On the Fiber Stretch in Shearing Deformations of Fibrous Soft Materials

The behavior of the fiber stretch in simple shear of soft materials fiber-reinforced with a single family of oriented parallel fibers is examined. The analysis is purely kinematical and the results are valid for both compressible and incompressible materials. It is shown that for a given amount of shear, for all fiber orientation angles in the range \(0 < \theta < \pi /4\), the fiber stretch increases with increasing \(\theta\) whereas in the range \(\pi /4 < \theta < \pi /2\), this is no longer the case and there is a particular fiber orientation for which the fiber stretch is a maximum. For a particular amount of shear corresponding to a special angle of shear (a “magic” angle of \(35.26^{\circ}\)), the fiber-orientation angle at which the fiber stretch is a maximum is its geometric complement namely a magic angle of \(54.74^{\circ}\). The results are also valid for torsion of a circular cylinder reinforced with a single family of helically wound fibers.     

Statistical Analysis of Scalar Dissipation Rate Transport in Turbulent Partially Premixed Flames: A Direct Numerical Simulation Study

Statistically planar turbulent premixed and partially premixed flames for different initial turbulence intensities are simulated for global equivalence ratios ??>?=?0.7 and ??>?=?1.0 using three-dimensional Direct Numerical Simulations (DNS) with simplified chemistry. For the simulations of partially premixed flames, a random distribution of equivalence ratio following a bimodal distribution of equivalence ratio is introduced in the unburned reactants ahead of the flame. The simulation parameters in all of the cases were chosen such that the combustion situation belongs to the thin reaction zones regime. The DNS data has been used to analyse the behaviour of the dissipation rate transports of both active and passive scalars (i.e. the fuel mass fraction Y _F and the mixture fraction ξ) in the context of Reynolds Averaged Navier–Stokes (RANS) simulations. The behaviours of the unclosed terms of the Favre averaged scalar dissipation rates of fuel mass fraction and mixture fraction (i.e. \(\widetilde {\varepsilon }_Y =\overline {\rho D\nabla Y_F^{\prime \prime } \cdot \nabla Y_F^{\prime \prime } } /\overline{\rho }\) and \(\widetilde {\varepsilon }_\xi =\overline {\rho D\nabla \xi ^{\prime \prime }\cdot \nabla \xi ^{\prime \prime }} /\overline {\rho })\) transport equations have been analysed in detail. In the case of the \(\widetilde {\varepsilon }_Y \) transport, it has been observed that the turbulent transport term of scalar dissipation rate remains small throughout the flame brush whereas the terms due to density variation, scalar–turbulence interaction, reaction rate and molecular dissipation remain the leading order contributors. The term arising due to density variation remains positive throughout the flame brush and the combined contribution of the reaction and molecular dissipation to the \(\widetilde {\varepsilon }_Y \) transport remains negative throughout the flame brush in all cases. However, the behaviour of scalar–turbulence interaction term of the \(\widetilde {\varepsilon }_Y \) transport equation is significantly affected by the relative strengths of turbulent straining and the straining due to chemical heat release. In the case of the \(\widetilde {\varepsilon }_\xi \) transport, the turbulent transport term remains small throughout the flame brush and the density variation term is found to be negligible in all cases, whilst the reaction rate term is exactly zero. The scalar–turbulence interaction term and molecular dissipation term remain the leading order contributors to the \(\widetilde {\varepsilon }_\xi \) transport throughout the flame brush in all cases that have been analysed in the present study. Performances of existing models for the unclosed terms of the transport equations of \(\widetilde {\varepsilon }_Y \) and \(\widetilde {\varepsilon }_\xi \) are assessed with respect to the corresponding quantities obtained from DNS data. Based on this exercise either suitable models have been identified or new models have been proposed for the accurate closure of the unclosed terms of both \(\widetilde {\varepsilon }_Y \) and \(\widetilde {\varepsilon }_\xi \) transport equations in the context of Reynolds Averaged Navier–Stokes (RANS) simulations.     

A semi-locally scaled eddy viscosity formulation for LES wall models and flows at high speeds

We show that the mean wall-shear stresses in wall-modeled large-eddy simulations (WMLES) of high-speed flows can be off by up to \(\approx 100\%\) with respect to a DNS benchmark when using the van-Driest-based damping function, i.e., the conventional damping function. Errors in the WMLES-predicted wall-shear stresses are often attributed to the so-called log-layer mismatch, which, albeit also an error in wall-shear stresses \(\tau _\mathrm{w}\), is an error of about \(15\%\). The larger error identified here cannot be removed using the previously developed remedies for the log-layer mismatch. This error may be removed by using the semi-local scaling, i.e., \(l_\nu =\mu /\sqrt{\rho \tau _\mathrm{w}}\), in the damping function, where \(\mu \) and \(\rho \) are the local mean dynamic viscosity and density, respectively.     

