Hyperbolic Approximation of the Navier-Stokes Equations for Viscous Mixed Flows

Simplified two-dimensional Navier-Stokes equations of the hyperbolic type are derived for viscous mixed (with transition through the sonic velocity) internal and external flows as a result of a special splitting of the pressure gradient in the predominant flow direction into hyperbolic and elliptic components. The application of these equations is illustrated with reference to the calculation of Laval nozzle flows and the problem of supersonic flow past blunt bodies. The hyperbolic approximation obtained adequately describes the interaction between the stream and surfaces for internal and external flows and can be used over a wide Mach number range at moderate and high Reynolds numbers. Examples of the calculation of viscous mixed flows in a Laval nozzle with large longitudinal throat curvature and in a shock layer in the neighborhood of a sphere and a large-aspect-ratio hemisphere-cylinder are given. The problem of determining the drag coefficient of cold and hot spheres is solved in a new formulation for supersonic air flow over a wide range of Reynolds numbers. In the case of low and moderate Reynolds numbers a drag reduction effect is detected when the surface of the sphere is cooled.     

Drag and Separation Flow Past Solid Sphere with Porous Shell at Moderate Reynolds Number

Axisymmetric viscous, two-dimensional steady and incompressible fluid flow past a solid sphere with porous shell at moderate Reynolds numbers is investigated numerically. There are two dimensionless parameters that govern the flow in this study: the Reynolds number based on the free stream fluid velocity and the diameter of the solid core, and the ratio of the porous shell thickness to the square root of its permeability. The flow in the free fluid region outside the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by a Darcy model. Using a commercially available computational fluid dynamics (CFD) package, drag coefficient and separation angle have been computed for flow past a solid sphere with a porous shell for Reynolds numbers of 50, 100, and 200, and for porous parameter in the range of 0.025–2.5. In all simulation cases, the ratio of b/a was fixed at 1.5; i.e., the ratio of outer shell radius to the inner core radius. A parametric equation relating the drag coefficient and separation point with the Reynolds number and porosity parameter were obtained by multiple linear regression. In the limit of very high permeability, the computed drag coefficient as well as the separation angle approaches that for a solid sphere of radius a, as expected. In the limit of very low permeability, the computed total drag coefficient approaches that for a solid sphere of radius b, as expected. The simulation results are presented in terms of viscous drag coefficient, separation angles and total drag coefficient. It was found that the total drag coefficient around the solid sphere as well as the separation angle are strongly governed by the porous shell permeability as well as the Reynolds number. The separation point shifts toward the rear stagnation point as the shell permeability is increased. Separation angle and drag coefficient for the special case of a solid sphere of radius r = a was found to be in good agreement with previous experimental results and with the standard drag curve.     

Isothermal flow past a blowing sphere

Steady, axisymmetric, isothermal, incompressible flow past a sphere with uniform blowing out of the surface is investigated for Reynolds numbers in the range 1 to 100 and surface velocities up to 10 times the free stream value. A stream-function-velocity formulation of the flow equations in spherical polar co-ordinates is used and the equations are solved by a Galerkin finite-element method. Reductions in the drag coefficients arising from blowing are computed and the effects on the viscous and pressure contributions to the drag considered. Changes in the surface pressure, surface vorticity and flow patterns for two values of the Reynolds number (1 and 40) are examined in greater detail. Particular attention is paid to the perturbation to the flow field far from the sphere.     

Flow around a sphere with material cross flow at low reynolds numbers

An analytical solution is carried out for the problem of the flow around a sphere with material cross flow at Reynolds numbers less than 1 and a blowing velocity less than the free stream velocity. The method of asymptotic expansions of Pearson and Proudman is used for the solution. Expressions are obtained for the distribution of the current and velocity component functions as well as for the aerodynamic drag coefficient of the sphere. It is shown that blowing diminishes the sphere drag, where its influence will increase as the Reynolds number grows.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 103–109, May–June, 1972.     

