Numerical prediction of locally forced turbulent separated and reattaching flow

An unsteady numerical simulation was performed for locally forced separated and reattaching flow over a backward-facing step. The local forcing was given to the separated and reattaching flow by means of a sinusoidally oscillating jet from a separation line. A version of the k––f_μ model was employed, in which the near-wall behavior without reference to distance and the nonequilibrium effect in the recirculation region were incorporated. The Reynolds number based on the step height (H) was fixed at Re_H=33 000, and the forcing frequency was varied in the range 0St_H2. The predicted results were compared and validated with the experimental data of Chun and Chun. It was shown that the unsteady locally forced separated and reattaching flows are predicted reasonably well with the k––f_μ model. To characterize the large-scale vortex evolution due to the local forcing, numerical flow visualizations were carried out.     

Thermo-mechanical modeling of turbulent heat transfer in gas–solid flows including particle collisions

A thermo-mechanical turbulence model is developed and used for predicting heat transfer in a gas–solid flow through a vertical pipe with constant wall heat flux. The new four-way interaction model makes use of the thermal k_θ–τ_θ equations, in addition to the hydrodynamic k–τ transport, and accounts for the particle–particle and particle–wall collisions through a Eulerian/Lagrangian formulation. The simulation results indicate that the level of thermal turbulence intensity and the heat transfer are strongly affected by the particle collisions. Inter-particle collisions attenuate the thermal turbulence intensity near the wall but somewhat amplify the temperature fluctuations in the pipe core region. The hydrodynamic-to-thermal times-scale ratio and the turbulent Prandtl number in the region near the wall increase due to the inter-particle collisions. The results also show that the use of a constant or the single-phase gas turbulent Prandtl number produces error in the thermal eddy diffusivity and thermal turbulent intensity fields. Simulation results also indicate that the inter-particle contact heat conduction during collision has no significant effect in the range of Reynolds number and particle diameter studied.     

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 effect of two-way coupling and inter-particle collisions on turbulence modulation in a vertical channel flow

Turbulence modulation due to its interaction with dispersed solid particles in a downward fully developed channel flow was studied. The Eulerian framework was used for the gas-phase, whereas the Lagrangian approach was used for the particle-phase. The steady-state equations of conservation of mass and momentum were used for the gas-phase, and the effect of turbulence on the flow-field was included via the standard k–ε model. The particle equation of motion included the drag, the Saffman lift and the gravity forces. Turbulence dispersion effect on the particles was simulated as a continuous Gaussian random field. The effects of particles on the flow were modeled by appropriate source terms in the momentum, k and ε equations. Particle–particle collisions and particle–wall collisions were accounted for in these simulations. Gas-phase velocities and turbulence kinetic energy in the presence of 2–100% mass loadings of two particle classes (50 μm glass and 70 μm copper) were evaluated, and the results were compared with the available experimental data and earlier numerical results. The simulation results showed that when the inter-particle collisions were important and was included in the computational model, the fluid turbulence was attenuated. The level of turbulence attenuation increased with particle mass loading, particle Stokes number, and the distance from the wall. When the inter-particle collisions were negligible and/or was neglected in the model, the fluid turbulence was augmented for the range of particle sizes considered.     

Traveling disturbance appearing in boundary layer transition in a Yawed cylinder

Boundary layers that develop over a body in fluid flow are in most cases three-dimensional owing to the spin, yaw, or surface curvature of the body. Therefore, the study of three-dimensional (3D) boundary-layer transition is essential to work in practical aerodynamics. The present investigation is concerned with the problem of 3D boundary layers over a yawed body. A yawed cylinder model that represents the leading edge portion of a swept wing and the mechanism of crossflow instability are investigated in detail using hot-wire velocimetry and a flow visualization technique. As a result, traveling disturbances having frequencies f₁ and f₂, which differ by about one order of magnitude, are detected in the transition region. The phase velocities and directions of travel of those disturbances are measured. Results for the low-frequency disturbance f₁ show qualitative coincidence with results numerically predicted for a crossflow unsteady disturbance. Nameley, F₁ travels nearly spanwise to the yawed cylinder and very close to the cylinder wall. The results for the high-frequency disturbance f₂ good agreement with the existing experimental results. The ₂ disturbance is found to be the high-frequency inflectional secondary instability that appears in 3D boundary layer transition in general. A two-stage transition process, where stationary crossflow vortices appear as the primary instability and a traveling inflectional disturbance is generated as a secondary instability, was observed. Secondary instability seems to play a major role in turbulent transition.     

