Steady Mean Flow Research Articles

Flow of a dusty fluid due to Wavy motion of a wall for moderately large Reynolds numbers

The two‐dimensional flow of a dusty fluid induced by sinusoidal wavy motion of an infinite wavy wall is considered for Reynolds numbers which are of magnitude greater than unity. While the velocity components of the fluid and the dust particles along the axial direction consist of a mean steady flow and a periodic flow, the transverse components of both the fluid and the dust consist only of a periodic flow. This is true both for the outer flow (the flow beyond the boundary layer) and the inner flow (boundary layer flow). It is found that the mean steady flow is proportional to the ratio 4π2a2/L2(a/L < <1) , where a and L are the amplitude and the wavelength of the wavy wall, respectively. Graphs of the velocity components, both for the outer flow and the inner flow for various values of mass ‐ concentration of the dust particles are drawn. It is found that the steady flow velocities of the fluid and the dust particles approach to a constant value. Certain interesting results regarding the axial and the transverse velocity components are also discussed.

Prediction of loaded airfoil unsteady aerodynamic gust response by a locally analytical method

A complete first order model is formulated to analyze the effects of steady loading on the incompressible unsteady aerodynamics generated by a two-dimensional gust convected with the steady mean flow past an arbitrary airfoil at finite nonzero angle of attack. A locally analytical solution is developed in which the discrete algebraic equations which represent the flow field equations are obtained from analytical solutions in individual grid elements. This model and locally analytical solution are then applied to a series of airfoil and flow configurations.

Prediction of loaded airfoil unsteady aerodynamic gust response by a locally analytical method

A complete first-order model is formulated to analyze the effects of steady loading on the incompressible unsteady aerodynamics generated by a two-dimensional gust convected with the steady mean flow past an arbitrary airfoil at finite nonzero angle of attack. A locally analytical solution is then developed in which the discrete algebraic equations which represent the flow field equations are obtained from analytical solutions in individual grid elements. The unsteady flow field is rotational and is linearized about the full potential steady flow past the airfoil. Thus, the effects of airfoil geometry and angle of attack are completely accounted for through the mean potential flow field. The steady flow is independent of the unsteady flow. However, the strong dependence of the unsteady flow on the steady effects of airfoil geometry and finite angle of attack are manifested in the unsteady boundary conditions which are coupled to the steady flow. A body-fitted computational grid is utilized. Analytical solutions to the transformed flow equations in individual grid elements are then developed, with the complete solution obtained by assembling these locally analytical solutions. This model and locally analytical solution are then applied to a series of airfoil and flow configurations. The results demonstrate that accurate predictions for the unsteady aerodynamic gust response are obtained only by including the coupled steady flow effects on the unsteady aerodynamics. Thus for cambered, or cambered and thick airfoils at zero or finite angle of attack, or a thin flat plate airfoil at a nonzero angle of attack, the model and solution developed herein accurately predict the gust response. It was also demonstrated that the classical small perturbation combined transverse and chordwise gust models yield accurate predictions only for the special case of a thin flat plate airfoil at zero angle of attack, i.e. only when the chordwise gust is zero.

Computational aeroacoustics as applied to the diffraction of sound by cylindrical bodies

