Properties of hypersonic separated flows at moderate reynolds numbers

The hypersonic rarefied flow past a flat plate with a transverse barrier and past a plate with a bend in the generator (a compression angle) is studied at Reynolds numbers Re≤10⁴. Direct statistical modeling (Monte Carlo method) is used to investigate the characteristics of the separated flow formed on the plate as a function of the Reynolds number, the surface temperature, the barrier dimensions, and the internal degrees of freedom of the molecules. The results obtained are compared with those for analogous high-Re flows. The possibility of using the similarity criteria derived for the continuum flow regime is considered. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 134–144, January–February, 2000. The study was carried out with the support of the Russian Foundation for Basic Research (project No. 97-01-00577) and the Program of State Support for Leading Scientific Schools (grant No. 96-15-9606).     

Calculation of two-dimensional shear-driven cavity flows at high reynolds numbers

The time-dependent Navier–Stokes equations are numerically integrated for two-dimensional incompressible viscous flow in a shear-driven square cavity. Using a time-splitting method and finite differences on a staggered mesh, the momentum and pressure equations are directly solved by a tensor product method where one finite difference direction is diagonalized by eigenvalue decomposition. The effects of increasing Reynolds number are studied and the developing boundary layer is captured by using a finely clustered mesh. At Re = 30000 the flow is in a continuously developing unsteady regime. Power spectrum plots indicate that the unsteady flow oscillates with one fundamental frequency and exhibits some characteristics of transition between laminar and turbulent states.     

Some two-dimensional flows at finite magnetic reynolds numbers

Flows at finite magnetic Reynolds numbers are characterized by a strong effect of the induced magnetic fields on the stream. In the present paper we determine the current distribution and estimate the influence of the Lorentz force component perpendicular to the stream in a two-dimensional channel with electrodes. We also estimate the influence of nonuniformities of the velocity in the stream path of an incompressible fluid when the characteristic magnetic Reynolds numbers     

Numerical investigation of overexpanded nozzle flows

The flow in a planar overexpanded nozzle with a slope discontinuity is studied numerically by means of two- (2D) and three-dimensional (3D) Reynolds-averaged Navier–Stokes simulations and is compared to experimental results. The nozzle pressure ratios (NPR) vary from 1.6 to 10. A good agreement is found between experimental and numerical results and two configurations are observed: under a certain critical NPR, the flow is shown to be asymmetrical with respect to the nozzle axis, while it is perfectly symmetrical for higher NPRs. The value of the critical NPR is found to be very dependent on the turbulence model. Finally, an hysteresis phenomenon is evidenced since the NPR at which the change of flow configuration occurs is different whether the NPR is increasing or decreasing in the nozzle.     

Optimal control of viscous flow at high reynolds numbers

The most promising and most highly developed method for reducing drag in aerodynamics remains control of the flow by blowing and suction. In practice the main control problems remain the reduction of separation and the protracting of the transition of the boundary layer. These problems are solved mainly by experimental methods [1]. Meanwhile the main theoretical question remains unanswered: what is the theoretical minimum drag attainable by control through blowing (or suction)? In the present study an answer is given to this question for the cage of laminar flow round a body by a viscous incompressible fluid at high Reynolds numbers.     

Calculation of oseen flows past a circular cylinder at low reynolds numbers

The velocity, pressure, vorticity and streamfunction are computed in the Oseen hydrodynamic field of an unbounded fluid past a circular cylinder in the Reynolds Number range going from 0.4 to 12. The boundary condition is satisfied by means of the method of least squares that determines suitable coefficients for Faxén series. Particular investigation is made of the wake region in which calculations are made of flow patterns, velocity and vorticity distributions. It is shown that, attached vortices arise at the rear of the cylinder at Reynolds Number Re=3.025. Calculated drag coefficients are in good agreement with known results of the works of several authors up to a Reynolds Number of 20.     

Numerical investigation of hypersonic step-flows

Hypersonic aerospace vehicles are exposed to extreme flight conditions with heavy contour loads during their mission. Especially at ridges and sharp corners, the wall heat flux and pressure may cause serious damage to the body. Sometimes, the surface material cannot resist the high loading and fails completely. In this work the laminar hypersonic flow over forward and backward facing steps is investigated by CFD techniques and the results are compared with experimental data. The selected flow conditions correspond to cold hypersonic flow according to the availability of experimental data. The Navier-Stokes equations in the high temperature gas approximation of a thermally perfect gas in local equilibrium serve as the model for the physical problem. A multiblock finite-volume method is used to discretize consistently all spatial derivatives appearing in the balance equations. A second order in space Godunov-type method is utilized for the non-diffusive part of the governing equations whereas centered differences are used for the diffusive part. Time integration is performed by a second order implicit scheme. In each time step, the resulting nonlinear system of equations is solved by Newton's method employing a relaxation scheme based on conjugate gradients for the linear equation system. The results obtained permit a close insight into the physics of the flow problems under consideration and by this provide valuable information for construction concepts of hypersonic vehicles. Besides a careful comparison of the numerical results with experimental data, numerical aspects like the grid influence are addressed. Received 9 November 1998 / Accepted 2 December 1999     

Optimization of body shape at small reynolds numbers

The problem of the optimization of the shape of a body in a stream of viscous liquid or gas was treated in [1–5]. The necessary conditions for a body to offer minimum resistance to the flow of a viscous gas past it were derived in [1], The necessary optimality conditions when the motion of the fluid is described by the approximate Stokes equations were derived in [2], The shape of a body of minimum resistance was found numerically in [3] in the Stokes approximation. The optimality conditions when the motion of the fluid is described by the Navier—Stokes equations were derived in [4, 5], and in [4] these conditions were extended to the case of a fluid whose motion is described in the boundary-layer approximation. The necessary optimality conditions when the motion of the fluid is described by the approximate Oseen equations were derived in [5] and an asymptotic analysis of the behavior of the optimum shape near the critical points was performed for arbitrary Reynolds numbers.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp, 87–93, January–February, 1978.     

