On near-wall turbulence-generating events in a turbulent boundary layer on a riblet surface

The boundary layer over a drag reducing riblet surface is investigated using hot-wire anemometry and flow visualisation. The concept of a riblet sublayer is introduced, and a definition is proposed in terms of a region of reduced turbulence energy production formed near the wall by the addition of riblets. The hot wire records are examined using a modified form of quadrant analysis, and results obtained over plain and riblet surfaces are compared. Close to the wall, the addition of riblets produces a marked reduction in the occurrence of ejection (2nd quadrant) events. A corresponding increase in the incidence of sweep (4th quadrant) events is accompanied by the development of a strong tendency toward a preferred event duration, and a preferred interval between events. These changes diminish rapidly with distance from the surface, becoming almost undetectable beyondy ⁺=40. They are discussed in the light of flow visualisation results, and interpreted in terms of mechanisms associated with the interaction between the riblets and the inner boundary layer flow structures. A conceptual model of the flow mechanisms in the riblet sublayer is proposed.     

Investigation of stationary flow of viscous gas in a thin three-dimensional shock layer

A three-dimensional shock layer near the blunt surface of a fairly smooth body is analyzed asymptotically. Equations of the first approximation are obtained and justified in various cases of the limit 1, 0, ( – 1)^–1M ^-2 0. These equations are simplified for the flow near the stagnation point of a body with double curvature and near the blunt leading edge of a sweptback wing. The results of some calculations are given and compared with the results of [17, 18] in the case of axisymmetric flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 115–126, September–October, 1980.     

Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

Stress-intensity factors for a semi-infinite plane crack with a wavy front

The stress-intensity factors for a semi-infinite plane crack with a wavy front are determined when the crack faces are subjected to normal and shearing tractions. The results are derived using asymptotic methods and are valid to O(²) where =A/1; A is the amplitude and is the wavelength of the wavy front. The normal and shearing tractions are in the form of line loads parallel to the crack front.The results are then used to evaluate, in a qualitative manner, the growth characteristics of a semi-infinite plance crack with a wavy front under combined mode loading. This provides a possible explanation of crack front segmentation observed experimentally.     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Indentation of the semi-infinite elastic solid by a concave rigid punch

A solution is obtained for the relationship between load, displacement and inner contact radius for an axisymmetric, spherically concave, rigid punch, indenting an elastic half-space. Analytic approximations are developed for the limiting cases in which the ratio of the inner and outer radii of the annular contact region is respectively small and close to unity. These approximations overlap well at intermediate values. The same method is applied to the conically concave punch and to a punch with a central hole. , , . , . . .     

Supersonic flow past axisymmetric flared cones

Systematic data on the determination of the aerodynamic characteristics of axisymmetric bodies with a break in the generating line (Fig. 1a, b) in supersonic flow at zero angle of attack are presented in [1, 2, and others]. A characteristic feature of the flow past such bodies is the appearance of an extensive separation zone dec in the region of the break in the generator when the break angle exceeds some minimum value _min, which for a turbulent boundary layer depends basically on the Mach number M at the body surface ahead of the separation zone. In this case, compression waves which change into the oblique compression shocks dc and cc, emanate both from the beginning of the separation zone (point c) and from the end of it (point d). These shocks, intersecting at the point c, form the triple shock configuration acd and acc for which we introduce the notationac[c, d]. The maximum value (_max) of the generator break angle is limited by the possibility of the existence of an attached compression shock, dc. According to these data a change in the generator break angle for the range _minmax of the angle does not disrupt the nature of the flow in the separation zone, but only alters the size of this zone.We shall examine the flow past cones with values of the generator break angles (_max) for which the attached shock dc cannot exist.     

Correlations between porous medium flow data and turbulent tube flow results for aqueous hydroxypropylguar solutions (Part 2)

Turbulent tube flow and the flow through a porous medium of aqueous hydroxypropylguar (HPG) solutions in concentrations from 100 wppm to 5000 wppm is investigated. Taking the rheological flow curves into account reveals that the effectiveness in turbulent tube flow and the efficiency for the flow through a porous medium both start at the same onset wall shear stress of 1.3 Pa. The similarity of the curves = ( _w ) and = ( _w ), respectively, leads to a simple linear relation / =k, where the constantk or proportionality depends uponc. This offers the possibility to deduce (for turbulent tube flow) from (for flow through a porous medium). In conjunction with rheological data, will reveal whether, and if yes to what extent, drag reduction will take place (even at high concentrations).The relation of our treatment to the model-based Deborah number concept is shown and a scale-up formula for the onset in turbulent tube flow is deduced as well.     