Lift force on rotating sphere at low Reynolds numbers and high rotational speeds

The lift force on an isolated rotating sphere in a uniform flow was investigated by means of a three-dimensional numerical simulation for low Reynolds numbers (based on the sphere diameter) (Re<68.4) and high dimensionless rotational speeds (Г5). The Navier-Stokes equations in Cartesian coordinate system were solved using a finite volume formulation based on SIMPLE procedure. The accuracy of the numerical simulation was tested through a comparison with available theoretical, numerical and experimental results at low Reynolds numbers, and it was found that they were in close agreement under the above mentioned ranges of the Reynolds number and rotational speed. From a detailed computation of the flow field around a rotational sphere in extended ranges of the Reynolds number and rotational speed, the results show that, with increasing the rotational speed or decreasing the Reynolds number, the lift coefficient increases. An empirical equation more accurate than those obtained by previous studies was obtained to describe both effects of the rotational speed and Reynolds number on the lift force on a sphere. It was found in calcttlations that the drag coefficient is not significantly affected by the rotation of the sphere. The ratio of the lift force to the drag force, both of which act on a sphere in a uniform flow at the same time, was investigated. For a small spherical particle such as one of about 100μm in diameter, even if the rotational speed reaches about 10^6 revolutions per minute, the lift force can be neglected as compared with the drag force.     

Drag of a rotating sphere

We consider the problem of steady incompressible viscous fluid flow about a rotating sphere, with the flow specified on a sphere of finite radius, which reduces to the solution of the complete Navier-Stokes equations.The dimensionless stream functions and circulai velocity are sought in the form of series in powers of the Reynolds numbers, which converge for small values of this number. Recurrence formulas are derived for determining the coefficients of these series. The pressure, rotational resistance torque, and drag are determined. It is established that the rotating sphere has higher drag than a stationary sphere. The leading term of the series in powers of the Reynolds number for the drag and resistive torque is calculated.     

Fluid-dynamic interaction between two spheres

Experiment of fluid-dynamic interaction between two spheres was conducted to obtain basic information concerning the two-phase flow, especially in dense phase. Two or three spheres were set up in a water tunnel in the longitudinal or transverse direction with Reynolds numbers less than 10³. The flow behind the sphere was visualized by the use of condense milk and change in vortex structure due to the interaction was observed in detail. Additionally, drag force on the sphere was measured by a pendulum method which was developed to detect small drag, and the range of distance in which the drag is affected by the interaction was shown.     

Hydrodynamic forces acting on a rigid fixed sphere in early transitional regimes

A spectral – spectral-element code is used to investigate the hydrodynamic forces acting on a fixed sphere placed in a uniform flow in the Reynolds number interval [10–320] covering the early stages of transition, i.e. the steady axisymmetric regime with detached flow, the steady non-axisymmetric and the unsteady periodic regimes of the sphere wake. The mentioned changes of regimes, shown by several authors to be related to a regular and a Hopf bifurcations in the wake, result in significant changes of hydrodynamic action of the flow on the sphere. In the present paper, we show that the loss of axisymmetry is accompanied not only by an onset of lift but also of a torque and we give accurate values of drag, lift and torque in the whole interval of investigated Reynolds numbers. Among other results show, moreover, that each bifurcation is accompanied also by a change of the trend of the drag versus Reynolds number dependence, the overall qualitative effect of instabilities being an increase of drag.     

Two-dimensional unsteady laminar flow of a power law fluid across a square cylinder

