Study of vortex breakdown by particle tracking velocimetry (PTV) Part 3: Time-dependent structure and development of breakdown-modes

This third part of the study deals with the time-dependent nature of vortex breakdown. The results show the unsteady velocity and vorticity field of the initiation and development of breakdown and transition between both predominant breakdown modes, the bubble and the spiral. During the development to breakdown, the generated amount of circumferential vorticity follows the theoretical prediction by Brown and Lopez (1990). This confirms the idea of positive feedback as the key-mechanism leading to vortex breakdown. We regard the bubble-type as the fundamental breakdown type, that is stationary and nearly axisymmetric. The circumferential vorticity is distributed in a form of an elliptical vortex-ring-like structure. Starting from this stage, an increase of volume flux to a higher Reynolds number leads to the transition to the spiral-type with an initial stretching of the vortex ring-like structure and a subsequent change to an asymmetric circumferential vorticity distribution. This in combination with the inductive effect causes the front stagnation point to be deflected radially away and later to rotate around the centerline. Consequently the approaching vortex core is radially deflected in opposite direction and evolves in a spiral path. The idea of a second positive feedback-mechanism gives a possible explanation for the transition. Following this theory the asymmetry of circumferential vorticity will trigger itself at a certain degree by the interaction with the inductively affected stagnation point and its influence on the approaching vortex core. This self-enhancing process will finally lead to the spiral-type breakdown in which the radial distance between rotating stagnation point and deflected vortex core is of the order of the characteristic vortex core radius. The reversed transition from the spiral to a stable bubble-type can be regenerated by decreasing the Reynolds number down to the value that corresponds to the stable bubble state. The flow structure evolves nearly in the time-reversed way as during transition from bubble towards the spiral.     

Study of vortex breakdown by particle tracking velocimetry (PTV)

Using a quasi three-dimensional instantaneous measurement technique, which combines particle tracking velocimetry (PTV) with volume scanning, first quantitative experimental results of the unsteady and asymmetric interior region of vortex breakdown were obtained. The study was carried out in a low speed flow through a cylindrical tube. A vortex was generated by a set of guidevanes and subjected to an adverse pressure gradient causing its breakdown. By scanning a pulsed illuminated planar laser light sheet, a set of meridional and azimuthal cuts of the flow was obtained. With PTV the recorded particle paths in the cuts were processed in order to obtain the instantaneous two-dimensional velocity field, mean streamlines and vorticity distribution. Moreover, the three-dimensional shape of the appearing breakdown, visualized with fluorescent dye, was reconstructed from the cuts. The results revealed that the shape of the bubble nearly equals the streamsurface of the stagnation point. According to the conditions in the water tunnel a single tilted vortex ring at the open rear part of the bubble dominates the interior flow structure of the bubble as first noted by Sarpkaya (1971). The vortical flow is bulged over the bubble, restored and intensified at the lower end. The gathered data lead to the conclusion that the vortex axis remains parallel to the centerline.     

Conditions of simulation of stagnation point heat transfer from a high-enthalpy flow

The problem of local simulation of stagnation point heat transfer to a blunt body is solved within the framework of boundary layer theory on the assumption that the simulation subsonic high-enthalpy flow is in equilibrium outside the boundary layer on the model, while the parameters of the natural flow are in equilibrium at the outer edge of the boundary layer on the body. The parameters of the simulating subsonic flow are expressed in terms of the total enthalpyH ₀, the stagnation point pressurep _w and the velocityV ₁ for the natural free-stream flow in the form of universal functions of the dimensionless modeling coefficients=R _m ^* /R _b ^* ( .<1),=V ₁/2H ₀ ( .<1) whereR _m ^* and R _b ^* are the effective radii of the model and the body at their stagnation points. Approximate conditions for modeling the heat transfer from a high-enthalpy (including hypersonic) flow to the stagnation point on a blunt body by means of hyposonic (M1) flows, corresponding to the case ²1, are obtained. The possibilities of complete local simulation of hypersonic nonequilibrium heat transfer to the stagnation point on a blunt body in the hyposonic dissociated air jets of a VGU-2 100-kilowatt induction plasma generator [4, 5] are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 172–180, January–February, 1993.     