Lagrangian-based numerical investigation of aerodynamic performance of an oscillating foil

The dynamic stall problem for blades is related to the general performance of wind turbines, where a varying flow field is introduced with a rapid change of the effective angle of attack (AOA). The objective of this work is to study the aerodynamic performance of a sinusoidally oscillating NACA0012 airfoil. The coupled \(k{-}\omega \) Menter’s shear stress transport (SST) turbulence model and \(\gamma {-}Re_{\uptheta }\) transition model were used for turbulence closure. Lagrangian coherent structures (LCS) were utilized to analyze the dynamic behavior of the flow structures. The computational results were supported by the experiments. The results indicated that this numerical method can well describe the dynamic stall process. For the case with reduced frequency \(K = 0.1\), the lift and drag coefficients increase constantly with increasing angle prior to dynamic stall. When the AOA reaches the stall angle, the lift and drag coefficients decline suddenly due to the interplay between the first leading- and trailing-edge vortex. With further increase of the AOA, both the lift and drag coefficients experience a secondary rise and fall process because of formation and shedding of the secondary vortex. The results also reveal that the dynamic behavior of the flow structures can be effectively identified using the finite-time Lyapunov exponent (FTLE) field. The influence of the reduced frequency on the flow structures and energy extraction efficiency in the dynamic stall process is further discussed. When the reduced frequency increases, the dynamic stall is delayed and the total energy extraction efficiency is enhanced. With \(K = 0.05\), the amplitude of the dynamic coefficients fluctuates more significantly in the poststall process than in the case of \(K = 0.1\).     

Uniaxial experimental study of the acoustic emission and deformation behavior of composite rock based on 3D digital image correlation (DIC)

In this paper, uniaxial compression tests were carried out on a series of composite rock specimens with different dip angles, which were made from two types of rock-like material with different strength. The acoustic emission technique was used to monitor the acoustic signal characteristics of composite rock specimens during the entire loading process. At the same time, an optical non-contact 3 D digital image correlation technique was used to study the evolution of axial strain field and the maximal strain field before and after the peak strength at different stress levels during the loading process. The effect of bedding plane inclination on the deformation and strength during uniaxial loading was analyzed. The methods of solving the elastic constants of hard and weak rock were described. The damage evolution process, deformation and failure mechanism, and failure mode during uniaxial loading were fully determined. The experimental results show that the θ = 0?–45?specimens had obvious plastic deformation during loading, and the brittleness of the θ = 60?–90?specimens gradually increased during the loading process. When the anisotropic angle θincreased from 0?to 90?, the peak strength, peak strain,and apparent elastic modulus all decreased initially and then increased. The failure mode of the composite rock specimen during uniaxial loading can be divided into three categories:tensile fracture across the discontinuities(θ = 0?–30?), slid-ing failure along the discontinuities(θ = 45?–75?), and tensile-split along the discontinuities(θ = 90?). The axial strain of the weak and hard rock layers in the composite rock specimen during the loading process was significantly different from that of the θ = 0?–45?specimens and was almost the same as that of the θ = 60?–90?specimens. As for the strain localization highlighted in the maximum principal strain field, the θ = 0?–30?specimens appeared in the rock matrix approximately parallel to the loading direction,while in the θ = 45?–90?specimens it appeared at the hard and weak rock layer interface.     

Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

In this paper, we consider the perturbed KdV equation with Fourier multiplier

$$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

with analytic data of size \(\varepsilon \). We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with \(\tilde{J}\) Diophantine frequencies, where the order of \(\tilde{J}\) is \(O(\frac{1}{\varepsilon })\). The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Töplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Töplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.

     

Glimpses of Flow Development and Degradation During Type B Drag Reduction by Aqueous Solutions of Polyacrylamide B1120