The two-dimensional and unsteady free stream flow of power law fluids past a long square cylinder has been investigated numerically in the range of conditions 60≤Re≤160 and 0.5≤n≤2.0. Over this range of Reynolds numbers, the flow is periodic in time. A semi-explicit finite volume method has been used on a non-uniform collocated grid arrangement to solve the governing equations. The global quantities such as drag coefficients, Strouhal number and the detailed kinematic variables like stream function, vorticity and so on, have been obtained for the above range of conditions. While, over this range of Reynolds number, the flow is known to be periodic in time for Newtonian fluids, a pseudo-periodic flow regime displaying more than one dominant frequency in the lift is observed for shear-thinning fluids. This seems to occur at Reynolds numbers of 120 and 140 for n=0.5 and 0.6, respectively. Broadly speaking, the smaller the value of the power law index, lower is the Reynolds number of the onset of the pseudo-periodic regime. This work is concerned only with the fully periodic regime and, therefore, the range of Reynolds numbers studied varies with the value of the power law index. Not withstanding this aspect, in particular here, the effects of Reynolds number and of the power law index have been elucidated in the unsteady laminar flow regime. The leading edge separation in shear-thinning fluids produces an increase in drag values with the increasing Reynolds number, while shear-thickening fluid behaviour delays this separation and shows the lowering of the drag coefficient with the Reynolds number. Also, the preliminary results suggest the transition from the steady to unsteady flow conditions to occur at lower Reynolds numbers in shear-thinning fluids than that in Newtonian fluids.     

Dynamic interaction of fixed dual spheres for several configurations and inflow conditions

The changes in force characteristics as well as the shedding patterns for various dual sphere configurations are studied. The Reynolds numbers considered are 300, 600 and two different inflow conditions are used: steady and pulsating. The sphere formations are defined by the separation distance D₀ between the spheres and the angle between the line connecting the centres of the spheres and the main flow direction, γ. The position of one of the spheres is varied in the range 0°–90° using a 15° increment. Two separation distances are studied; 1.5D and 3D. The method used for the simulations is the Volume of Solid (VOS) approach, a method based on Volume of Fluid (VOF). A major conclusion from this work is that the sphere interaction alters the wake dynamics by obstructing the vortex shedding (generating a steady wake or a wake with lower Strouhal number) and by changing the direction of the lift force so that it in most cases is directed in the plane containing the sphere centres. The results also show that changing the inflow condition gives the same relative change in drag and lift as for a single sphere. The drag is substantially reduced by placing the sphere downstream in a tandem arrangement and slightly increased in a side-by-side arrangement. However, the effect is decreased by increasing separation distance and increasing Reynolds number.     

The oseen flow of finite clusters of spheres and the wake effect on the drag factor

In this paper, the wake effect on drag factor in the axisymmetric Oseen flow of the finite clusters of equally spaced spheres with same size is studied. Putting the Oseen lets on the centres of all the spheres, the series solution of the problem is obtained. By truncating the infinite series and applying the collocation method to solve a set of the linear algebraic equations, the approximate solution of the Oseen flow of finite clusters of spheres and the drag factor for each sphere are presented. The effect of the sphere number and spacing on the drag factor of each sphere under different Reynolds numbers are calculated and the wake effect as well as the shielding effect and the end effect are revealed. The influence of various parameters on the effects is considered and compared with the corresponding results of the Stokes flow. The convergence of the method is also studied numerically in this paper.     

Drag forces of interacting spheres in power-law fluids

Drag forces of interacting particles suspended in power-law fluid flows were investigated in this study. The drag forces of interacting spheres were directly measured by using a micro-force measuring system. The tested particles include a pair of interacting spheres in tandem and individual spheres in a cubic matrix of multi-sphere in flows with the particle Reynolds number from 0.7 to 23. Aqueous carboxymethycellulose (CMC) solutions and glycerin solutions were used as the fluid media in which the interacting spheres were suspended. The range of power-law index varied from 0.6 to 1.0. In conjunction to the drag force measurements, the flow patterns and velocity fields of power-law flows over a pair of interacting spheres were also obtained from the laser assisted flow visualization and numerical simulation.

Both experimental and computational results suggest that, while the drag force of an isolated sphere depends on the power-index, the drag coefficient ratio of an interacting sphere is independent from the power-law index but strongly depends on the separation distance and the particle Reynolds number. Our study also shows that the drag force of a particle in an assemblage is strongly positions dependent, with a maximum difference up to 38%.     

