptote at large Ri_x. Critical values of Ri_x to distinguish the various convection regimes are determined for different Pr and B.     

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).     

The slip and drift model of a wheel with tyre compared to some other attempts in this field

The Slip and Drift Model (SDM) of a wheel with tyre, introduced in 1969, has brought the formally correct fundamental solution to the two-dimensional problem of a wheel motion: longitudinal force-lateral force-slip-drift (slip angle) relationship. It has taken into account the tyre elasticity and the specifics of soil behaviour. In some recent attempts to compile a modified model, the SDM was either ignored or presented inadequately. The analysis of two such attempts from 1981 and 1987 has revealed that their kinematic formulae reproduce the basic formulae of the SDM and that their static equilibrium formulae, as far as they are based on the resolution of the soil shear stress into x, y components, are not satisfactory as a matter of principle.     

The effect of axial stretch on cavitation in an elastic cylinder

The phenomenon of cavitation in an elastic cylinder subjected to a combined axial stretch λ_z and radial traction P is examined. One possible response of the cylinder to this loading, for all λ_z; and P, is a pure homogeneous deformation. However, for some materials, the homogeneous deformation becomes unstable at certain critical values of (λ_z, P), and a second deformation bifurcates from the homogeneous one; this latter deformation involves a cylindrical cavity co-axial with the elastic cylinder. We determine the values of (λ_z, P) at which this happens. The results are displayed by constructing a curve in the (λ_z, P)-plane which divides this plane into regions where the homogeneous deformation is stable and unstable.     

A study of properties of vortex stretching and enstrophy generation in numerical and laboratory turbulence

Dynamically relevant alignments are used in order to show that regions with weak vorticity are not structureless, non-Gaussian and dynamically not passive. for example, the structure of vorticity in quasi-homogeneous/isotropic turbulent flows is associated with strong alignment between vorticity ω and the eigenvectors of the rate of strain tensor λ_i (especially — but not only — between ω and λ₂) rather than with intense vorticity only. Consequently, much larger regions of turbulent flow than just those with intense vorticity are spatially structured. The whole flow field — even with the weakest measurable enstrophy — is strongly non-Gaussian, which among other things is manifested in strong alignment between vorticity and the vortex stretching vector W_i ≡ ω_jS_ij. It is shown that the quasi-two-dimensional regions corresponding to large cos(ω, λ₂) are qualitatively different from purely two-dimensional ones, e.g. in that they possess essentially nonvanishing enstrophy generation, which is larger than its mean for the whole field.