Control of low-speed turbulent separated flow using jet vortex generators

A parametric study has been performed with jet vortex generators to determine their effectiveness in controlling flow separation associated with low-speed turbulent flow over a two-dimensional rearward-facing ramp. Results indicate that flow-separation control can be accomplished, with the level of control achieved being a function of jet speed, jet orientation (with respect to the free-stream direction), and jet location (distance from the separation region in the free-stream direction). Compared to slot blowing, jet vortex generators can provide an equivalent level of flow control over a larger spanwise region (for constant jet flow area and speed).Nomenclature C _p pressure coefficient, 2(P-P_)/V ² - C _Q total flow coefficient, Q/ v - D ₀ jet orifice diameter - Q total volumetric flow rate - R Reynolds number based on momentum thickness - u fluctuating velocity component in the free-stream (x) direction - V free-stream flow speed - VR ratio of jet speed to free-stream flow speed - x coordinate along the wall in the free-stream direction - jet inclination angle (angle between the jet axis and the wall) - jet azimuthal angle (angle between the jet axis and the free-stream direction in a horizontal plane) - boundary-layer thickness - momentum thickness - lateral distance between jet orifices A version of this paper was presented at the 12th Symposium on Turbulence, University of Missouri-Rolla, 24–26 Sept. 1990     

Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.     

Characteristics of single and clusters of bursting events in the inner layer

A new method of identification of bursting events in a turbulent boundary layer is presented and applied to VITA. The philosophy adapted here is similar to that of Tiederman (1988). It consists of the classification of individual events into groups corresponding to multiple breakups of the same streak (Bogard and Tiederman, 1986). The strategy is however different, in that it uses differences in the signatures of the events belonging to the same cluster. The iterative procedure finds out these differences and classifies events which have similar signatures. It is found that the distribution of the frequency of groups of events identified this way differs from the distribution of the solitary ones. The multiple shear layer events (MSLs) have strikingly different signatures compared to single shear layer events (SSLs). These differences are mostly present near the wall and disappear progressively at the beginning part of the constant shear layer. The contribution of the MSLs to the Reynolds shear stress is larger. The differences in the conditional averages of the streamwise and wall normal velocities are highly suggestive that the regeneration mechanism of the clusters of events near the wall is dynamically different.Part of this study was carried out while the author was invited to Ecole Polytechnique Fédérale de Lausanne for a period of nine months. He would like to thank Prof. I. L. Rhyming for this very kind invitation. Part of the data presented here was obtained by Dr. M. Q. Feng. Financial support of the Délégation Générale de l'Armement via the Direction des Recherches, Etudes et Techniques is gratefully acknowledged. Thanks are finally due to the referees for their interest in this work and their comments.     

Simultaneous measurement of size and velocity of microbubbles moving in an opaque tube using an X-ray particle tracking velocimetry technique

An X-ray particle tracking velocimetry (PTV) technique was developed to simultaneously measure the sizes and velocities of microbubbles in a fluid without optical aberration. This technique is based on a combination of in-line X-ray holography and PTV. The X-ray PTV technique uses a configuration similar to that of conventional optical imaging techniques, and is easy to implement. In the present work, microbubbles generated from a fine wire by electrical heating were used as tracer particles. The X-ray PTV technique simultaneously recorded size and velocity data for microbubbles (_b=10–60 m) moving upward in an opaque tube (inner diameter =2.7 mm). Due to the different refractive indices of water and air, phase contrast X-ray images clearly show the exact size and shape of overlapped microbubbles. In all of the working fluids tested (deionised water and 0.01 M and 0.10 M NaCl solutions), the measured terminal velocity of the microbubbles rising through the solution was proportional to the square of the bubble diameter. The proposed technique can be used to extract useful information on the behaviour of various bio/microscale fluid flows that are not amenable to analysis using conventional methods.     

Experimental investigation of the resistance to the flow of a conducting fluid in flat insulated ducts in the presence of a transverse magnetic field with allowance for fringe effects and wall roughness

The distribution of pressure, velocity, and electrical potential has been investigated for a mercury flow in insulated rectangular ducts with a large side ratio (Hartmann-type flow). The ranges of variation of the Reynolds, Hartmann, and Stewart numbers were 7·10²R5·10⁵, 0H490, and 0N24, respectively. Special attention is given to the sections of the channel where the flow enters and leaves the magnetic field. In these zones the pressure is sharply nonuniform and the velocity profiles in a plane perpendicular to the field acquire an M shape. A relation is established between the length of the entrance section, where the flow is three-dimensional, and the MHD similarity criteria. It is shown that ducts which are hydraulically smooth in the absence of a magnetic field become increasingly rough as the field grows stronger. Data are obtained on the resistance coefficient for a stabilized flow measured in a magnetic field and on the dependence of the critical Reynolds number on the Hartmann number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 10–21, July–August, 1971.     