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.     

Turbulent boundary layer on a permeable surface

Several theoretical [1–4] and experimental [5–7] studies have been devoted to the study of the effect of distributed injection of a gaseous substance on the characteristics of the turbulent boundary layer. The primary study has been made of flow past a flat plate with gas injection. The theoretical methods are based primarily on the semiempirical theories of Prandtl [1] and Karman [2].In contrast with the previous studies, the present paper proposes a power law for the mixing length; this makes it possible to obtain velocity profiles which degenerate to the known power profiles [8] in the case of flow without blowing and heat transfer. This approach yields analytic results for flows with moderate pressure gradient.Notation x, y coordinates - U, V velocity components - density - T temperature - h enthalpy - H total enthalpy - c mass concentration - , , D coefficients of molecular viscosity, thermal conductivity, diffusion - c_p specific heat - adiabatic exponent - r distance from axis of symmetry to surface - boundary layer thickness - U velocity in stream core - friction - c_f friction coefficient - P Prandtl number - S Schmidt number - S_t Stanton number - M Mach number - j=0 plane case - j=1 axisymmetric case The indices 1 injected gas - 2 mainstream gas - w quantities at the wall - core of boundary layer - 0 flow of incompressible gas without injection - v=0 flow of compressible gas without injection - ^* quantities at the edge of the laminar sublayer - quantities at the initial section - turbulent transport coefficients     

Approximate method for calculating turbulent boundary layer with positive pressure gradient

A single-parameter integral method is proposed for calculating the turbulent boundary layer with positive pressure gradient which makes it possible to calculate the friction, thermal flux, and layer thickness both ahead of the separation point and in some region behind the separation point.Notation u velocity - density - ^* displacement thickness - ^** momentum thickness - energy thickness - M Mach number - r radius - dynamic viscosity - c_p specific heat at constant pressure - Reynolds number based on initial boundary layer thickness - P Prandtl number - p₁ static pressure at point of initial interaction - p₂ static pressure at pressureplateau - p₀ stagnation pressure - T₀ stagnation temperature - I enthalpy - T_e recovery temperature - T_w ⁰ temperature factor - H form parameter - r₁ recovery coefficient Indices 0 denotes initial section of boundary layer - 1 parameters taken at edge of boundary layer - w parameters taken at the wall temperature - * parameters referred to flow on a flat plate with =0     

A Packet of Elastic Plates Joined Along a Periodic System of Segments

A packet of infinite thin elastic plates joined along a periodic system of collinear segments is considered. The elastic properties and thicknesses of the plates can be different. The plates are loaded by tensile forces at infinity. An algorithm for determining the complex potentials governing stresses in the plates is constructed by solving the Riemann matrix boundaryvalue problem. The stressintensity factors are found and their graphs are given.     