Numerical solution of viscous flows using integral equation methods

A formulation of the boundary element method for the solution of non-zero Reynolds number incompressible flows in which the non-linear terms are lumped together to form a forcing function is presented. Solutions can be obtained at low to moderate Reynolds numbers. The method was tested using the flow of a fluid in a two-dimensional converging channel (Hamel flow) for which an exact solution is available. An axisymmetric formulation is demonstrated by examining the drag experienced by a sphere held stationary in uniform flow. Performance of the method was satisfactory. New results for an axisymmetric free jet at zero Reynolds number obtained using the boundary element method are also included. The method is ideal for this type of free-surface problem.     

Steady flow of a power-law non-Newtonian fluid across an unconfined square cylinder

A two-dimensional flow of a non-Newtonian power-law fluid directed normally to a horizontal cylinder with a square cross section is considered in the present paper. The problem is investigated numerically with a finite volume method by using the commercial code Ansys Fluent with a very large computational domain so that the flow could be considered unbounded. The investigation covers the power-law index from 0.1 to 2.0 and the Reynolds number range from 0.001 to 45.000. It is found that the drag coefficient for low Reynolds numbers and low power-law index (n ≤ 0.5) obeys the relationship C_D = A/Re. An equation for the quantity A as a function of the power-law index is derived. The drag coefficient becomes almost independent of the power-law index at high Reynolds numbers and the wake length changes nonlinearly with the Reynolds number and power-law index.     

Numerical study of the turbulent channel flow under space-dependent electromagnetic force control at different Reynolds numbers

In this paper, the control of turbulent channel flow by space-dependent electromagnetic force and the mechanism of drag reduction are investigated with the direct numerical simulation(DNS) methods for different Reynolds numbers. A formulation is derived to express the relation between the drag and the Reynolds shear stress. With the application of optimal electromagnetic force, the in-depth relations among characteristic structures in the flow field, mean Reynolds shear stress, and the effect of drag reduction for different Reynolds numbers are discussed. The results indicate that the maximum drag reductions can be obtained with an optimal combination of parameters for each case of different Reynolds numbers. The regular quasi-streamwise vortex structures, which appear in the flow field, have the same period with that of the electromagnetic force.These structures suppress the random velocity fluctuations, which leads to the absolute value of mean Reynolds shear stress decreasing and the distribution of that moving away from the wall. Moreover, the wave number of optimal electromagnetic force increases,and the scale of the regular quasi-streamwise vortex structures decreases as the Reynolds number increases. Therefore, the rate of drag reduction decreases with the increase in the Reynolds number since the scale of the regular quasi-streamwise vortex structures decreases.     

Interaction of a jet exhausting from a body with an opposing supersonic stream of rarefied gas

The experimental investigation of supersonic flow past a sphere with a jet exhausting from the front point of the sphere into the flow at large [1] and moderate [2] Reynolds numbers Re has revealed an effect of shielding from the oncoming stream, this leading to a decrease in the drag coefficient of the sphere and of the energy flux to it. A numerical simulation of the flow has been made in the case of supersonic flow past a sphere with a sonic jet from a nozzle situated on the symmetry axis in the continuum regime [3]. In the present paper, this problem is investigated for flow of a rarefied gas on the basis of numerical solution of a model kinetic equation for a monatomic gas.     

Motion of tracer particles in supersonic flows

 Factors that may act on particle motion in high-speed flow are investigated. The classical expressions of drag coefficient C _D for a sphere are reviewed. Then, a drag expression is proposed, extending Cunningham’s method to higher velocities and Knudsen numbers. This law, valid from continuum to free molecule conditions, for Re≲200 and M≲1 (where Re and M are, respectively, the Reynolds and Mach numbers based on relative velocity), is used to compare calculated and experimental values of the drag coefficient, as well as the particle velocities across an oblique shock wave. Calculated results are found to be in agreement with experiments. Received: 3 June 1997/Accepted: 16 August 1998     

Study on aerodynamic force acting on a sphere with and without boundary layer trips around the critical Reynolds number with a magnetic suspension and balance system

