scholarly journals The steady oblique path of buoyancy-driven disks and spheres

2012 ◽  
Vol 707 ◽  
pp. 24-36 ◽  
Author(s):  
David Fabre ◽  
Joël Tchoufag ◽  
Jacques Magnaudet
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Density Ratio ◽  
Steady Motion ◽  
The Body ◽  
Body Motion ◽  
Angle Of Incidence ◽  
Weakly Nonlinear ◽  
Numerical Studies ◽  
Flow Past

AbstractWe consider the steady motion of disks of various thicknesses in a weakly viscous flow, in the case where the angle of incidence $\ensuremath{\alpha} $ (defined as that between the disk axis and its velocity) is small. We derive the structure of the steady flow past the body and the associated hydrodynamic force and torque through a weakly nonlinear expansion of the flow with respect to $\ensuremath{\alpha} $. When buoyancy drives the body motion, we obtain a solution corresponding to an oblique path with a non-zero incidence by requiring the torque to vanish and the hydrodynamic and net buoyancy forces to balance each other. This oblique solution is shown to arise through a bifurcation at a critical Reynolds number ${\mathit{Re}}^{\mathit{SO}} $ which does not depend upon the body-to-fluid density ratio and is distinct from the critical Reynolds number ${\mathit{Re}}^{\mathit{SS}} $ corresponding to the steady bifurcation of the flow past the body held fixed with $\ensuremath{\alpha} = 0$. We then apply the same approach to the related problem of a sphere that weakly rotates about an axis perpendicular to its path and show that an oblique path sets in at a critical Reynolds number ${\mathit{Re}}^{\mathit{SO}} $ slightly lower than ${\mathit{Re}}^{\mathit{SS}} $, in agreement with available numerical studies.

Hypersonic Swirling Flow past Blunt Bodies

1973 ◽  
Vol 24 (4) ◽  
pp. 241-251 ◽  
Author(s):  
Roger Smith
Keyword(s):  
High Speed ◽  
Swirling Flow ◽  
Pressure Coefficient ◽  
Density Ratio ◽  
Perturbation Expansion ◽  
The Body ◽  
Constant Density ◽  
Flow Past ◽  
Speed Flow ◽  
Blunt Bodies

SummaryThe effect of swirl on the high speed flow past blunt bodies is analysed by assuming constant density in the region between the shock wave and the body. For small swirl the stand-off distance is only slightly affected, but it is shown that there is a critical value of the swirl parameter which, if exceeded, will cause a jump in the position of the shock. This is demonstrated by solving the full constant-density equations for the flow past a sphere and by a perturbation expansion in powers of the density ratio across the shock for a more general body shape. The perturbation solution shows that the pressure coefficient on the body is constant at the critical swirl number.

A new method of solving Oseen’s equations and its application to the flow past an inclined elliptic cylinder

1954 ◽  
Vol 224 (1157) ◽  
pp. 141-160 ◽  
Keyword(s):  
Reynolds Number ◽  
Elliptic Cylinder ◽  
Cylindrical Body ◽  
Thickness Ratio ◽  
Successive Approximations ◽  
Angle Of Incidence ◽  
Mathieu Functions ◽  
Flow Past ◽  
Transcendental Functions ◽  
Lift And Drag

In this paper is developed a general method of solving Oseen’s linearized equations for a two-dimensional steady flow of a viscous fluid past an arbitrary cylindrical body. The method is based on the fact that the velocity in the neighbourhood of the cylinder can be generally expressed in terms of a pair of analytic functions, the determination of which from the appropriate boundary condition can be effected by successive approximations in powers of the Reynolds number, R . The method enables one to obtain the velocity distribution near the cylinder and the lift and drag acting on it in the form of power series in R , without recourse to manipulation of higher transcendental functions such as Bessel and Mathieu functions for circular and elliptic cylinders, respectively. As an example of the application of the method, the uniform flow past an elliptic cylinder at an arbitrary angle of incidence is considered. Analytical expressions for the lift and drag coefficients are obtained, which are correct to the order of R , the lowest order terms being O ( R -1 ) and numerical calculations are carried out for the thickness ratio t = 0, 0.1, 0.5, 1 and the Reynolds number R = 0.1, 1. It is found that drag increases slightly with increase of either thickness ratio or angle of incidence, and that lift decreases with increase of thickness ratio while, as a function of the angle of incidence, it has a maximum at about 45°.