Jet flow field during screech

Measurements were made of the flow field structure and the near field parameters of a jet exhausting from a sonic nozzle with a 1.27 cm exit diameter. Compressed air was used for obtaining stagnation pressures up to 5 atmospheres. The jet exhausted vertically from a settling chamber into an acoustically insulated room and through an insulated duct out through the roof. Measurements were made with several different reflecting surfaces at the nozzle exit as well as an insulating surface. Schlieren pictures at 500,000 frames/s were taken. Overall sound pressure level, impact pressure level downstream, and sound frequency analyzer measurements were made.It was found that with a reflecting surface there was a radial oscillation of the jet which had the same frequency as the dominant sound (screech) frequency emitted by the jet. No axial motion of the inviscid part of the flow structure was detected. The insulated surface at the nozzle exit appeared to shift the dominant frequencies of the sound generated into the region above the audible (>16 KHz). A reflecting surface yielded pure tones (screech) with one or two harmonics. The dominant (screech) frequency decreased as the stagnation pressure increased. The screech frequency was found to be approximately inversely proportional to the length of the first shock cell.Nomenclature C ₀ speed of sound in ambient gas - D diameter of nozzle exit - f frequency of pure tone (screech frequency) - L ₁ length of first cell, distance between nozzle exit plane and intersection of shock with shear layer - M Mach number based on isentropic expansion to ambient pressure - P ₀ stagnation chamber pressure - P _a ambient pressure - P _i impact pressure - R _LB distance from nozzle centerline to left boundary of jet - R _RB distance from nozzle centerline to right boundary of jet - t time - period of screech, 1/f - X _E axial distance of eddy from nozzle exit plane - X _I axial distance of third cell shock intersection from nozzle exit plane - Y _I transverse distance of third cell shock intersection from nozzle centerline     

Asymmetric snap-buckling of a column restrained by a stiff wire

Summary A relatively simple example of asymmetric snap-through buckling in a continuous structure is the nonlinear problem of a cantilevered column restrained at its tip by a stiff wire, which is inclined at an acute angle to the column centerline, and loaded at its tip by a force perpendicular to the centerline. A parameter called , which is the nondimensional ratio of the flexural rigidity of the column to the combined extensional stiffness of the wire and the column centerline, determines the essential features of the buckling. If is zero, or is small compared to unity, the bending of the column is small enough to justify the use of linear bending theory for the column. Hence, even though the constraint is nonlinear, the solution to this problem is obtained in closed form. The critical point for the structure is found to be an asymmetric branching point for =0, while for positive, the critical point is a snap-through type. The effect of is similar to that induced by initial imperfections in more complex structures. For very small , the critical load is markedly decreased from the value for =0. Moreover, the graph of the load vs. tip deflection has the appearance of having an acute discontinuity in slope at the critical point for very small, although it is actually found that the graph has a horizontal tangent there.

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Sommario Un esempio relativamente semplice di collasso asimmetrico per carico di punta in una struttura continua è il problema non lineare di una colonna incastrata ad una estremità e vincolata all'altra estremità da un filo rigido che forma un angolo acuto con l'asse della colonna e caricata a quella estremità con una forza perpendicolare allo stesso asse. Un parametro chiamato , che è il rapporto adimensionale fra la rigidità flessionale della colonna e la rigidezza longitudinale del filo e dell'asse della colonna, determina le caratteristiche essenziali del cedimento. Se è zero o è piccolo rispetto all'unità, l'inflessione della colonna è sufficientemente piccola per giustificare l'uso della teoria lineare di inflessione per la colonna. Di conseguenza anche se il vincolo non è lineare la soluzione del problema è ottenuta in forma compatta. Il punto critico della struttura si trova nel punto di biforcazione asimmetrica per =0, mentre per positivo, il punto critico rappresenta un punto di collasso. L'effetto di è simile a quello prodotto da imperfezioni iniziali in strutture più complesse. Per molto piccolo il carico critico è notevolmente ridotto rispetto al valore per =0. Inoltre il grafico del carico in funzione della curvatura all'estremità sembra avere una netta discontinuità nella pendenza dal punto di biforcazione per molto piccolo benchè, in realtà si trovi che il grafico ha lì una tangente orizzontale.

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This research was supported by the Advanced Research Projects Agency (Project DEFENDER) and was monitored by the U.S. Army Research Office, Durham, under Contract DA-31-124-ARO-D-257.     