An unsymmetrical van der pol equation with stable subharmonics

Possible stable subharmonic solutions of the equation

ÿ − k(1 + 2cy − y²)ÿ + Y = bkμ cos μt, c > 0

, klarge, are discussed by the techniques used by J.E. Littlewood for van der Pol's equation in Acta Math. 97 (1957), that is the case of the above equation with c = 0 and

, k large. Their variation as c increases is also considered briefly.     

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%.     

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 interaction between a plane shear layer and a slender body

The interaction of a plane shear layer with a thin flat plate located in the nonlinear region of the shear layer has been investigated experimentally. The shear layer's velocity ratio is 0.375, and its Reynolds number is ΔUθ₀/μ = 625. It is found that for different angles of attack of the plate, the mixing layer is deflected toward the slower stream. In addition, the plate attenuates turbulent fluctuations of the shear layer structures.     

Permeability of periodic arrays of spheres

For periodic arrays of spheres the permeability is obtained numerically as a function of the dimensionless wave number kD in the flow direction, where D is the sphere diameter, k = 2π/λ is the wave number, and λ is the distance between the spheres in the flow direction. Our numerical results for the solids fraction of 0.45 show that for kD < 6.5 the permeability increases with increasing kD. But, it decreases for 6.5 < kD < 8.5 and reaches a local minimum at kD  8.5, and then increases again with increasing kD. Since the Fourier spectrum of the area fraction is zero for kD = 8.98, this result suggests that the area fraction plays an important role in determining the dependence of permeability on the distance between the spheres in the flow direction. For smaller solids fractions, the positions of the local maximum and minimum of permeability shift to slightly smaller kD’s.     

Group properties of utt=[F(u)ux]x

The group properties and the associated Lie algebra are developed for the subject quasilinear wave equation, for arbitrary f[fεC²(R), f > 0, f ≠ 0]. From the resulting information sets of explicit invariant solutions are constructed for wave propagation in gases and for the transonic equation.     

DNS of heat transfer in turbulent and transitional channel flow obstructed by rectangular prisms

Direct numerical simulation (DNS) of heat transfer in a channel flow obstructed by rectangular prisms has been performed for Re_τ = 80–20, where Re_τ is based on the friction velocity, the channel half width and the kinematic viscosity. The molecular Prandtl number is set to be 0.71. The flow remains unsteady down to Re_τ = 40 owing to the disturbance induced by the prism. For Re_τ = 30 and 20, the flow results in a steady laminar flow. In the vicinity of the prism, the three-dimensional complex vortices are generated and heat transfer is enhanced. The Reynolds number effect on the time-averaged vortex structure and the local Nusselt number are investigated. The mechanism of the heat transfer enhancement is discussed. In addition, the mean flow parameters such as the friction factor and the Nusselt number are examined in comparison with existing DNS and experimental data.     

Effect of swirling flow on unburned ratio and nitrogen oxide concentration in a spray combustion system

The effect of swirling flow on the unburned ratio and NO concentration in exhausted gas was studied for slurry [coal-water mixture (CWM)] spray combustion with variations of swirl numbers. A numerical analysis for CWM combustion was performed for axisymmetric flow in a cylindrical geometry. First, to check the performance of three previous k-ε turbulence models modified with swirling flow, velocity components of isothermal swirling jets were measured by laser-Doppler anemometry (LDA) and compared with predicted results. The two modified models gave more reliable results than the conventional one. Next, as the swirl number could not be estimated by the angular momentum derived from the vane angle of the circular swirler, the reduction rate of the tangential momentum flux through the tube of the circular swirler was measured and calculated. Both measured and predicted results showed that when the swirl number S′ given by the vane angle was 2.0, the effective swirl number S_eff decreased by about 60% to S′. To take the results mentioned above into consideration, effects of swirl number on both the exhausted NO concentration and unburned ratio were investigated. The predicted unburned ratio showed good agreement with the experimental results. Both experimental and calculated results showed that the optimum operating conditions controlling the exhausted NO concentration and unburned ratio in this spray combustion system were obtained when the swirl number S_eff was about 0.5.     