Flow development and degradation during Type B turbulent drag reduction by 0.10 to 10 wppm solutions of a partially-hydrolysed polyacrylamide B1120 of MW \(=\) 18x10⁶ was studied in a smooth pipe of ID \(=\) 4.60 mm and L/D \(=\) 210 at Reynolds numbers from 10000 to 80000 and wall shear stresses Tw from 8 to 600 Pa. B1120 solutions exhibited facets of a Type B ladder, including segments roughly parallel to, but displaced upward from, the P-K line; those that attained asymptotic maximum drag reduction at low Re√ f but departed downwards into the polymeric regime at a higher retro-onset Re√ f; and segments at MDR for all Re√ f. Axial flow enhancement profiles of S\(^{\prime }\) vs L/D reflected a superposition of flow development and polymer degradation effects, the former increasing and the latter diminishing S\(^{\prime }\) with increasing distance downstream. Solutions that induced normalized flow enhancements S\(^{\prime }\)/S\(^{\prime }_{\mathrm {m}} <\) 0.4 developed akin to solvent, with L_e,p/D \(=\) L_e,n/D \(<\) 42.3, while those at maximum drag reduction showed entrance lengths L_e,m/D \(\sim \) 117, roughly 3 times the solvent L_e,n/D. Degradation kinetics were inferred by first detecting a falloff point (Re√f^∧, S\(^{{\prime }\wedge }\)), of maximum observed flow enhancement, for each polymer solution. A plot of S\(^{{\prime }\wedge }\)vs C revealed S\(^{{\prime }\wedge }\)linear in C at low C, with lower bound [S\(^{\prime }\)] \(=\) 5.0 wppm^??1, and S\(^{{\prime }\wedge }\) independent of C at high C, with upper bound S\(^{\prime }_{\mathrm {m}} =\) 15.9. The ratio S\(^{\prime }\)/S\(^{{\prime }\wedge }\) in any pipe section was interpreted to be the undegraded fraction of original polymer therein. Semi-log plots of (S\(^{\prime }\)/S\(^{{\prime }\wedge }\)) at a section vs transit time from pipe entrance thereto revealed first order kinetics, from which apparent degradation rate constants k_deg s^??1 and entrance severities ?ln(S\(^{\prime }\)/S\(^{{\prime }\wedge }\))₀ were extracted. At constant C, k_deg increased linearly with increasing wall shear stress Tw, and at constant Tw, k_deg was independent of C, providing a B1120 degradation modulus (k_deg/Tw) \(=\) (0.012 \(\pm \) 0.001) (Pa s)^??1 for 8 \(<\) Tw Pa \(<\) 600, 0.30 \(<\) C wppm \(<\) 10. Entrance severities were negligible below a threshold Twe \(\sim \) 30 Pa and increased linearly with increasing Tw for Tw \(>\) Twe. The foregoing methods were applied to Type A drag reduction by 0.10 to 10 wppm solutions of a polyethyleneoxide PEO P309, MW \(=\) 11x10⁶, in a smooth pipe of ID \(=\) 7.77 mm and L/D \(=\) 220 at Re from 4000 to 115000. P309 solutions that induced S\(^{\prime }\)/S\(^{\prime }_{\mathrm {m}} <\) 0.4 developed akin to solvent, with L_e,p/D \(=\) L_e,n/D \(<\) 23, while those at MDR had entrance lengths L_e,m/D \(\sim \) 93, roughly 4 times the solvent L_e,n/D. P309 solutions described a Type A fan distorted by polymer degradation. A typical trajectory departed the P-K line at an onset point Re√ f* followed by ascending and descending polymeric regime segments separated by a falloff point Re√f^∧, of maximum flow enhancement; for all P309 solutions, onset Re√ f* = 550 \(\pm \) 100 and falloff Re√f^∧ = 2550 \(\pm \) 250, the interval between them delineating Type A drag reduction unaffected by degradation. A plot of falloff S\(^{{\prime }\wedge }\) vs C for PEO P309 solutions bore a striking resemblance to the analogous S\(^{{\prime }\wedge }\) vs C plot for solutions of PAMH B1120, indicating that the initial Type A drag reduction by P309 after onset at Re√ f* had evolved to Type B drag reduction by falloff at Re√f^∧. Presuming that Type B behaviour persisted past falloff permitted inference of P309 degradation kinetics; k_deg was found to increase linearly with increasing Tw at constant C and was independent of C at constant Tw, providing a P309 degradation modulus (k_deg/Tw) \(=\) (0.011 \(\pm \) 0.002) (Pa s)^??1 for 4 \(<\) Tw Pa \(<\) 400, 0.10 \(<\) C wppm < 5.0. Comparisons between the present degradation kinetics and previous literature showed (k_deg/Tw) data from laboratory pipes of D \(\sim \) 0.01 m to lie on a simple extension of (k_deg/Tw) data from pipelines of D \(\sim \) 0.1 m and 1.0 m, along a power-law relation (k_deg/Tw) \(=\) 10^??5.4.D^??1.6. Intrinsic slips derived from PAMH B1120 and PEO P309-at-falloff experiments were compared with previous examples from Type B drag reduction by polymers with vinylic and glycosidic backbones, showing: (i) For a given polymer, [S\(^{\prime }\)] was independent of Re√ f and pipe ID, implying insensitivity to both micro- and macro-scales of turbulence; and (ii) [S\(^{\prime }\)] increased linearly with increasing polymer chain contour length Lc, the proportionality constant \(\beta =\) 0.053 \(\pm \) 0.036 enabling estimation of flow enhancement S\(^{\prime } =\) C.Lc.β for all Type B drag reduction by polymers.     

Direct Numerical Simulation of Head-On Quenching of Statistically Planar Turbulent Premixed Methane-Air Flames Using a Detailed Chemical Mechanism

A three-dimensional compressible Direct Numerical Simulation (DNS) analysis has been carried out for head-on quenching of a statistically planar stoichiometric methane-air flame by an isothermal inert wall. A multi-step chemical mechanism for methane-air combustion is used for the purpose of detailed chemistry DNS. For head-on quenching of stoichiometric methane-air flames, the mass fractions of major reactant species such as methane and oxygen tend to vanish at the wall during flame quenching. The absence of \(\text {OH}\) at the wall gives rise to accumulation of carbon monoxide during flame quenching because \(\text {CO}\) cannot be oxidised anymore. Furthermore, it has been found that low-temperature reactions give rise to accumulation of \(\text {HO}_{2}\) and \(\mathrm {H}_{2}\mathrm {O}_{2}\) at the wall during flame quenching. Moreover, these low temperature reactions are responsible for non-zero heat release rate at the wall during flame-wall interaction. In order to perform an in-depth comparison between simple and detailed chemistry DNS results, a corresponding simulation has been carried out for the same turbulence parameters for a representative single-step Arrhenius type irreversible chemical mechanism. In the corresponding simple chemistry simulation, heat release rate vanishes once the flame reaches a threshold distance from the wall. The distributions of reaction progress variable c and non-dimensional temperature T are found to be identical to each other away from the wall for the simple chemistry simulation but this equality does not hold during head-on quenching. The inequality between c (defined based on \(\text {CH}_{4}\) mass fraction) and T holds both away from and close to the wall for the detailed chemistry simulation but it becomes particularly prominent in the near-wall region. The temporal evolutions of wall heat flux and wall Peclet number (i.e. normalised wall-normal distance of \(T = 0.9\) isosurface) for both simple and detailed chemistry laminar and turbulent cases have been found to be qualitatively similar. However, small differences have been observed in the numerical values of the maximum normalised wall heat flux magnitude \(\left ({\Phi }_{\max } \right )_{\mathrm {L}}\) and the minimum Peclet number \((Pe_{\min })_{\mathrm {L}}\) obtained from simple and detailed chemistry based laminar head-on quenching calculations. Detailed explanations have been provided for the observed differences in behaviours of \(\left ({\Phi }_{\max }\right )_{\mathrm {L}}\) and \((Pe_{\min })_{\mathrm {L}}\). The usual Flame Surface Density (FSD) and scalar dissipation rate (SDR) based reaction rate closures do not adequately predict the mean reaction rate of reaction progress variable in the near-wall region for both simple and detailed chemistry simulations. It has been found that recently proposed FSD and SDR based reaction rate closures based on a-priori DNS analysis of simple chemistry data perform satisfactorily also for the detailed chemistry case both away from and close to the wall without any adjustment to the model parameters.     