MHD flow in the vicinity of the stagnation point in a purely azimuthal magnetic field

Axisymmetric MHD flow in the vicinity of the stagnation point in the presence of a purely azimuthal nonhomogeneous magnetic field B {0, B, 0} is studied. This problem belongs to the class of MHD problems whose solutions are known as solutions of the layer type [1]. This class also includes, in particular, the classical exact solutions of the Navier-Stokes equations.The approximate solutions of the analogous MHD problems for the limiting cases of large and small values of the diffusion number ==/ have been considered in [2–5]. In this case it is possible to divide the flow into the so-called viscous and current layers, for each of which the approximate equations, simpler than the exact equations, are solved numerically or in quadratures. Using this technique it is possible to avoid the basic mathematical difficulty, which is that the sought solution of the boundary-value problem must be selected from a family of two-parameter solutions. The approximate method permits dividing the problem into two stages (corresponding to the two boundary layers) in each of which one unknown parameter is determined (in place of their simultaneous determination by direct integration of the basic equations).The drawback of the approximate methods [2–5] is their nonapplicability in the most interesting case, when the thicknesses of the current and viscous layers are of comparable magnitude, i. e., when the kinematic and magnetic viscosities ( and ) are quantities of the same order. We should also note the poor accuracy of the methods in the framework of the considered approximations for a comparatively large volume of the calculations required, which, in turn, prevents obtaining more exact solutions.The present paper presents a numerical integration of the equations describing MHD flow in the vicinity of the stagnation point over a wide range of S and numbers (Alfvén and diffusion numbers), without the assumption of their smallness, with preliminary determination of the unknowns at the zero of the derivatives of the sought functions with the aid of the method of asymptotic integration.A critical value of the Alfvén number is found, for which the retardation of the fluid by the magnetic field (for the first considered configuration of the magnetic field) at the wall is so intense that the friction vanishes everywhere on the surface of the solid body. It is also found that with further increase of the number S a region of reverse flow appears near the wall, which is separated from the remaining flow by a plane on which the z-component of the velocity is equal to zero.     

Problem of Oil Displacement by Gas Mixtures

The system of equations of one-dimensional flow of a multicomponent mixture with phase transitions through a porous medium is considered. In particular, this system describes the processes of enhanced oil recovery by injection of gases. For this system self-similar solutions of the Riemann problem of discontinuity breakdown are constructed by splitting the problem into physicochemical and hydrodynamic problems.Main elements of the procedure for constructing the solution are illustrated with reference to a four-component system with constant distribution coefficients and the solutions obtained by different methods are compared. It is shown that the approach proposed is also effective for a system with a greater number of components.     

Determination of rough-surface skin friction coefficients from wake profile measurements

A technique for determining the skin friction coefficients from wake profile measurements is described, and is applied to symmetric turbine airfoils with rough surfaces, which operate in a compressible, high-speed flow environment. The procedure involves the measurement of profiles of streamwise momentum in the wakes which form downstream of different airfoils with different levels of surface roughness. Different physical phenomena which affect such wake profiles are discussed and related to different effects, such as surface roughness, form drag, flow separation zones, and laminar-to-turbulent transition. With the same inlet experimental condition for each case, overall skin friction coefficients for the rough airfoils are determined. Resulting values increase considerably as the magnitude of equivalent sandgrain roughness size increases.List of symbols A test airfoil surface area - A _i test section inlet area - A _e test section exit area - c chord length of airfoil - C _f/2 skin friction coefficient - (C _f/2)_smooth skin friction coefficient for smooth airfoil - (C _f/2)_rough skin friction coefficient for rough airfoil - F _s force from surface shear stress - F _p force from form drag due to airfoil blockage and separated flow - F _w force imposed by the top and bottom test section walls - F _s,smooth force from surface shear stress for smooth airfoil - F _s,rough force from surface shear stress for rough airfoil - h height of test section - k roughness height - k _s equivalent sand grain roughness - L total length of airfoil surface from leading edge to trailing edge - p airfoil passage effective pitch - P _o stagnation pressure - P _oe exit local stagnation pressure - P _oe, exit freestream stagnation pressure - P _oi inlet stagnation pressure - P _s static pressure - P _se exit static pressure - s distance along airfoil surface from leading edge - u local streamwise velocity - u _i local streamwise velocity at test section inlet - u _e local streamwise velocity at test section exit - u local freestream streamwise velocity at test section exit - w width of test section - x linear distance along airfoil centerline from airfoil leading edge - y normal coordinate measured from airfoil centerline Greek symbols ratio of specific heats - _s roughness parameter - _i local static air density at test section inlet - _e local static air density at test section exit - local static air density in freestream at test section exit     

