Projection conditions on the vorticity in viscous incompressible flows

The problem of establishing appropriate conditions for the vorticity transport equation is considered. It is shown that, in viscous incompressible flows, the boundary conditions on the velocity imply conditions of an integral type on the vorticity. These conditions determine a projection of the vorticity field on the linear manifold of the harmonic vector fields. Some computational consequences of the above result in two-dimensional calculations by means of the nonprimitive variables, stream function and vorticity, are examined. As an example of the application of the discrete analogue of the projection conditions, numerical solutions of the driven cavity problem are reported.     

Influence of the initial spatially inhomogeneous temperature perturbations on vibrational relaxation dynamics

The vibrational relaxation of a nonequilibrium molecular gas (T_V>T) plays an important role in the physics of gas lasers, laser chemistry [1], and plasma chemistry [2]. This paper is devoted to an analysis of the dynamics of V-T relaxation with spatially inhomogeneous perturbations of the translational temperature taken into account.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 6, pp. 77–80, November–December, 1984.     

Influence of the initial conditions on axisymmetric jets in a parallel flow

An experimental investigation was made of the initial-section flow of axisymmetric helium, air, and freon-12 jets in a parallel air flow for two different velocity profiles at the nozzle exit near the boundary of the jet. In one case, the velocity profile was determined by boundary layers on the nozzle walls; in the other case, it was produced artificially by means of a honeycomb of tubes of variable length. Measurements were made of the profiles of the mean and the pulsation velocity and the temperature. The flow was also photographed. The investigations showed that, depending on the initial conditions, the intensity of mixing of the jets in the initial section at Reynolds numbers Re 10⁴ (calculated using the jet diameter) can change from the level determined by molecular diffusion to the level characteristic of developed turbulent flow. The flow structure in the annular mixing layer also depends strongly on the initial conditions. The observed ordered structures in the mixing layer are related to a section of development of perturbations near the nozzle. The ordered structures are strongly influenced by the effect on the jet of acoustic vibrations from an external source. When the initial velocity profile is produced by the honeycomb, the transition to developed turbulence may be due to the development of long-wavelength perturbations or to the development of small-scale turbulence generated by the flow over the end of the honeycomb.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 18–24, July–August, 1980.We thank V. M. levlev and K. I. Artamonov for assistance and for discussing the work.     

Role of initial conditions in the dynamics of a double pendulum at low energies

We consider the low energy dynamics of the double pendulum. Low energy implies energies close to the critical value required to make the outer pendulum rotate. All the known interesting results for the double pendulum are at high energies, that is, energies higher than that required to make both pendulums rotate. We show that interesting behavior can occur at low energies as well by which we mean energies just sufficient to make the outer pendulum rotate. A harmonic balance and the Lindstedt–Poincare analysis at the low energies establish that at small, but finite amplitude; the two normal modes behave differently. While the frequency of the “in-phase” mode is almost unchanged with increasing amplitude, the frequency of the “out-of-phase” mode drops sharply. Numerical analysis verifies this analytic result and since the perturbation theory indicates a mode softening for the out-of-phase mode at a critical amplitude, we did a careful numerical analysis of the low energy region just above the threshold for onset of rotation for the outlying pendulum. We find chaotic behavior, but the chaos is a strong function of the initial condition.     

Interplay of surface geometry and vorticity dynamics in incompressible flows on curved surfaces

Incompressible viscous flows on curved surfaces are considered with respect to the interplay of surface geometry, curvature, and vorticity dynamics. Free flows and cylindrical wakes over a Gaussian bump are numerically solved using a surface vorticitystream function formulation. Numerical simulations show that the Gaussian curvature can generate vorticity, and non-uniformity of the Gaussian curvature is the main cause.In the cylindrical wake, the bump dominated by the positive Gaussian curvature can significantly affect the vortex street by forming velocity depression and changing vorticity transport. The results may provide possibilities for manipulating surface flows through local change in the surface geometry.     