Design of a pulse-type strain gauge balance for a long-test-duration hypersonic shock tunnel

When the measurement of aerodynamic forces is conducted in a hypersonic shock tunnel, the inertial forces lead to low-frequency vibrations of the model, and its motion cannot be addressed through digital filtering because a sufficient number of cycles cannot be obtained during a tunnel run. This finding implies restrictions on the model size and mass as the natural frequencies are inversely proportional to the length scale of the model. Therefore, the force measurement still has many problems, particularly for large and heavy models. Different structures of a strain gauge balance (SGB) are proposed and designed, and the measurement element is further optimized to overcome the difficulties encountered during the measurement of aerodynamic forces in a shock tunnel. The motivation for this study is to assess the structural performance of the SGB used in a long-test-duration JF12 hypersonic shock tunnel, which has more than 100 ms of test time. Force tests were conducted for a large-scale cone with a \(10^{\circ }\) semivertex angle and a length of 0.75 m in the JF12 long-test-duration shock tunnel. The finite element method was used for the analysis of the vibrational characteristics of the Model-Balance-Sting System (MBSS) to ensure a sufficient number of cycles, particularly for the axial force signal during a shock tunnel run. The higher-stiffness SGB used in the test shows good performance, wherein the frequency of the MBSS increases because of the stiff construction of the balance. The experimental results are compared with the data obtained in another wind tunnel and exhibit good agreement at \(M = 7\) and \(\alpha =5^\circ \).     

井眼轨迹控制工具主轴载荷与造斜能力关系研究

井眼轨迹控制工具是随钻实时完成导向功能的一种导向式钻井工具.在复杂工况下,主轴承受钻压、扭矩和偏置机构作用力等,十分复杂.在课题组前期研究的基础上,对样机主轴的力学行为开展了进一步的研究,在建立主轴静力学模型的基础上,分析了工具外壳刚度、偏置机构安装位置等因素对主轴力学行为与造斜能力的影响规律.研究发现,主轴下端偏转角、偏心机构作用力和最大截面弯矩会随着偏心位移增加呈线性增加;随着外壳刚度的增加和偏置机构安装的位置与上支撑轴承之间距离的增加,工具造斜能力也会增强;通过分析偏置机构安装位置和外壳刚度对主轴力学行为的影响规律发现,随着外壳刚度的增大和偏置机构安装位置与上支撑轴承之间距离的增大,主轴最大截面弯矩也会增大.最后,给出了最佳的外壳刚度与偏置机构安装位置建议.     

A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions

A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as \(R_{{i}}\) (i =  water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms \(\lambda _\mathrm{wo}\) and \(\lambda _\mathrm{ow}\) are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient \(I>0\) which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio \(\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} \) is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms \(R_{{i}}\) and the fluid–fluid interaction term R.     

The PELskin project-part V: towards the control of the flow around aerofoils at high angle of attack using a self-activated deployable flap

During the flight of birds, it is often possible to notice that some of the primaries and covert feathers on the upper side of the wing pop-up under critical flight conditions, such as the landing approach or when stalking their prey (see Fig. 1) . It is often conjectured that the feathers pop up plays an aerodynamic role by limiting the spread of flow separation . A combined experimental and numerical study was conducted to shed some light on the physical mechanism determining the feathers self actuation and their effective role in controlling the flow field in nominally stalled conditions. In particular, we have considered a NACA0020 aerofoil, equipped with a flexible flap at low chord Reynolds numbers. A parametric study has been conducted on the effects of the length, natural frequency, and position of the flap. A configuration with a single flap hinged on the suction side at 70 % of the chord size c (from the leading edge), with a length of \(L=0.2c\) matching the shedding frequency of vortices at stall condition has been found to be optimum in delivering maximum aerodynamic efficiency and lift gains. Flow evolution both during a ramp-up motion (incidence angle from \(\alpha _0=0\) to \(\alpha _s=20^\circ\) with a reduced frequency of \(k= 0.12\, U_{\infty }/c\), \(U_{\infty }\) being the free stream velocity magnitude), and at a static stalled condition (\(\alpha =20^\circ\)) were analysed with and without the flap. A significant increase of the mean lift after a ramp-up manoeuvre is observed in presence of the flap. Stall dynamics (i.e., lift overshoot and oscillations) are altered and the simulations reveal a periodic re-generation cycle composed of a leading edge vortex that lift the flap during his passage, and an ejection generated by the relaxing of the flap in its equilibrium position. The flap movement in turns avoid the interaction between leading and trailing edge vortices when lift up and push the trailing edge vortex downstream when relaxing back. This cyclic behaviour is clearly shown by the periodic variation of the lift about the average value, and also from the periodic motion of the flap. A comparison with the experiments shows a similar but somewhat higher non-dimensional frequency of the flap oscillation. By assuming that the cycle frequency scales inversely with the boundary layer thickness, one can explain the higher frequencies observed in the experiments which were run at a Reynolds number about one order of magnitude higher than in the simulations. In addition, in experiments the periodic re-generation cycle decays after 3–4 periods ultimately leading to the full stall of the aerofoil. In contrast, the 2D simulations show that the cycle can become self-sustained without any decay when the flap parameters are accurately tuned.     