Global bifurcations in externally excited two-degree-of-freedom nonlinear systems

In this study we examine the global dynamics associated with a generic two-degree-of-freedom (2-DOF), coupled nonlinear system that is externally excited. The method of averaging is used to obtain the second order approximation of the response of the system in the presence of one-one internal resonance and subharmonic external resonance. This system can describe a variety of physical phenomena such as the motion of an initially deflected shallow arch, pitching vibrations in a nonlinear vibration absorber, nonlinear response of suspended cables etc. Using a perturbation method developed by Kovai and Wiggins (1992), we show the existence of Silnikov type homoclinic orbits which may lead to chaotic behavior in this system. Here two different cases are examined and conditions are obtained for the existence of Silnikov type chaos.An earlier version of this paper was presented in the workshop on Applications of Pattern Formation at the Fields Institute of Mathematical Sciences, Waterloo, Canada, March 1993.     

Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

In this paper, we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions.Two typical particular cases of our results are the following. On one hand, we prove that the equation (1) {ie77-01} where {ie77-02} and {ie77-03} satisfies the maximum principle for solutions in H ¹()L(), i.e., that two solutions u ₁, u ₂H¹() L() of (1) such that u ₁u₂ on , satisfy u ₁u₂ in . This implies in particular the uniqueness of the solution of (1) in H ₀ ¹ ()L().On the other hand, we prove that the equation (2) {ie77-04} where fH^–1() and g(u)>0, g(0)=0, satisfies the maximum principle for solutions uH¹() such that g(u)¦Du|{²L¹(). Again this implies the uniqueness of the solution of (2) in the class uH ₀ ¹ () with g(u)¦Du|{²L¹().In both cases, the method of proof consists in making a certain change of function u=(v) in equation (1) or (2), and in proving that the transformed equation, which is of the form (3) {ie77-05}satisfies a certain structure condition, which using ((v₁ -v ₂)⁺)ⁿ for some n>0 as a test function, allows us to prove the maximum principle.     

A Robust High-Resolution Split-Type Compact FD Scheme for Spatial Direct Numerical Simulation of Boundary-Layer Transition

In this paper an efficient split-type Finite-Difference (FD) scheme with high modal resolution – most important for the streamwise convection terms that cause wave transport and interaction – is derived for a mixed Fourier-spectral/FD method that is designed for the spatial direct numerical simulation (DNS) of boundary-layer transition and turbulence. Using a relatively simple but thorough and instructive modal analysis we discuss some principal trouble sources of the related FD discretization. The new scheme is based on a 6th-order compact FD discretization in streamwise and wall-normal direction and the classical 4th-order Runge–Kutta time-integration scheme with symmetrical final corrector step. Exemplary results of a fundamental-(K-) type breakdown simulation of a strongly decelerated Falkner–Skan boundary layer (Hartree parameter _H = – 0.18) using 70 mega grid points in space are presented up to the early turbulent regime (Re_,turb 820). The adverse pressure gradient gives rise to local separation zones during the breakdown stage and intensifies final breakdown by strong amplification of (background) disturbances thus enabling rapid transition at moderate Reynolds number. The appearance and dynamics of small-scale vortical structures in early turbulence basically similar to the large-scale structures at transition can be observed corroborating Kachanov's hypothesis on the importance of the K-regime of breakdown for coherent structures in turbulence.     