The problem of the diffraction of sound by two-dimensional rigid cylindrical objects has been formulated in generalized curvilinear coordinates. The governing equations for the aeroacoustic field are derived by perturbing the unsteady Euler equations and then extracting the steady mean flow equations from them. The governing equations are then solved by an explicit, four-stage Runge-Kutta, time-marching, finite-volume numerical scheme. The scheme has been implemented with far-field radiation and time-accurate wall boundary conditions. The cases of diffraction by a knife edge and a cylinder have been computed and compared against their corresponding analytical solutions in order to demonstrate the prediction capability of the numerical scheme. The acoustic field resulting from the interaction of the incident, reflected, and diffracted fields has been predicted very well in both the shadow and the illuminated zones. The scheme has also been applied to compute the diffraction of plane waves as well as that of a monopole acoustic field by a GA(W)-1 airfoil. I. Introduction W HEN sound waves impinge upon large rigid obstacles in their path, the resultant aeroacoustic field comprises of the incident waves, the reflected waves, and the diffracted waves. The diffraction of sound waves in the shadow zone plays a key role in assessing the acoustic shielding capability of aerodynamic bodies and forms a problem of fundamental importance in the field of aeroacoustics . The various fluctuating quantities that describe the aeroacoustic field are governed by the Euler equations which are, of course, nonlinear. Even if the governing equations are linearized, it is extremely difficult to obtain analytical solutions in the cases of the realistic twodimensional obstacles such as airfoils without some unacceptable drastic simplifications. The difficulty can be essentially attributed to the complexities associated with the treatment of the boundary conditions. These difficulties pertaining to the analytical solutions have been appreciated by many investigators14 and have led to the emergence of a new field of computational aeroacoustics (CAA). Progress in the methods of CAA has now advanced because of the considerable advances in the fields of computational fluid dynamics (CFD) and computer technology. The aim of both CFD and CAA is to find solutions to the practical problems numerically, but the main differences between the two lie in the nature of the problems one deals with and in the type of the information one wishes to obtain. CFD treats steady as well as unsteady problems, whereas all the CAA problems are, by definition, unsteady. In current CFD problems, interests are focused primarily on the flowfield in the immediate vicinity of an aerodynamic body, and the objectives are to determine aerodynamic quantities such as pressure distributions, skin friction, lift and drag, etc. On the other hand, in CAA problems, attention is directed to both the near-field acoustic vibrations and the far-field noise radiation away from the body, and the objectives are to find the directivity and the spectral characteristics of the acoustic field. Therefore, while the well-established techniques of CFD can be utilized in order to develop the

Generation of steady flow by an oscillatory acceleration and an oscillatory wall temperature

It is possible to generate a mean steady flow through the interaction of an oscillary wall temperature and an oscillatory acceleration.

Finite amplitude, steady Rossby waves and mean flows: analytical illustrations of the Charney-Drazin non-acceleration theorem

Finite amplitude, steady Rossby waves and mean flows: analytical illustrations of the Charney-Drazin non-acceleration theorem

Finite amplitude, steady Rossby waves and mean flows: Analytical illustrations of the Charney‐Drazin non‐acceleration theorem

AbstractExact steady‐state solutions of the nonlinear quasi‐geostrophic equations are used to illustrate some important repercussions of the Charney—Drazin theorem on the non‐acceleration of zonal mean flows by steady conservative waves. The solutions, which are similar to those discussed by Kuo in 1959, consist of spatially non‐uniform zonal flows and barotropic and baroclinic Rossby waves each having the same three‐dimensional wavenumber; a uniform zonal flow is also allowed. Two examples, both consisting of an internal jet zonal flow, a barotropic Rossby wave and a baroclinic Rossby wave, are considered. The absence of wave/mean interaction in these steady‐state solutions is clearly reflected in the absence of poleward eddy potential vorticity fluxes, but is disguised by the existence of non‐zero poleward eddy heat and zonal momentum fluxes and active Lorenz energy cycles. The solutions thus exhibit the same behaviour as the linearized examples discussed previously by various workers, and emphasize the danger of inferring from incomplete vorticity, energy or heat budget analyses that wave forcing (or maintenance) of Eulerian mean flows by eddy motion is taking place.One of the solutions (which is valid in a laterally open domain) is examined as a possible theoretical model of the steady waves seen in rotating annulus experiments. The spatial variations of the zonal mean flow and the eddy heat and momentum fluxes are in reasonable agreement with the results of a numerical simulation of a steady annulus wave, but the analytical solution is unsatisfactory in some other respects. It is clear that dissipative effects must be represented in any complete theoretical model of these steady waves.

Pulsatory Flow in Curved Pipes of Rectangular Cross-section : 1st Report, Nurmerical Analysis of Laminar Flow in Square-Sectioned Pipes

Numerical analysis was made of a fully-developed laminar flow in curved pipes of square cross-section under conditions wherein an oscillatory component of flow was superimposed on a steady mean flow. Velocity profiles, stream lines of secondary flow and distribution of wall shearing stresses were calculated in a wide range of various parameters, i.e. mean Dean number D^-, Womersley number α and flow rate ratio η. It was found that one to four pairs of vortices were induced in the cross section in a cycle at moderate values of α. The kinetic energy of the secondary flow and the resistance factor were described.

Oscillatory Flow in Curved Pipes of Rectangular Cross-section : 1st Report, Numerical Analysis of Laminar Flow in Square-sectioned Pipes

Numerical analysis was made of a fully-developed laminar flow in curved pipes of square cross-section under conditions wherein an oscillatory component of flow was superimposed on a steady mean flow. Velocity profiles, stream lines of secondary flow and distribution of wall shearing stresses were calculated in a wide range of various parameters, i.e. mean Dean number D^-, Womersley number α and flow rate ratio η. It was found that one to four pairs of vortices were induced in the cross section in a cycle at moderate values of α. The kinetic energy of the secondary flow and the resistance factor were described.