Measurements of turbulent inclined plane dual jets

Measurements of mean velocities, flow direction, velocity fluctuations and Reynolds shear stress were made with a split film probe of hot wire anemometer to investigate the interactions created by two air jets issuing from two identical plane inclined nozzles. The reverse flow was detected by using the split film probe and observed by flow visualization. Experimental results with an inclined angle of 9° are presented in the paper. Some experimental results with an inclined angle of 27° are presented to investigate the effect of inclination on the flow field.Mean velocities approach self-preservation in both the converging region and the combining region. Velocity fluctuations and Reynolds shear stress approach self-preservation in the combining region only. The spreads of jet and the square of the decay of maximum mean velocity increase linearly as the distance from the nozzle exit increases.List of symbols D nozzle width - h nozzle height - J momentum of jet - J ₀ momentum of jet at nozzle exit - M mass flow rate - M ₀ mass flow rate at nozzle exit - S nozzle spacing - U, V mean velocities in the X and Y axis respectively - U _m maximum axial velocity - U ₀ axial velocity at nozzle exit - u, v velocity fluctuations in the X and Y axis respectively - u, v r.m.s. of u and v - Reynolds shear stress - X, Y Cartesian coordinates - X _m , Y _m coordinates at the location of maximum axial velocity - y _0.5 distance from the location of maximum axial velocity to the location where the velocity is half of maximum axial velocity - inclined angle - yY/S - Reynolds stress correlation coefficient - C.P combining point - max maximum value - M.P merging point - o nozzle exit plane - V.C vortex center     

Self-similar motion of a gas heated by a nonequilibrium continuous radiation spectrum