A uniformly valid solution for the hypersonic flow past blunted bodies

1968 ◽  
Vol 31 (2) ◽  
pp. 397-415 ◽  
Author(s):  
W. Schneider
Keyword(s):  
Flow Field ◽  
Hypersonic Flow ◽  
Local Thermodynamic Equilibrium ◽  
Density Ratio ◽  
Intersection Point ◽  
The Body ◽  
Boundary Values ◽  
Stagnation Region ◽  
Flow Past ◽  
Valid Solution

The plane and axisymmetric hypersonic flow past blunted bodies is investigated as an inverse problem (shock shape given). The fluid may behave as a real gas in local thermodynamic equilibrium. Viscosity and heat conduction are neglected. An analytical solution uniformly valid in the whole flow field (from the stagnation region up to large distances from the body nose) is given. The solution is based on two main assumptions: (i) the density ratio ε across the shock is very small, (ii) the pressure at a pointPof the disturbed flow field isnotvery small compared with the pressure immediately behind the shock in the intersection point of the shock surface with its normal throughP.TermsO(ε) are neglected in comparison with 1, but it is not necessary for the shock layer to be thin. The change of velocity along streamlines is taken into account. In order to calculate the flow quantities one has to evaluate only two integrals (equations (49) and (53) together with the boundary values (5) and (10)). The application of the solution is illustrated and the accuracy is tested in some examples.

Dynamics of axisymmetric bodies rising along a zigzag path

2008 ◽  
Vol 606 ◽  
pp. 209-223 ◽  
Author(s):  
PEDRO C. FERNANDES ◽  
PATRICIA ERN ◽  
FRÉDÉRIC RISSO ◽  
JACQUES MAGNAUDET
Keyword(s):  
Aspect Ratio ◽  
Transverse Direction ◽  
Transverse Velocity ◽  
Density Ratio ◽  
Transverse Component ◽  
The Body ◽  
Force Balance ◽  
Body Motion ◽  
Major Influence ◽  
Inertia Force

The forces and torques governing the planar zigzag motion of thick, slightly buoyant disks rising freely in a liquid at rest are determined by applying the generalized Kirchhoff equations to experimental measurements of the body motion performed for a single body-to-fluid density ratio ρs/ρf ≈ 1. The evolution of the amplitude and phase of the various contributions is discussed as a function of the two control parameters, i.e. the body aspect ratio (the diameter-to-thickness ratio χ = d/h ranges from 2 to 10) and the Reynolds number (100 < Re < 330), Re being based on the rise velocity and diameter of the body. The body oscillatory behaviour is found to be governed by the force balance along the transverse direction and the torque balance. In the transverse direction, the wake-induced force is mainly balanced by two forces that depend on the body inclination, i.e. the inertia force generated by the body rotation and the transverse component of the buoyancy force. The torque balance is dominated by the wake-induced torque and the restoring added-mass torque generated by the transverse velocity component. The results show a major influence of the aspect ratio on the relative magnitude and phase of the various contributions to the hydrodynamic loads. The vortical transverse force scales as fo = (ρf − ρs)ghπd2 whereas the vortical torque involves two contributions, one scaling as fod and the other as f1d with f1 = χfo. Using this normalization, the amplitudes and phases of the vortical loads are made independent of the aspect ratio, the amplitudes evolving as (Re/Rec1 − 1)1/2, where Rec1 is the threshold of the first instability of the wake behind the corresponding body held fixed in a uniform stream.

scholarly journals Generalized two-dimensional Lagally theorem with free vortices and its application to fluid–body interaction problems

2012 ◽  
Vol 698 ◽  
pp. 73-92 ◽  
Author(s):  
C. T. Wu ◽  
F.-L. Yang ◽  
D. L. Young
Keyword(s):  
Hydrodynamic Force ◽  
Density Ratio ◽  
Vortex Pair ◽  
The Body ◽  
Body Motion ◽  
Liquid Density ◽  
Two Dimensional ◽  
Fixed Target ◽  
Two Dimensional Flow ◽  
Neutrally Buoyant