k-ω-γ模式对转捩影响因素的预测性能研究

高超声速边界层转捩的准确预测对飞行器的防热、减阻至关重要,而影响高超声速边界层转捩的因素众多.从模式角度出发研究边界层转捩的影响因素,采用k-ω-γ 转捩模式对5°圆锥的边界层转捩进行了数值分析,计算了不同头部钝度、来流雷诺数和湍流度情况下的边界层转捩,并与实验结果进行了对比. 研究结果表明:k-ω-γ 转捩模式基本能够反映头部钝度、来流雷诺数、来流湍流度对高超声速圆锥边界层转捩的影响规律,但对转捩后的热流峰值预测不准;从模式构造角度分析发现,雷诺数越高或头部钝度越小,层流区边界层越薄,k-ω-γ 转捩模式中第一、第二模态时间尺度增大,因此转捩起始位置提前;来流湍流度越大,等效脉动动能初值越大,导致层流区发展过程中等效脉动动能越大,因此转捩易于发生.      

Laminar free-convective heat transfer from a vertical isothermal plate to water at low temperatures with variable physical properties

All studies concerning laminar free convection along a vertical isothermal plate in water at low temperatures have been conducted assuming constant dynamic viscosity and thermal conductivity both taken at ambient or film temperature. In this study the problem has been treated taking into account the temperature dependence of all water physical properties. The results are obtained with the numerical solution of the boundary layer equations. The variation of μ and k with temperature has a small influence on wall heat transfer but a strong influence on wall shear stress. These quantities show a significant reduction at density extremum.     

Flow and heat transfer in a rotating cavity with a radial inflow of fluid Part 1: The flow structure

Flow visualization has been conducted in a rotating cavity, comprising two steel discs and a peripheral polycarbonate shroud, for dimensionless flow rates of air up to |C_w|8000 and rotational Reynolds number up to Re_φ10⁶. For all the experiments, the ratio of the inner to outer radii of the discs was 0.1 and the ratio of the axial clearance between the discs to their outer radii was 0.133; five different shroud geometries were tested. The flow visualization has confirmed that the flow structure comprises a source region near the shroud, laminar or turbulent Ekman layers on the discs, a sink layer near the centre of the cavity, and an interior core of rotating fluid. Above a certain flow rate, this structure was found to be unstable; heating one disc tended to stabilize the flow. For isothermal flow, measurements of the size of the source region were in good agreement with values predicted from a simple theoretical model.     

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.     

Forced convection from an isolated heat source in a channel with porous medium

A numerical study is made of flow and heat transfer characteristics of forced convection in a channel that is partially filled with a porous medium. The flow geometry models convective cooling process in a printed circuit board system with a porous insert.The channel walls are assumed to be adiabatic. Comprehensive numerical solutions are acquired to the governing Navier-Stokes equations, using the Brinkman-Forchheimer-extended Darcy model for the regions of porous media. Details of flow and thermal fields are examined over ranges of the principal parameters; i.e., the Reynolds number Re, the Darcy number Da (≡K/H²), the thickness of the porous substrate S, and the ratio of thermal conductivities R_k (≡k_eff/k). Two types of the location of the porous block are considered. The maximum temperature at the heat source and the associated pressure drop are presented for varying Re, Da, S, and R_k. The results illustrate that as S increases or Da decreases, the fluid flow rate increases. Also, as R_k increases for fixed Da, heat transfer rates are augmented. Explicit influences of Re on the flow and heat transport characteristics are also scrutinized. Assessment is made of the utility of using a porous insert by cross comparing the gain in heat transport against the increase in pressure drop.     

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).