In this paper we discuss the motion of the vapor formed during the evaporation of a solid by a continuous radiation spectrum. The vapor is assumed to be heated by this radiation to a temperature T much higher than the phase-transition temperature T_v and much higher than the temperature T_i at which significant ionization of the vapor begins. in the case, T_v and T_i can be neglected (as can the heat of evaporation Q_v and the energy Q_i expended on ionization). As a result of this motion, the vapor has a density ? much lower than the densityρ ₀ of the solid. It can therefore be assumed that the heating wave moves through an absolutely cold and infinitely dense gas. At the same time, the vapor temperature is assumed low enough that reradiation can be neglected. The radiation-absorption coefficient η for the ionized vapor can be described by a power-law dependence on T and ? for certain ranges of T, ε, and the photon energy ε. In this case, the motion of the gas is a self-similar problem. The spectrum and angular distribution of the incident radiation [φ (ε, θ)] and the η and ε dependences can be arbitrary. A system of ordinary differential equations is found and solved. Intense radiation incident on a solid surface will evaporate the solid. If the absorption coefficient η of the vapor and the flux density q of the radiation are high enough, the escaping vapor will be heated to a high temperature in a relatively short time. This temperature will not only be much higher than the evaporation temperature T_v, but it will also be higher than the “ionization temperature” T_i. If the internal energy per unit mass of the vapor is much higher than the heat of evaporation Q_v and the energy Q_i expended on ionization, and if the vapor density ? is much lower than the initial densityρ ₀ as a result of its escape, then the problem of the motion and heating of the vapor can be simplified through the assumptions. (0.1) $$T_v = T_i = Q_v = Q_i = 0,\rho _0 = \infty (v_0 = 1/\rho _0 = 0)$$ (here and below, v is the specific volume). We can therefore assume that the heating wave moves through an infinitely dense and absolutely cold gas. In the region of multiple and complete ionization, the ionized-vapor absorption coefficient η, associated with free-free electron transitions in the field of ions, and bound-free transitions from the higherlying states of atoms and ions, has an approximately power-law dependence on T and ? [1], or on p and ? (p is the pressure): Here k and K are numerical coefficients which depend on the substance and on the ranges of T, ?, and ε in which (0.2) is used. For a completely ionized gas, we have α=3/2, β=1,a=?5/2, b=?3/2, and \( - \bar 1/2\) when ε?T; or α=3/2, β=1,a=3/2, and b=?1/2 when when ε ? T. We assume that (0.2) holds for any T, for approximation (0.1). We assume the ratio of specific heats γ to be constant for a certain temperature range in the range of multiple and complete ionization. With these simplifying approximations, the problem of the planar, transient flow of a gas heated by a beam of monochromatic radiation is a self-similar problem. It has been studied in [2,3]. It is shown below that the analogous problem of the motion of a gas heated by a nonequilibrium continuous radiation spectrum is also self-similar. For a partially ionized gas, approximation (0.2) is usualy satisfied only for the long-wavelength part of the incident spectrum. For the short-wavelength part of the spectrum (that is, for photons whose energy is close to or greater than the ionization potential characteristics of the ions for the given temperature range, and which are capable of direct photoionization of these ions from the ground or first excited status), the absorption coefficient is usually much smaller (by several orders of magnitude). This “hard” radiation penetrates a short distance into the solid, causing intense heating of a thin surface layer of small mass. An afterionization wave propagates through the substance, moving under the influence of the radiation flux in the hard part of the spectrum; if the temperature of the surface layer is close to the source temperature T_e, and reradiation becomes important, there will also be a thermal wave [1]. Since the energy expended in heating is large in these waves, their propagation velocity is small (in comparison with that of the wave of evaporation, initial ionization, and heating of the plasma by the long-wavelength part of the spectrum), even if the hard and soft parts of the incidence spectrum have comparable energies (E_h and E_s). Also, the intense reradiation by the thermal wave in the hard part of the spectrum increases its propagation velocity. Finally, the energy in the short-wavelength part of the spectrum may in general be small because of self-adsorption in the source itself (for example, adsorption of the short-wavelength radiation in the cold working gas ahed of a shock wave front in an explosive source [4]). Accordingly, the heating waves for the various parts of the source spectrum may propagate differently Since the mass of the surface layer heated by the short-wave-length part of the spectrum is small, the pressure produced as a result of of the disintegration of the surface layer is small when E_h is of the order order of E_s or, especially, when E_h?E_s; that is, the hydrodynamic effects of the heating and surface-layer disintegration on the motion and and heating of the deep layers heated by the “basic” part of the spectrum can also be neglecred. The high temperature and low density of this layer only facilitate the penetration of the long-wavelength part of the spectrum into the deeper layers; however, because of the small mass of this layer, even this phenomenon has little effect on the hydrodynamic processes in the deeper layers. Accordingly, Eq. (0.2) can frequently be assumed valid for the basic part of the spectrum in the case of a partially ionized gas, also; the rest of the spectrum may simply be neglected. These restrictions on the applicability of the self-similar problem are generally removed in the case of a completely ionized gas. A state close to that of complete ionization arises when two ionization potentials typical of a given temperature range are greatly different (this occurs, for example, in the case of the alkaline metals, and also when one atomic shell has been essentially ionized, while another has not yet started to be ionized; e. g., the L- and K-shells or the M- and L-shells). We consider here the case in which the heating is caused by nonequilibrium radiation, that is, radiation such that the intrinisic radiation of the vapor may be neglected. This is a valid assumption when the vapor temperature is considerably below the source temperature T_e, or, more accurately, when the following condition holds (for a Planckian source spectrum): (0.3) $$W\sigma T_e^4 \chi \left( {\frac{{\varepsilon _1 }}{{T_e }},\frac{{\varepsilon _2 }}{{T_e }}} \right) \gg \sigma \Upsilon ^4 \chi \left( {\frac{{\varepsilon _1 }}{T},\frac{{\varepsilon _2 }}{T}} \right)$$ Here W is the source-radiation dilution coefficient due to geometric factors, σ is the Stefan-Boltzmann constant, ?₁ and ?₂ are the boundaries of the “basic part” of the spectrum, and χ is the fraction of the spectral energy of a Planckian source with a temperature T_e or T for photous with energies ?₁≤?≤?₂. We note that the boundaries ?₁ and ?₂ for the source and vapor-radiation spectra are sometimes slightly differnt, but condition (0.3) can be easily modified for this situation or for a non-Planckian source spectrum. For our problem, the radiation intensity J=J (m, t, ε, θ) is a function of four variables: the time t, the Lagrangian mass coordinate m, the photon energy ε, and the angle θ between the direction of motion and the beam direction. The intensity J_o=J (o, t, ε, θ) of the radiation incident on the boundary m=0 is assumed to be a given function. In the self-similar problem, J can be represented as (0.4) $$J = t^\lambda J(mt^{ - n} ,\varepsilon ,\theta )$$ . This can be done (when conditions (0.1)–(0.3) are satisfied) when J_o can be represented by (0.5) $$J_0 = t^\lambda \psi (\varepsilon ,\theta )(\varepsilon _1 \leqslant \varepsilon \leqslant \varepsilon _2 ,\theta _1 \leqslant \theta \leqslant \theta _2 )$$ If the source spectrum is Planckian, condition (0.5) requires that T_e=const. In this case, the power-law time dependence of the intensity J_o may reflect, for example, motion of the radiation source toward the irradiated surface; in this case, however, the limiting angle θ₂ of the incident radiation also changes (usually, θ₁=0). As before, the problem is self-similar if these angles θ₂(t) are always small; that is, if the radiation is almost completely unidirectional. The arbitrary nature of the function ψ(ε, ч), which shows the spectrum and angular distribution of thesource radiation, and the arbitrary nature of the function φ (ε), which shows the dependence of the absorption coefficient on the photon energy, permit us to analyze the effects of these functions on the heating and motion of the substance for the case of the self-similar solution.     