A phase field approach with a reaction pathways-based potential to model reconstructive martensitic transformations with a large number of variants

A pathway tree is constructed by recursively duplicating a single reconstructive martensitic transformation path with respect to lattice symmetries and point-group rotations. An energy potential built on this pathway is implemented in a phase-field technique in large strain framework, with the transformational strain as the order parameter. A specific splitting between non-dissipative elastic behavior and the dissipative evolution of the order parameter allows for the modeling of acoustic waves during rapid transformations. A simple toy-model transition from hexa- to square-lattice successfully demonstrates the possibility to model reconstructive martensitic transformations for a large number of variants (more than one hundred). Pure traction applied to our toy-model shows that variants can nucleate into previously created variants, with a hierarchical nucleation of variants spanning over five levels of transformation.     

General expressions for unsteady forces in a lattice of profiles

A large number of studies have been devoted to the unsteady flow of a viscid incompressible fluid past a lattice of thin profiles and the determination of the resulting aerodynamic forces and moments. For example, in the particular case of the motion of a lattice with stagger with zero phase shift of the oscillations between neighboring profiles, Haskind [1] determined the unsteady lift force and moment. Popescu [2] suggested expressions for the force and moment in the case when =0 and =0, using the method of conformal mapping. Samoilovich [3] obtained equations for the unsteady lift force and moment by the method of the acceleration potential for phase shift =0 and = of the oscillations between neighboring profiles. Musatov [4] used an electronic digital computer to calculate the overall unsteady aerodynamic characteristics of a grid by the vortex method, taking into account the amplitude of the oscillations and the initial circulation for =m (m1). Gorelov [5] determined the coefficients of the over-all unsteady aerodynamic force and moment of a profile in a lattice with the stagger and any value of =m. He used a method based on the unsteady flow past an isolated profile with subsequent account for the interference of the profiles in the lattice.In the following we find general expressions for the unsteady lift force and moment acting on a lattice moving in an incompressible fluid with the constant velocity U. These formulas generalize the known formulas for the isolated profile [6]. The profiles of a staggered grid (Section 1) are considered to be thin and slightly curved, and perform oscillations with a phase shift of the oscillations between neighboring profiles. The method of separation of singularities is used to obtain the solution in closed form. The coefficients of the expansion of the complex velocity in a series in the derivatives of a function are calculated. An integral equation relative to the unknown tangential velocity component in the wake is derived (Section 2), and its analytic solution is given (Section 3). For =0 the solution coincides with the solution obtained earlier in [7]. Expressions are obtained for the forces and moments (Section 4) in the form of four terms. The first two terms determine the force and moment for motion with constant circulation, and the last two determine these characteristics for motion with variable circulation. The suction force acting at the leading edges of the profiles is found in a general form. Particular cases of closely and widely spaced lattices are considered. Computational results are presented.     

New method of solving Lame-Helmholtz equation and ellipsoidal wave functions

Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lamé-Helmholtz equation still remains to be tackled. Arscott and Moglich method of double-series expansion as well as Malurkar nonlinear integral equation are incapable of reaching the final explicit solution.Our main result consists in obtaining analytic expressions for ellipsoidal wave functions of four species (i=1,2,3,4) including the well known Lam(α) functions E_ci(snα),E_z1(snα) as special cases. This is effected by deriving two Integra-differential equations with variable coefficients and solving them by integral transform. Generalizing Riemann’s idea of P function, we introduce D function to express their transformation properties.     

Turbulent flow of non-Newtonian substances

Summary A unique shear stress-shear rate relationship exists for laminar flow of any time independent substance in a tube, whereas this is not the case for turbulent flow. In order to obtain a unique relationship for turbulent flow, a new approach based on the elementary theoretical interpretation of experimental data is adopted in the present paper. In particular, wall shear stress is found to be a unique function of a new turbulent pseudo shear rate term. In this relationship therè are two parameters which characterize a given substance — the limiting viscosity at high shear rateµ _m and a factor _m which takes into account modification of turbulent structure by the non-Newtonian properties. Both of these parameters must be determined experimentally. Methods of predicting pressure gradients and of scaling up are outlined. In applying the approach to suspensions in which the solid phase has a density greater than that of the liquid medium, it may be important to determine the increment in shear stress equivalent to the energy required to maintain the solid particles in suspension.The validity of this approach is confirmed by data for the flow of a variety of substances including kaolin suspensions and Carbopol solutions in tubes ranging in diameter from 1.5 to 20 mm. Nomenclature C volume fraction solid in suspension - D tube diameter - f Darcy-Weisbach friction factor - g gravitational acceleration - K _s proportionality constant defined by eq. [10] - L length of tube - P pressure - Re Reynolds number - t exponent defined by eq. [1] - V mean velocity - V ^* volume of particles in pipe lengthL - W settling velocity of particles - _m factor defined by eq. [1] - shear rate - turbulent pseudo shear rate defined by eqs. [8] and [9] - _w wall shear stress - ( _w) _s increment in wall shear stress due to presence of settling particles - µ _m limiting viscosity at high rate of shear - ₁ density of carrier liquid - _m density of mixture - _s density of solid Professor of Chemical Engineering, University of Toronto and scientific advisor to Worthington (Canada) Ltd.With 8 figures     