Deformation to breaking of deep water gravity waves

The results of laboratory observations of the deformation of deep water gravity waves leading to wave breaking are reported. The specially developed visualization technique which was used is described. A preliminary analysis of the results has led to similar conclusions than recently developed theories. As a main fact, the observed wave breaking appears as the result of, first, a modulational instability which causes the local wave steepness to approach a maximum and, second, a rapidly growing instability leading directly to the breaking.List of symbols L total wave length - H total wave height - crest elevation above still water level - trough depression below still water level - wave steepness =H/L - crest steepness =/L - trough steepness =/L - F ₁ forward horizontal length from zero-upcross point (A) to wave crest - F ₂ backward horizontal length from wave crest to zero-downcross point (B) - crest front steepness =/F ₁ - crest rear steepness =/F ₂ - vertical asymmetry factor=F ₂/F ₁ (describing the wave asymmetry with respect to a vertical axis through the wave crest) - µ horizontal asymmetry factor=/H (describing the wave asymmetry with respect to a horizontal axis: SWL) - T ₀ wavemaker period - L ₀ theoretical wave length of a small amplitude sinusoïdal wave generated at T _inf0 ^sup–1 frequency - ₀ average wave height     

A particle transport problem with a quadratic nonlinearity and general boundary conditions: The stationary solution

Summary We study a stationary, nonlinear, particle transport problem in slab geometry with general boundary conditions. The existence and uniqueness of the solution is proved by means of fixed point techniques, provided that the source term is sufficiently small.

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Sommario Si studia un problema stazionario nonlineare di particelle in geometria piana con condizioni al contorno generali. L'esistenza e unicitá della soluzione è dimostrata con tecniche di punto fisso purchè il termine di sorgente sia sufficientemente piccolo.

     

Reflection and diffraction of SH waves due to a line source in a heterogeneous medium

The field due to a line source of harmonic SH waves embedded in a semi infinite medium whose density and rigidity vary exponentially with depth is derived in the integral form. The displacement due to diffraction at any point in the shadow zone is obtained and, by the saddle point method of evaluation of the integral, the field at any point in the illuminated region is also found. Finally, geometrical interpretation is given to the different rays arriving in the illuminated as well as in the shadow zone.Nomenclature b shear wave velocity on the free surface - C wave velocity - H _v ⁽¹⁾ (p), H _v ⁽²⁾ (p) Hankel's function of the first and second kind respectively - k Fourier transform parameter with respect to x - v the displacement - fourier transform of v with respect to x - X grazing angle - , small positive constants - positive constant, –/2 - ₀ coefficient of rigidity at the free surface - coefficient of rigidity - _is values of _i at the saddle point (i=1, 2, 3, 4) - the density of the medium - ₀ the density of the medium at the free surface - /2 frequency of vibration     

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.     

Diffusion and Electrical Processes in the Turbulent Boundary Layer and in the Neighborhood of the Stagnation Point

The problem of electric current (engine current) formation in aircraft jet engine ducts as a result of the development of electrical diffusion boundary layers on the surfaces of the duct and internal engine components is investigated. It is assumed that the outer flow containing electrons and positive ions is quasi-neutral and that the electrical quasi-neutrality is violated (and the electric engine current develops) in the wall flow zone as a result of the difference between the electron and ion diffusion coefficients. The problem of the development of an electrical diffusion boundary layer inside the turbulent gasdynamic boundary layer on a plane surface is formulated and solved. The engine current distribution along the duct is found for various values of a turbulent viscosity on the boundary of the gasdynamic boundary layer which affect the laminar-turbulent transition point.The electrical diffusion processes that occurs when an electrically quasi-neutral hydrodynamic stream impinges on a plane surface (simulation of the flow in the neighborhood of a stagnation point) is studied. In this case the Navier-Stokes equations have a self-similar solution. It is shown that the system of electrohydrodynamic equations also has a self-similar solution. The electrical parameter fields are determined and the engine current is found on the basis of this solution.