Aerodynamic force acting on a sphere for five kinds of boundary layer trips around the critical Reynolds number, together with the force on a smooth sphere, was successfully measured. This was achieved using JAXA’s 60-cm Magnetic Suspension and Balance System after performing detailed simulations and adjusting the sphere mass and its control parameters. The minimum drag coefficient of a smooth sphere was evaluated around 0.19 in the support-interference-free condition. No hysteresis was observed for the drag coefficient in the critical range for tested sphere with boundary layer trips. Using three serially connected 2nd-order Butterworth low-pass filters, an inertia force oscillating at less than 15 Hz was evaluated from the measured model position, and the unsteady aerodynamic force acting on the sphere was also evaluated with reasonable accuracy. Two kinds of oscillatory aerodynamic forces appeared in the critical range depending on the sphere surface condition: a force rotating around an axis parallel to the uniform flow for both a smooth sphere and a sphere with axially symmetric 0.17-mm-high backward step, and an oscillating force in the plane including the axis parallel to the flow for a sphere with axially symmetric step implemented with 0.35–mm-thick tape with wrinkles acting as small vortex generators. There was also observed a force irregularly rotating through less than 180° in the range about a sphere axis parallel to the flow for a smooth sphere in the supercritical range.     

Unsteady force measurements in sphere flow from subcritical to supercritical Reynolds numbers

The flow over a smooth sphere is examined in the Reynolds number range of 5.0 × 10⁴ < Re < 5.0 × 10⁵ via measurements of the fluctuating forces and particle image velocimetry measurements in a planar cut of the velocity field. Comprehensive studies of the statistics and spectra of the forces are presented for a range of subcritical and supercritical Reynolds numbers. While the subcritical lateral force spectra are dominated by activity corresponding to the large-scale vortex shedding frequency at a Strouhal number of approximately 0.18, there is no such peak apparent in the supercritical spectra, although resolution effects may become important in this region. Nor does the large-scale vortex shedding appear to have a significant effect on the drag force fluctuations at either sub- or super-critical Reynolds numbers. A simple double spring model is shown to capture the main features of the lateral force spectra. The low-frequency force fluctuations observed in earlier computational studies are shown to have important implications for statistical convergence, and in particular, the apparent mean side force observed in earlier studies. At least one thousand dimensionless time units are required for reasonable estimates of the second and higher moments below the critical Reynolds number and even more for supercritical flow, stringent conditions for computational studies. Lastly, investigation of the relationship between the motion of the instantaneous wake shape, defined via the local position where the streamwise velocity is equal to half the freestream value, and the in-plane lateral force for subcritical flow reveals a significant negative correlation throughout the near wake, which is shown to be related to a structure inferred to arise from the large-scale vortex shedding convecting downstream at 61% of the freestream velocity. In addition to its utility in understanding basic sphere flow, the apparatus is also a testbed that will be used in future studies, examining the effect of both static and dynamic changes to the surface morphology.     

Stochastic equations for continuum and determination of hydraulic drag coefficients for smooth flat plate and smooth round tube with taking into account intensity and scale of turbulent flow

The stochastic equations of continuum are used for determining the hydraulic drag coefficients. As a result, the formulas for the hydraulic drag coefficients dependent on the turbulence intensity and scale instead of only on the Reynolds number are proposed for the classic flows of an incompressible fluid along a smooth flat plate and a round smooth tube. It is shown that the new expressions for the classical drag coefficients, which depend only on the Reynolds number, should be obtained from these new general formulas if to use the well-known experimental data for the initial turbulence. It is found that the limitations of classical empirical and semiempirical formulas for the hydraulic drag coefficients and their deviation from the experimental data depend on different parameters of initial fluctuations in the flow for different experiments in a wide range of Reynolds numbers. On the basis of these new dependencies, it is possible to explain that the differences between the experimental results for the fixed Reynolds number are caused by the difference in the values of flow fluctuations for each experiment instead of only due to the systematic error in the processing of experiments. Accordingly, the obtained general dependencies for the smooth flat plate and the smooth round tube can serve as the basis for clarifying the results of experiments and the experimental formulas, which used for continuum flows in different devices.