The fixed point and eigenelement of sum and product about concave and convex operators

In this paper, we obtain the sufficient conditions under which there exists the fixed point of sum and product about concave and – convex operators in the positive cone of linear semi-order space, and the iterative procedure and error estimate can be given. The relation between eigenvalue and eigenelement will also be studied in this paper.     

Quadratic systems of conservation laws with generic behavior at infinity

We identify quadratic systems of conservation laws with generic behavior at infinity, where the genericity conditions derive naturally when studying weak solutions of conservation laws. Namely, we identify quadratic models for which the vector field associated with the viscosity admissibility criterion has properties at infinity that are true for an open and dense subset of the set of all planar quadratic vector fields in the metric associated with the Euclidian space of coefficients. We determine the boundaries of the regions containing, generic models in the parameter space of coefficients of quadratic models. We show that when crossing the boundaries of nongeneric models transversally, the Poincaré compactification of the corresponding vector field undergoes either a saddle node or a transcritical bifurcation at infinity. For quadratic models with a bounded elliptic region we calculate the loci of nongeneric models assuming the viscosity matrix to be the identity. We obtain a two-parameter normal form for such models and show that the boundaries that determine generic models in the two-dimensional parameter space correspond to the Schaeffer-Shearer classification of models with an isolated umbilic point. Since the loci of nongeneric models are invariant under the equivalence transformations that preserve weak solutions of conservation laws, understanding their behavior at infinity promises to provide an insight into a general classification of quadratic conservation laws.     

Character of the boundary layer near its origin on a body rotating in a fluid at rest

The evolution of the boundary layer on bodies of revolution rotating about the symmetry axis in a fluid at rest is largely determined by the position of its origin with respect to the axis of rotation. If the origin of the boundary layer coincides with a pole of the rotating body, then under fairly general assumptions as to the shape of the body the boundary layer has a nonzero thickness in the vicinity of the pole, and the flow in it is described by a particular self-similar solution of the boundary-layer equations [1, 2]. The applicability of existing approximate methods for calculation of the boundary layer [2, 3] is restricted to this case. The results of the present article refer to the case in which the boundary originates at the leading edge at a finite distance from the rotation axis. The behavior of the solution of the boundary-layer equations near the edge is determined. A transformation of variables that reduces the system of boundary-layer equations to a form suitable for analysis and solution is derived. This transformation is used to obtain universal equations determining the local behavior of the boundary layer in the vicinity of its origin in both of the cases indicated above.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1976.     

Recording and evaluation methods of PIV investigations on a helicopter rotor model

Three-component particle image velocimetry measurements at moderate speeds and observation distances can now be accomplished on a routine basis. This article discusses the experiment performed on a 4 m-diameter model rotor in the 6-m×8-m open test section of the Large Low Speed Facility of the German–Dutch Wind Tunnels. More than half a terabyte of raw data were recorded at various positions on the advancing and retreating sides of the rotor in order to obtain detailed measurements of the trailing vortex in the frame of an international project. This paper addresses measuring techniques and possible sources of errors and presents a limited number of cases for the purpose of illustrating the solutions to numerous technical challenges relating to the acquisition and analysis of vortical flows.List of symbols C _T thrust coefficient (T/² R ⁴) - M magnification - r _c radius of vortex core (mm) - R rotor radius (m) - T thrust (N) - u,v,w velocity components in x, y and z coordinates (m/s) - (u,v,w)_wt velocity components in wind tunnel coordinates (m/s) - U _max maximum in-plane velocity component (m/s) - W _max maximum out-of-plane velocity component (m/s) - x,y,z particle image velocimetry (PIV) frame coordinates (m) - (x,y,z)_wt wind tunnel coordinates (m) - t time delay (s) - Z light sheet thickness (mm) - Z light sheet thickness (mm) - rotor rotation frequency (rad/s) - rotor azimuth angle during recording (deg) vortex age - rotor shaft angle (deg) - _x displacement measurement error - advance ratio (V/R) - air density (kg/m³) - circulation (m²/s) - _z vorticity (s^–1) Abbreviations AFDD Aeroflightdynamics Directorate - BVI blade–vortex interaction - DLR Deutches zentrum für Luft- und Raumfahrt - DNW German–Dutch Wind tunnel - HART HHC aeroacoustic rotor test - LLF large low speed facility - NASA National Aeronautics and Space Administration - ONERA Office National dEtudes e de Recherches Aerospatiales - RANS Reynolds-averaged Navier–Stokes - SPR stereo pattern recognition - 3C-PIV three-component particle image velocimetry     

Constitutive equations for two‐step thermoelastic phase transformations

A number of hypotheses on the mechanical behavior of shape memory alloys such as titanium nickelide in twostep (martensitic and rhombohedral) phase transformations are formulated on the basis of experimental data. A system of relations linking stresses, strains, temperature, and phase composition in such transitions is proposed.     