Face-Centered and Volume-Centered Discrete Analogs of the Exterior Differential Equations Governing Porous Medium Flow I: Theory

This paper presents a physics-oriented approach to approximate the continuum equations governing porous media flow by discrete analogs. To that end, the continuity equation and Darcys law are reformulated using exterior differential forms. This way the derivation of a system of algebraic equations (the discrete analog) on a finite-volume mesh can be accomplished by simple and elegant translation rules. In the discrete analog the information about the conductivities of the porous medium and the metric of the mesh are represented in one matrix: the discrete dual. The discrete dual of the block-centered finite difference method is presented first. Since this method has limited applicability with respect to anisotropy and non-rectangular grid blocks, the finite element dual is introduced as an alternative. Application of a domain decomposition technique yields the face-centered finite element method. Since calculations based on pressures in volume centers are sometimes preferable, a volume-centered approximation of the face-centered approximation is presented too.     

Unsteady effects in a thin viscous shock layer near a three-dimensional stagnation point of a body moving with given acceleration and deceleration

An unsteady viscous shock layer near a stagnation point is studied. The Navier-Stokes equations are analyzed in the limit 1, Re₀ , df/dt = ^n-mF(t/^m). The Reynolds number Re₀ is defined in the paper by Eq. (1.3) (df/dt is the velocity of the body with respect to an inertial frame of reference moving with the original steady velocity –V'_t8, ₂ = ( – 1)/( + 1)). Various flow regimes in the case 1, l, n max(2m, m + 1), m 0, where ² = 1/Re₀ are analyzed. Equations are derived that generalize the asymptotic analysis to the case of a viscous unsteady flow of gas in a thin three-dimensional shock layer. The problem of a thin unsteady viscous shock layer near the stagnation point of a body with two curvatures is formulated. Examples of numerical solution are given for different ratios of the principal curvatures of the body, the wall temperature, the parameters of the original steady flow, and the acceleration and deceleration regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 100–111, March–April, 1981.I thank Yu. D. Shevelev for a fruitful discussion of the work.     

Shear fracture of polystyrene melts and solutions

We find that symptoms of polymer melt fracture, such as a time-dependent decrease in apparent sample modulus and apparent slip, can be induced by oscillatory torsional shearing flow of polystyrene melts and solutions, even when the polymer molecular weight is below the entanglement threshold, and thre strain amplidute is as low as 3%. Visualization of samples during and after fracture show crack and bubble formation, as well as delamination of the polymer from the rheometer tools. For polystyrene melts, the critical stress for fracture is ^* 0.1–1.0 MPa, depending on polymer molecular weight and temperature, and for solutions it is as low as 5 × 10³ Pa. Since constitutive instabilities require the viscoelastic properties to be highly nonlinear, our observations of melt fracture in unentangled polymers at shearing strains well within the linear viscoelastic range rule out this mechanism for some of our experiments, and show that melt fracture is not always caused by constitutive instabilities.     

Asymptotic theory for calculating heat fluxes near the corner of a body

A method is presented for calculating the distribution of the thermal fluxes, friction stresses, and pressure near the corner point of a body contour in whose vicinity the outer supersonic flow passes through an expansion wave. The method is based on a study of the asymptotic solutions of the Navier-Stokes equations as the Reynolds number R approaches infinity for the flow region in which the longitudinal gradients of the flow functions are large, invalidating conventional boundary layer theory. This problem was examined in part in [1], in which the distribution of the friction and pressure in a region with length on the order of a few thicknesses of the approaching boundary layer was obtained in the first approximation. The leading term of the expansion for the thermal flux to the surface of the body vanishes for a value of the Prandtl number equal to unity and for other values of the Prandtl number does not match directly with its value in the undisturbed boundary layer.The thermal-flux distribution is obtained for values of the Prandtl number approaching unity. For this purpose it was necessary to consider a more general double passage to the limit as 1 and 0 for a finite value of the parameter B=[(–1)/] [–ln ^1/4/]^1/4 characterizing the ratio of the effects of thermal conduction, viscous dissipation, and convection. The solution obtained previously [1] corresponds to the particular case B and therefore for actual values of R=10⁴–10⁶, ~ 0.7 overestimates considerably the effect of the dissipative term on heat transfer, although even in first approximation it describes the pressure distribution well and the friction distribution satisfactorily. For smooth matching of the solutions with the corresponding flow functions in the undisturbed boundary layer it was necessary to introduce a flow region with free interaction for the expansion flow. Equations and boundary conditions which describe the flow as a whole are presented. Examples are given of numerical calculations and comparison with experiment.     