Neutrally buoyant bubbles used as flow tracers in air

Research has been performed to determine the accuracy of neutrally buoyant and near-neutrally-buoyant bubbles used as flow tracers in an incompressible potential flowfield. Experimental and computational results are presented to evaluate the quantitative accuracy of neutrally buoyant bubbles using a commercially available helium bubble generation system. A two-dimensional experiment was conducted to determine actual bubble trajectories in the stagnation region of a NACA 0012 airfoil at 0° angle of attack. A computational scheme evaluating the equation of motion for a single bubble was also used to determine the factors which affect a bubble's trajectory. The theoretical and computational analysis have shown that neutrally buoyant bubbles will trace complex flow patterns faithfully in the flowfield of interest. Experimental analysis revealed that the use of bubbles generated by the commercially available system to trace flow patterns should be limited to qualitative measurements unless care is taken to ensure neutral buoyancy.Nomenclature a _c centripetal acceleration - c model chord - c _D bubble drag coefficient - D bubble diameter - g acceleration due to gravity - g _v acceleration due to gravity vector - h trajectory deviation normalization parameter - K nondimensional inertia parameter, - m _f mass of fluid - m _p mass of bubble - p static pressure - r radial distance, bubble radius - R gas constant - Re free-stream Reynolds number, - Re _p bubble slip Reynolds number, - S cross-sectional area of sphere - T temperature - t time - u streamwise velocity component - U free-stream velocity - v _f fluid velocity vector - V _p bubble velocity vector - x _p bubble position vector - y _b bubble trajectory y/c - y _s streamline y/c - model angle of attack - bubble solution surface tension - potential vortex strength - _bfs bubble solution density - fluid density - bubble density - bubble wall thickness - fluid viscosity     

A mixing length model for turbulent flow in constant cross section ducts

In this paper, a study is made of the damping influence of the wall on turbulent fluid flow. By considering the oscillation of the whole of the boundary, van Driest's original hypothesis has been extended to obtain the wall damping factor in flow in a duct of constant cross section. The damping factor is used in conjunction with mixing length expressions to obtain the velocity field. Particular examples considered are plane parallel flow and axisymmetric flow in a pipe and in an annulus.

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Ein Modell für die Mischungslänge von turbulenten Strömungen in Rohren mit konstantem Querschnitt

Zusammenfassung In dieser Arbeit wurde der dämpfende Wandeinfluß in turbulenten Strömungen untersucht. Unter Berücksichtigung der Schwingungen in der gesamten Grenzschicht wurde die ursprüngliche Theorie von van Driest erweitert und ein Dämpfungsfaktor an der Wand in Rohrströmungen mit konstantem Querschnitt ermittelt. Dieser Dämpfungsfaktor diente in Verbindung mit Ausdrücken für die Mischungslänge zur Bestimmung des Geschwindigkeitsfeldes. Ausgewählte Beispiele waren die ebene Parallelströmung sowie die Zylinderströmung in einem Rohr und einem Ringspalt.

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Nomenclature A, A^*(=Au/v) Parameter defined in text - b, b^*(=bu/v) semi-width of parallel plate channel - c(= 1/A) parameter defined in text - E[, /2] complete elliptic integral of the second kind - d damping factor - F, G, H functions - l, l^*(=/v) mixing length - M_O, _O functions - r, r^*(=ru/v) radius - A real part of function - R, S, T, U functions - u, u^*(=u/u) velocity in flow direction Z - friction velocity - x, y, z co-ordinates (z in flow direction) - y^*(=yu/v) non-dimensional wall distance - fluid density - , _eff kinematic viscosity, effective kinematic viscosity - phase angle, or polar coordinate angle - shear stress - (=r/r_W) radius ratio - angular velocity Suffixes w wall value - far from a wall     

Rheology of concentrated coagulating suspensions in non-aqueous media

The rheology of concentrated coagulating suspensions is analysed on the basis of the following model: (i) at low shear rates, the shear is not distributed homogeneously but limited to certain shear planes; (ii) the energy dissipation during steady flow is due primarily to the overcoming of viscous drag by the suspended particles during motion caused by encounters of particles in the shear planes. This model is called the giant floc model.With increasing shear rate the distance between successive shear planes diminishes, approaching the suspended particles' diameter at average shear stresses of 88–117 Pa in suspensions of 78 µm particles (glass ballotini coated by a hydrophobic layer) in glycerol — water mixtures, at solid volume fractions between 0.35 and 0.40. Smaller particles form a more persistent coagulation structure. The average force necessary to separate two touching 78 µm particles is too large to be accounted for by London-van der Waels forces; thus coagulation is attributed to bridging connections between polymer chains protruding from the hydrophobic coatings.The frictional ratio of the glass particles in these suspensions is of the order of 10. Coagulation leads to build-up of larger structural units at lower shear rates; on doubling the shear rate the average distance between the shear planes decreases by a factor of 0.81 to 0.88. A inter-shear plane distance - A Hamaker constant - b radius of primary particles - f frictional ratio - F _A attractive force between two particles - g acceleration due to gravity - H distance between the surfaces of two particles - K proportionality constant in power law - l fraction of distance by which a moving particle entrains its neighbours - l effective length of inner cylinder in the rheometer - M torque experienced by inner cylinder during measurements - n exponent in power law - n ₀ ,n ₁ ,n ₂ constants in extended power law - NC _hex number of contacts, per mm², between particles in adjacent layers with an average degree of occupation, assuming a hexagonal arrangement of the particles within the layers - NC _cub asNC _hex, but with a cubical arrangement - p () d increase of slippage probability when the shear stress increases from to + d - q average coordination number of a particle in a coagulate - R _i radius of inner cylinder of rheometer - R _u radius of outer cylinder of rheometer - t _i time during which particlei moves - t ₀ time during which a particle bordering a shear plane moves from its rectilinear course, on meeting another particle - u angle between the direction of motion, and the line connecting the centers of two successive particles bordering a shear plane - V _A attractive energy between two particles - x, y, z Cartesian coordinates:x — the direction of motion;y — the direction of the velocity gradient - y ₀ ,z ₀ y, z value of a particle meeting another particle, when both are far removed from each other - y ₀ spread iny ₀ values - —2/n - ₀ capture efficiency - shear rate - average shear rate calculated for a Newtonian liquid - _i distance by which particlei moves - ₀ distance by which a particle bordering a shear plane moves from its rectilinear course, when it encounters another particle - square root of area occupied by a particle bordering a shear plane, in this plane - _c energy dissipated during one encounter of two particles bordering a shear plane - _p energy dissipated by one particle - energy dissipated per unit of volume and time during steady flow - viscosity - _app calculated as if the liquid is Newtonian - ₀ viscosity of suspension medium - _PL lim - [] intrinsic viscosity - _diff - _{diff, rel diff}/ ₀ - standard deviation of distribution ofy ₀ values - shear stress - _n average shear stress at the highest values applied - mass average particle diameter - _n number average particle diameter - solid volume fraction - _eff effective solid volume fraction in Dougherty-Krieger relation - _max maximum solid volume fraction permitting flow - _i angular velocity of inner cylinder in rheometer during measurements     