Effect of compressibility on the development of taylor-görtler vortices at high reynolds numbers

The development of disturbances in viscous compressible flows caused by centrifugal forces is investigated. On the basis of an asymptotic analysis of the Navier-Stokes equations at high Reynolds and Görtler numbers, mathematical models describing the development of three-dimensional unstable vortex structures are constructed. Various linear boundary-value problems are analytically solved. One type of boundary layer instability is that generated by a centrifugal force field. This kind of instability can manifest itself in the flow past concave surfaces or, in general, in flows with streamlines of positive curvature [1, 2]. Instability-driven Görtler vortices have been the subject of much research which was reviewed, for example, in [2–4].     

The effects of nozzle geometry on waterjet breakup at high Reynolds numbers

Waterjet breakup is traditionally considered to follow the Ohnesorge classification. In this classification, high Reynolds number waterjets are considered to atomize quickly after discharge. By generating a constricted waterjet where the water flow stays detached all the way through the nozzle, we have observed the first wind-induced breakup mode at high Reynolds numbers. Such a peculiar behavior, however, was not observed in non-constricted waterjets. Our results indicate that, constricted jets do not follow the Ohnesorge classification, in contrast to the non-constricted waterjets. We discuss the impact of nozzle geometry on the characteristics of waterjets and support our discussion by numerical simulations.List of symbols Z Ohnesorge number - _L water dynamic viscosity - air–water surface tension - _L water density - _g air density - d_j waterjet diameter at the nozzle outlet - d₀ nozzle capillary diameter - U_L flow velocity - We_L Weber number based on water density - We_g Weber number based on air density - Re_L Reynolds number     

Numerical investigation of transient nozzle flow

Abstract. The starting process of two-dimensional and axisymmetric nozzle flows has been investigated numerically. Special attention has been paid to the early phase of the starting process and to the appearance of a strong secondary shock wave. For both cases, shock intensities and velocities are obtained and discussed. The flow evolution in the axisymmetric case is proved to be more complex and the transient starting process is slower than in the plane case. Finally, the effects of changing the nozzle angle and the incident shock wave Mach number on the transient flow are addressed. It is shown that a faster start-up can be induced either by decreasing the nozzle angle or increasing the Mach number of the incident shock wave. Received 16 November 2001 / Accepted 24 September 2002 / Published online 4 December 2002 Correspondence to:A.-S. Mouronval (e-mail: mouronv@coria.fr)     

Hypersonic flow over blunt edges at low reynolds numbers

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 161–168, July–August, 1989.     

Parametric investigation of particle acceleration in high enthalpy conical nozzle flows for coating applications

A modified cold gas-dynamic spray technique is under development by using shock tunnel technology, which can enhance the coating quality by increasing the solid particle velocity up to 1,500 m/s. The particle diameter typically amounts to 10 μm. A theoretical model based on gas-particle flows is employed to describe the behaviour of the flow and the solid particles. This quasi-1D model is capable to consider non-equilibrium effects of the gas phase due to high reservoir temperatures, and the influence of wall friction and heat transfer averaged over the nozzle cross section. This model is used for the design and optimization of the nozzle geometry by a parametric study, which results in a conical nozzle with a half opening angle of 2.8° and a length of 325 mm. Particles for coating are injected at about 55 mm downstream of the throat. A shock tunnel facility has been set up at the Shock Wave Laboratory for performing an experimental study of this new technique. The theoretical performance of this setup is evaluated by the KASIMIR simulation software and the quasi-1D method described in this paper. The high reservoir conditions required to achieve particle velocities of 1,500 m/s can be realized by using either a very high driver pressure of about 600 bar for air as driver gas or a relatively low driver pressure of about 200 bar for helium as driver gas.      

Steady rotational motions of fluid masses at low reynolds numbers

The problem of steady rotational motions of an incompressible viscous fluid at small Reynolds numbers (R<1) between surfaces of rotationspecified in parametric form (1.3) is considered. Solutions of the Navier-Stokes equation are sought using the method of expansion in terms of the small parameter R. It is shown that in the case of twice continuously differentiable surfaces the solution of the problem in the first approximation enables the friction force torque on a fixed solid surface (1.3) to be determined with accuracy to R². Some solutions of Eq. (1.2) and the corresponding families of surfaces of revolution (1.3) are studied.     

Statistics of velocity gradients in turbulence at moderate reynolds numbers

A direct numerical simulation of the Navier-Stokes equation is performed in order to investigate the small scale structure of turbulence at moderately large microscale Reynolds numbers 40–140, using the spectral method with 340³ modes starting from a high-symmetric flow. It is shown that the small scale motion is statistically isotropic. The probability density distribution of the velocity is Gaussian, while those of the velocity gradients and the vorticity are not Gaussian but have long exponential tails. The moments of the velocity gradients are expressed in terms of the gamma function, and the ratio of the moments of the velocity gradients of successive orders increases linearly with the order. A comparison is made with a laboratory experiment.