Turbulent boundary layer over a smooth wall with widely separated transverse square cavities

Laser-Doppler velocimetry (LDV) measurements and flow visualizations are used to study a turbulent boundary layer over a smooth wall with transverse square cavities at two values of the momentum thickness Reynolds number (R =400 and 1300). The cavities are spaced 20 element widths apart in the streamwise direction. Flow visualizations reveal a significant communication between the cavities and the overlying shear layer, with frequent inflows and ejections of fluid to and from cavities. There is evidence to suggest that quasi-streamwise near-wall vortices are responsible for the ejections of fluid out of the cavities. The wall shear stress, which is measured accurately, increases sharply immediately downstream of the cavity. This increase is followed by a sudden decrease and a slower return to the smooth wall value. Integration of the wall shear stress in the streamwise direction indicates that there is an increase in drag of 3.4% at bothR .Nomenclature C _f skin friction coefficient - C _fsw friction coefficient for a continuous smooth wall - k height of the cavity - k ⁺ ku / - R Reynolds number based on momentum thickness (U ₁ /v) - Rx Reynolds number based on streamwise distance (U ₁ x/) - s streamwise distance between two cavities - t time - t ⁺ tu ₂ / - U ₁ freestream velocity - mean velocity inx direction - u,v,w rms turbulent intensities inx,y andz directions - u local friction velocity - u _sw friction velocity for a continuous smooth wall - w width of the cavity - x streamwise co-ordinate measured from the downstream edge of the cavity - y co-ordinate normal to the wall - z spanwise co-ordinate - y ⁺ yu / - boundary layer thickness - ₀ boundary layer thickness near the upstream edge of the cavity - _i thickness of internal layer - kinematic viscosity of water - ⁺ zu / - momentum thickness     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

The onset of Taylor-like vortices in the flow induced by an impulsively started rotating cylinder

The onset of instability in the flow by an impulsively started rotating cylinder is analyzed under linear theory. It is well-known that at the critical Taylor number T_c=1695 the secondary flow in form of Taylor vortices sets in under the narrow-gap approximation. Here the dimensionless critical time _c to mark the onset of instability for TT_c is presented as a function of the Taylor number T. Available experimental data of water indicate that deviation of the velocity profiles from the primary flow occurs starting from a certain time 4_c. It seems evident that during _c4_c the secondary flow is very weak and the primary state of time-dependent annular Couette flow is maintained.     

A Capillary Microstructure of the Wetting Front

This article reports the experimental results of a study of the wetting-front microscale structure formed only by capillary forces in homogeneous and random etched glass capillary models. In the homogeneous model, water propagates through the capillary system, evenly filling the capillaries across the direction of flow. Air is trapped by the pinch-off mechanism inside the pore bodies in the form of individual bubbles. The experiments specified three consecutive steps of the pinch-off mechanism, film flow, snap-off, and interface movement. In the random model, both the bypass and pinch-off, forming bypass/cut-off mechanism, create residual air structure. Bypass traps air inside large capillary-pore aggregates which are bounded by small-diameter capillaries in where pinch-off traps air in the adjacent pores. An analysis of the residual air distribution versus depth below the surface in the homogeneous and random micromodels made it possible to identify three successive zones, namely a transition zone, a transmission zone, and a wetting-and-front zone. In the transition zone, the residual air content increases with depth from zero to the constant value in the transmission zone where it remains practically constant. The capillary processes within the wetting-and-front combined zone govern air replacement with wetting and formation of the transmission zone.     

Irrotational outflow of liquid from a stream through a lateral hole of finite depth

An algorithm is constructed for numerical determination of the flow parameters and coefficient of contraction of a jet in the case of irrotational lateral outflow of liquid from a semiinfinite stream through a nozzle of finite depth situated at an arbitrary angle to the mainstream flow. The solution is based on the use of N. E. Zhukovskii's method and the Schwarz-Christoffel formula. The results of calculations for a nozzle situated at an angle = /2 ± , where = /6, are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 162–164, January–February, 1977.     

Elliptic Two-Dimensional Invariant Tori for the Planetary Three-Body Problem

The spatial planetary three-body problem (i.e., one star and two planets, modelled by three massive points, interacting through gravity in a three dimensional space) is considered. It is proved that, near the limiting stable solutions given by the two planets revolving around the star on Keplerian ellipses with small eccentricity and small non-zero mutual inclination, the system affords two-dimensional, elliptic, quasi-periodic solutions, provided the masses of the planets are small enough compared to the mass of the star and provided the osculating Keplerian major semi-axes belong to a two-dimensional set of density close to one.     

Boundary kinetic effects in the problem of slow flow over an evaporating heat-conducting spherical particle

It is shown that for sufficiently large values of the thermal conductivity of the condensed phase' as compared with the thermal conductivity of the vapor (/' Kn) the effects associated with the presence of a Knudsen layer on the evaporating surface must be taken into account in order to obtain a solution of the problem of a spherical particle in a slow (Re, 1) continuum (Knudsen number Kn 1) flow of its own vapor. The drag is calculated for various types of boundary conditions on the particle surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 134–139, March–April, 1987.In conclusion the authors wish to thank V. S. Galkin and M. N. Kogan for useful discussions.     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².