I. Simple pulsatile flow in an artery with a constriction

A smooth isolated, axisymmetric occlusion in a straight vascular tube is a tractable problem for pulsatile flow calculations via finite-difference approximations to the Navier-Stokes equation. Steady flow depends on the Reynolds number and two geometric parameters which describe the stenosis. The mere addition of a simple harmonic to the mean flow adds two more parameters. One is the reduced frequency, or Stokes number, and the other ε, the ratio of unsteady to steady flux. After describing steady stenosis flow examples, the dynamic patterns of pulsatile flow are illustrated indicating the inadequacy of basing hypotheses of atherosclerosis on mean (steady) flow.

Heat and mass transfer in rotating fluid with Hall current, II

Effects of Hall current on the hydromagnetic free-convection resulting from the combined effects of thermal and mass diffusion of an electrically conducting liquid past an infinite vertical porous plate in a rotating frame of reference have been analysed. As the mean steady flow has been presented in Part I, only the solutions for the transient primary and secondary velocity profiles, transient temperature profiles in the boundary layer and the transient skin friction, the transient rate of heat transfer on the plate are presented in this work. The influence of the different parameters entering into the problem are discussed.

Oscillatory free-convective flow of an elastico-viscous fluid past an impulsively started infinite vertical porous plate, II

In this work we present the two-dimensional free-convective flow of an elastico-viscous fluid past an infinite vertical porous plate for the Stokes problem when the flow is subjected to a constant suction velocity through the porous plate. As the mean steady flow has been presented in Part I, obly the solution for the transient velocity profiles, transient temperature profiles, the amplitude and the phase of the skin-friction and the rate of heat transfer are presented in this work. As in the case of mean steady flow, the influence of the various parameters on the unsteady flow field is discussed for both the cases cooling and heating of the porous plate by free-convection currents.

NUMERICAL MODEL FOR DYNAMIC AND THERMAL DEVELOPMENTS OF A PULSED LAMINAR DUCTED FLOW

Abstract A numerical model is constructed to analyze the simultaneous development of the dynamic and thermal regimes of an incompressible viscous ducted flow under the influence of sinusoidal pulsations. The Navier-Stokes and energy equations are solved by a finite-difference method and by using asymptotic developments for the different dynamic and thermal functions. The model is validated by obtaining the Poiseuille regime for steady mean flow and by verifying Uchida's solutions for the unsteady dynamic components in fully developed flow. The model allows the description of Richardson's effect and depicts the existence of an annular effect for the pulsed temperature term.

Pulsatile entrance flow in a curved pipe

An experimental investitition was made of the entry flow in a curved pipe under conditions wherein a pulsatile component of flow was superimposed on a steady mean flow. Two experiments were conducted, one in a pipe of curvature ratio for αw = 12·5 and κ = 372. Laser-velocimetry measurements of the axial and secondary velocities were made throughout the cross-section at different instants of time within the cycle and at several axial locations downstream from the pipe entrance.The flow in the first experiment was found to be quasisteady, with axial and secondary velocities varying in time at the several axial stations proportionally to the instantaneous mean velocity, but essentially the same in character as low-Dean-number steady entry flow. The flow in the second experiment was more complex, with separation of the axial flow appearing at the inner bend during deceleration, starting downstream and propagating upstream toward the pipe entrance. Helical motions imbedded within the Dean circulation were also observed, and during certain portions of the cycle the secondary motions within the central core took on a jet-like structure.It suggested that the classifications employed by Smith to describe the different regimes of fully-developed pulsatile flow in curved pipes may also be useful to distinguish between different entry-flow regimes.

Numerical Model for Dynamic and Thermal Developments of a Pulsed Laminar Ducted Flow

Abstract A numerical model is constructed to analyze the simultaneous development of the dynamic and thermal regimes of an incompressible viscous ducted flow under the influence of sinusoidal pulsations. The Navier-Stokes and energy equations are solved by a finite-difference method and by using asymptotic developments for the different dynamic and thermal functions. The model is validated by obtaining the Poiseuille regime for steady mean flow and by verifying Uchida's solutions for the unsteady dynamic components in fully developed flow. The model allows the description of Richardson's effect and depicts the existence of an annular effect for the pulsed temperature term.