Contact Symmetry Algebras of Scalar Ordinary Differential Equations

Using Lie's classification of irreducible contact transformations in thecomplex plane, we show thata third-order scalar ordinary differential equation (ODE)admits an irreducible contact symmetry algebra if and only if it is transformableto q ⁽³⁾=0 via a local contact transformation. This result coupled with the classification of third-order ODEs with respect to point symmetriesprovide an explanation of symmetry breaking for third-order ODEs. Indeed, ingeneral, the point symmetry algebra of a third-order ODE is not asubalgebra of the seven-dimensional point symmetry algebra of q ⁽³⁾=0.However, the contact symmetry algebra of any third-order ODE, except forthird-order linear ODEs with four- and five-dimensional pointsymmetry algebras, is shown to be a subalgebra of the ten-dimensional contact symmetryalgebra of q ⁽³⁾==0. We also show that a fourth-orderscalar ODE cannot admit an irreducible contact symmetry algebra. Furthermore, weclassify completely scalar nth-order (n5) ODEs which admitnontrivial contact symmetry algebras.     

On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations

We consider nonlinear wave and Klein-Gordon equations with general nonlinear terms, localized in space. Conditions are found which provide asymptotic stability of stationary solutions in local energy norms. These conditions are formulated in terms of spectral properties of the Schrödinger operator corresponding to the linearized problem. They are natural extensions to partial differential equations of the known Lyapunov condition. For the nonlinear wave equation in three-dimensional space we find asymptotic expansions, as t, of the solutions which are close enough to a stationary asymptotically stable solution.     

Reversible stress-induced martensitic phase transformations in a bi-atomic crystal

In an earlier work, Elliott et al. [2006a, Stability of crystalline solids—II: application to temperature-induced martensitic phase transformations in bi-atomic crystals. Journal of the Mechanics and Physics of Solids 54(1), 193-232], the authors used temperature-dependent atomic potentials and path-following bifurcation techniques to solve the nonlinear equilibrium equations and find the temperature-induced martensitic phase transformations in stress-free, perfect, equi-atomic binary B2 crystals. Using the same theoretical framework, the current work adds the influence of stress to study the model's stress-induced martensitic phase transformations.The imposition of a uniaxial Biot stress on the austenite (B2) crystal, lowers the symmetry of the problem, compared to the stress-free case, and leads to a large number of stable equilibrium paths. To determine which ones are possible reversible martensitic transformations, we use the (kinematic) concept of the maximal Ericksen-Pitteri neighborhood (max EPN) to select those equilibrium paths with lattice deformations that are closest, with respect to lattice-invariant shear, to the austenite phase and thus capable of a reversible transformation. It turns out that for our chosen parameters only one stable structure (distorted αIrV) is found within the max EPN of the austenite in an appropriate stress window. The energy density of the corresponding configurations shows features of a stress-induced phase transformation between the higher symmetry austenite and lower symmetry martensite paths and suggests the existence of hysteretic stress-strain loops under isothermal load-unload conditions. Although the perfect crystal model developed in this work over-predicts many key material properties, such as the transformation stress and the Clausious-Clapeyron slope, when compared to real experimental values (based on actual polycrystalline specimens with defects), it is—to the authors' knowledge—the first atomistic model that has been demonstrated to capture all essential trends and behavior observed in shape memory alloys.     

Capillary Imbibition and Pore Characterisation in Cement Pastes

The kinetics of capillary imbibition in ordinary Portland cement pastes has been studied experimentally and theoretically. Nuclear magnetic resonance stray field imaging (STRAFI) has been used to record water concentration profiles for various ingress times. The profiles follow a t law and thus a master curve can be formed using the Boltzmann transformation. The distribution of pore sizes within the sample as measured by NMR cryoporometry shows a prominent peak at 100Å. A computer model of the pore structure was developed consisting of a lattice of interconnecting pores with a size distribution consistent with the cryoporometry results. The Hagen–Poiseuille law was used to describe the kinetics of the water in this pore structure. The best agreement between the computer simulations and the experimental master curve was obtained by using a narrower range of pore sizes than indicated by the cryoporometry results.