Wall boundary conditions for inviscid compressible flows on unstructured meshes

In this paper we revisit the problem of implementing wall boundary conditions for the Euler equations of gas dynamics in the context of unstructured meshes. Both (a) strong formulation, where the zero normal velocity on the wall is enforced explicitly and (b) weak formulation, where the zero normal velocity on the wall is enforced through the flux function are discussed. Taking advantage of both approaches, mixed procedures are defined. The new wall boundary treatments are accurate and can be applied to any approximate Riemann solver. Numerical comparisons for various flow regimes, from subsonic to supersonic, and for various approximate Riemann solvers point out that the mixed boundary procedures drastically improve the accuracy. © 1998 John Wiley & Sons, Ltd.     

Influence of boundary conditions on dynamics and stability of thin cylindrical shells conveying fluid

This paper is concerned with the effect of boundary constraints of thin cylindrical shells containing flowing fluid, and investigates the quantitative relation between the fundamental natural frequency and mode shape and the geometric parameters and the flow velocity in detail. The results show that although axial displacements are smaller than radical displacements, the effect of axial constraints is significant for simply supported systems. It makes the lowest critical velocity of instability increase by 40%, and the minimum frequencies by 50%, even for long shells (L/a=20.0). On the contrary, the influence of rotation constraint (w/x=O) is much smaller.     

Linear response of a plasma in a magnetic field to localized initial vorticity

It is established that for three-dimensional disturbances the long-range effect is observed even in the absence of boundaries. The problem of the evolution of the electrodynamic and gas dynamic disturbances created by a localized vorticity source is considered. It is shown that acoustic disturbances of a nonlocal nature are formed. The spatial structure of the electric potential and the nonlocal electric field created by localized initial vorticity at a finite value of the Hall parameter is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 109–114, March–April, 1990.     

Effect of initial conditions on a circular jet

 Two circular jets are considered. One issues from a contraction with a laminar top-hat velocity profile. The other exits from a pipe with a fully developed turbulent mean velocity profile. In spite of the significantly different initial conditions, spectra of axial and radial velocity fluctuations in the far field (x≳30d, where d is the jet diameter) collapse over scales ranging from the Kolmogorov length scale to a length scale characterizing the organized motion. This agreement is consistent with the observation that, at the measurement station, the large-scale anisotropy is unchanged between the two flows. Received: 21 September 2000/Accepted: 5 March 2001     

Influence of vorticity on the heat transfer to blunt bodies in hypersonic flight

When blunt bodies are in hypersonic flight, a high-entropy layer of gas with nonzero vorticity is formed near their surface. The transverse gradients of the entropy, density, and gas velocity in the layer are high, which makes it necessary to take into account its absorption by the boundary layer of finite thickness . This vortex interaction is usually accompanied by an increase in the heat flux q and the frictional stress on the wall compared with their values as calculated in accordance with the classical scheme of a thin boundary layer, when the parameters on the outer edge of the boundary layer are set equal to the inviscid parameters on the body. This effect has been investigated on the side surface of slender (with angle 1 to the undisturbed flow) blunt bodies in a hypersonic stream [1–3]. It is shown in the present paper that the effect can have a stronger and even qualitative influence on the flow over blunt bodies with 1 if the radius of curvature R_s of the detached shock wave on the axis is small compared with the midsection radius R of the body. It is shown that the distributions of the heat fluxes with allowance for the vorticity of the inviscid shock layer are similar in the case of slightly blunt (r₀/R 0) cones with half-angles less than a critical _*.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 50–57, March–April, 1981.     

The transport of vorticity on rotating Bernoulli surfaces of any geometry in compressible,inviscid flow with applications to turbomachine blade rows

The transport of vorticity is analysed for a compressible, inviscid flow which is steady relative to a reference frame rotating at constant angular velocity. It is shown that Helmholtz's vorticity convection theorem does not apply to this flow but nevertheless the vortex lines are transported on a streamsurface which therefore corresponds to the familiar Bernoulli surface.Explicit, integrated results are obtained for Bernoulli surfaces of any geometry. The transport of the normal component of vorticity is obtained for the general case in closed form, whereas the transport of the streamwise component is closed in form for some cases but involves a time difference integral over the bounding streamlines.Application is made to a turbomachine blade row where the flow between two consecutive blades is examined. Explicit results are obtained for the streamwise vorticity development in the axial flow configuration in terms of the traverse time integral for a particle, taken around the blade profile. The more general mixed flow configuration is also examined where a closed result is obtained only for the incompressible case.