Elastico-viscous boundary-layer flow on the surface of a sphere

Summary The Prandtl boundary-layer theory is extended for an idealized elastico-viscous liquid. The boundary layer equations are solved approximately by Kármán-Pohlhausen technique for the case of a sphere. It is shown that the increase in the elasticity of the liquid causes a shift in the point of separation towards the forward stagnation point.

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Zusammenfassung Die Prandtlsche Grenzschicht-Theorie wird für eine idealisierte viskoelastische Flüssigkeit erweitert. Die Grenzschichtgleichungen werden für den Fall einer angeströmten Kugel näherungsweise mit Hilfe der Kármán-Pohlhausen-Methode gelöst. Es wird gezeigt, daß das Anwachsen der Flüssigkeitselastizität eine Verschiebung des Ablösepunktes auf den vorderen Staupunkt hin zur Folge hat.

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Nomenclature b ^ik arbitrary contravariant tensor - D non-dimensional boundary layer thickness - g _ik metric tensor of a fixed coordinate system - K curvature at any point on the generating curve - K ₀ elastico-viscous parameter - p arbitrary hydrostatic pressure - p _ik stress tensor - p ^ik part of stress tensor associated with the change of shape of material - R radius of the sphere - r radius of any transverse cross-section of the sphere - t time - U potential velocity around the body - U stream-velocity at a large distance from the body - u, w velocity components along (x, z) directions respectively - x distance measured along a generating line from the forward stagnation point - z distance measured along a normal to the surface - non-dimensional elastico-viscous parameter - density of the liquid - boundary layer thickness - convected time derivative - ₀ limiting viscosity for very small changes in deformation velocity - angle measured along the transverse direction - x/R - v kinematic coefficient of viscosity - T _s shearing stress on the surface of the sphere With 2 figures and 1 table     

Turbulent wall pressure fluctuations over wavy surfaces

Measurements of the spectral characteristics of the wall pressure fluctuations produced by a turbulent boundary layer flow over solid sinusoidal surfaces of moderate wave amplitude to wave-length ratios have been obtained. The wave amplitudes were sufficiently small so that the flow remained attatched. The results show that the root mean square pressure level reaches a maximum on the adverse pressure gradient side of the wave at a position somewhat before the trough. Spectral analysis of the pressure fluctuations in narrow frequency bands reveals considerable differences in low and high frequency behavior. At low frequencies, the peak fluctuation amplitude was found at the trough whereas at high frequencies, the peak occurs just after the crest and a minimum is found at the trough. Pressure fluctuations having streamwise correlation lengths on the order of or larger than the wavelength of the surface do not return to their equilibrium (crest) amplitudes as they travel the length of a wave. Pressure fluctuations having streamwise correlation lengths about one order of magnitude less than a wavelength return exactly to their equilibrium amplitudes. Two-point correlation measurements show a decrease in longitudinal coherence on the adverse pressure gradient side of the wave at low frequencies and a considerable increase over a broad frequency range on the positive pressure gradient side. No change is found in the lateral coherence.List of symbols C _f skin friction coefficient - C _p pressure coefficient - C _n Fourier amplitudes of the pressure coefficient - C _dp pressure drag coefficient - d pinhole diameter - f frequency - h half the crest to trough distance - h ₊ nondimensional wave amplitude = - k _n wavenumber = - k fundamental wavenumber = - l _p pressure correlation length - p _s mean surface pressure - P ambient pressure - p fluctuating pressure - p ² mean square pressure - q dynamic head = 1/2 U ² - R space-time correlation - P Reynolds number based on wavelength = - R Reynolds number based on momentum thickness = - t time - R free stream velocity - U mean streamwise velocity - U _e streamwise velocity at the edge of the boundary layer - u ^* friction velocity = - x streamwise coordinate - y wall-normal coordinate - z spanwise coordinate - ⁺ non-dimensional wavelength = ^*) - phase of the cross-spectral density - ^* boundary layer displacement thickness - _long longitudinal coherency - _lat lateral coherency - wavelength of wavy surface - v kinematic viscosity - radian frequency = 2 f - spectral or cross-spectral density - _n phase of the Fourier series - density - time delay - _w wall shear stress - boundary layer momentum thickness