Meridional circulation in rotating stars. II - Mean steady motions in rotationally and tidally distorted stars. III - The reflection effect in close binaries

This paper deals with the problem of circulatory currents in the radiative envelope of a nonmagnetic star, which is a synchronously rotating, detached component of a double star. Uniformly valid solutions are found when departure from spherical symmetry is not too large. To the lowest possible order in the ratio of the centrifugal force to gravity at the equator, the velocity field has three nonvanishing components. Actually, the mean steady flow is the superposition of (i) rotationally driven currents (which are symmetrical with respect to both the rotation axis and the equator), (ii) tidally driven currents (which are symmetrical with respect to the tidal axis only), and (iii) motions having no particular symmetry that result from the continuous interaction (via the Coriolis force) between these two meridional circulations and overall rotation.

Free convection and mass transfer effects on the oscillatory flow of a dissipative fluid past an infinite vertical porous plate (I)

The effects of free convection and mass transfer on the oscillatory flow of an incompressible, dissipative, viscous fluid past an infinite vertical, porous plate with constant suction, is studied. The solution of the problem is obtained with assumption that there exists a “mean steady flow” and on it superimposed the unsteady oscillatory flow. The mean steady flow is studied in this paper and the effects of Grashof number (Gr), modified Grashof number(Gc), Eckert number(E) and Schmidt number(Sc) are discussed for the case of air (P = 0·71). The study of the expression of the unsteady parts of the velocity, temperature and related quantities will be given in a subsequent paper.

Free convection effects on the hydromagnetic oscillatory flow in the stokes problem past an infinite porous vertical limiting surface with constant suction, II

Unsteady hydromagnetic boundary layer flow of a viscous incompressible and electrically conducting fluid past an infinite vertical non-conducting porous limiting surface in presence of a transverse magnetic field, is considered when the limiting surface is moving impulsively in its own plane and is subjected to a constant suction. The free stream oscillates in time about a constant mean value and the magnetic Reynolds number is taken to be small enough so that the induced magnetic field is negligible. As the mean steady flow has been presented in Part I, only the solutions for the transient velocity profiles, transient temperature profiles, the amplitude and the phase of the skin friction and the rate of the heat transfer are presented in this work. The influence of the various parameters entering into the problem, especially of the magnetic parameterM, is extensively discussed. A comparative study with hydrodynamic case (M=0) is also made.

The influence of mean shear on unsteady aperture flow, with application to acoustical diffraction and self-sustained cavity oscillations

This paper discusses the linearized theory of unsteady flow through a two-dimensional aperture in a thin plate in the presence of a grazing mean flow on one side of the plate. The mean shear layer is modelled by a vortex sheet, and it is predicted that at low mean-flow Mach numbers there is a transfer of energy from the mean flow to the disturbed motion of the vortex sheet provided (i) the Kutta condition is imposed at the leading edge of the aperture, resulting in the unsteady shedding of vorticity from the edge, and (ii) the width of the aperture 2s satisfies ½ < 2s/λ < 1.1, where λ is the hydrodynamic wavelength of the disturbance on the vortex sheet within the aperture. The theory is used to examine the effect of mean shear on the diffraction of sound by a perforated screen, and to predict the spontaneous excitation and suppression of self-sustained oscillations in a wall-cavity beneath a nominally steady mean flow. In the latter case support for the proposed theory is provided by a favourable comparison of theoretical results with experimental data available in the literature.

Free convection and mass transfer effects on the hydro-magnetic oscillatory flow past an infinite vertical porous plate

Two-dimensional unsteady free convection and mass transfer, flow of an incompressible viscous dissipative and electrically conducting fluid, past an infinite, vertical porous plate, is considered, when the flow, is subjected in the action of uniform transverse magnetic field. The magnetic Reynolds number is taken to be small enough so that the induced magnetic field is negligible. The solution of the problem is obtained in the form of power series of Eckert numberE, which is very small for incompressible fluids. Analytical expressions for the velocity field and temperature field are given, as well as for the skin friction and the rate of heat transfer for the case of the mean steady flow and for the unsteady one. The influence of the magnetic parameter,M, modified Grashof numberG c , Schmidt numberS c and frequency ω, on the flow field, is discussed with the help of graphs, when the plate is being cooled, by the free convection currents (G r ,E>0), or heated (G r ,E<0). A comparative study with hydrodynamic case (M=0) and the hydromagnetic one (M≠0) is also made whenever necessary.