AbstractThe Lagally theorem describes the unsteady hydrodynamic force on a rigid body exhibiting arbitrary motion in an inviscid and incompressible fluid by the properties of the singularities employed to generate the flow and the body motion and to meet the boundary condition. So far, only sources and dipoles have been considered, and the present work extends the theorem to include free vortices in a two-dimensional flow. The present extension is validated by reproducing the system dynamics or the force evolution of three literature problems: (i) a free cylinder interacting with a free vortex; (ii) the moving Föppl problem; and (iii) a cylinder in constant normal approach to a fixed identical cylinder. This work further extends the bifurcation analysis on the moving Föppl problem by including the solid-to-liquid density ratio as a new parameter, in addition to the system total impulse and the vortex strength. We then apply the theorem to the problem where a moving Föppl system is made to approach a fixed or a free neutrally buoyant target cylinder of identical size from far away. The force developed on each cylinder is examined with respect to the vortex pair configuration and the target mobility. When approaching a fixed target, a greater force is developed if the vortex pair has stronger circulation and larger structure. The mobility of the target cylinder, however, can modify the hydrodynamic force by reducing its magnitude and reversing the force ordering with respect to the vortex pair configuration found for the case with fixed target. Possible mechanisms for such a change of force nature are given based on the currently derived equation of motion.

Transitions and instabilities of flow in a symmetric channel with a suddenly expanded and contracted part

2001 ◽  
Vol 434 ◽  
pp. 355-369 ◽  
Author(s):  
J. MIZUSHIMA ◽  
Y. SHIOTANI
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Nonlinear Stability ◽  
Critical Reynolds Number ◽  
Reynolds Numbers ◽  
Critical Conditions ◽  
Stable Steady State ◽  
Weakly Nonlinear ◽  
Symmetric Channel ◽  
Steady State Solutions

Transitions and instabilities of two-dimensional flow in a symmetric channel with a suddenly expanded and contracted part are investigated numerically by three different methods, i.e. the time marching method for dynamical equations, the SOR iterative method and the finite-element method for steady-state equations. Linear and weakly nonlinear stability theories are applied to the flow. The transitions are confirmed experimentally by flow visualizations. It is known that the flow is steady and symmetric at low Reynolds numbers, becomes asymmetric at a critical Reynolds number, regains the symmetry at another critical Reynolds number and becomes oscillatory at very large Reynolds numbers. Multiple stable steady-state solutions are found in some cases, which lead to a hysteresis. The critical conditions for the existence of the multiple stable steady-state solutions are determined numerically and compared with the results of the linear and weakly nonlinear stability analyses. An exchange of modes for oscillatory instabilities is found to occur in the flow as the aspect ratio, the ratio of the length of the expanded part to its width, is varied, and its relation with the impinging free-shear-layer instability (IFLSI) is discussed.

The effects of surface roughness and tunnel blockage on the flow past spheres

1974 ◽  
Vol 65 (1) ◽  
pp. 113-125 ◽  
Author(s):  
Elmar Achenbach
Keyword(s):  
Surface Roughness ◽  
Reynolds Number ◽  
Drag Coefficient ◽  
Critical Reynolds Number ◽  
Measurement Techniques ◽  
Flow Conditions ◽  
Number Range ◽  
Flow Past ◽  
Reynolds Number Range ◽  
Shedding Frequency

The effect of surface roughness on the flow past spheres has been investigated over the Reynolds number range 5 × 104 < Re < 6 × 106. The drag coefficient has been determined as a function of the Reynolds number for five surface roughnesses. With increasing roughness parameter the critical Reynolds number decreases. At the same time the transcritical drag coefficient rises, having a maximum value of 0·4.The vortex shedding frequency has been measured under subcritical flow conditions. It was found that the Strouhal number for each of the various roughness conditions was equal to its value for a smooth sphere. Beyond the critical Reynolds number no prevailing shedding frequency could be detected by the measurement techniques employed.The drag coefficient of a sphere under the blockage conditions 0·5 < ds/dt < 0·92 has been determined over the Reynolds number range 3 × 104 < Re < 2 × 106. Increasing blockage causes an increase in both the drag coefficient and the critical Reynolds number. The characteristic quantities were referred to the flow conditions in the smallest cross-section between sphere and tube. In addition the effect of the turbulence level on the flow past a sphere under various blockage conditions was studied.

Oscillatory motion and wake instability of freely rising axisymmetric bodies

2007 ◽  
Vol 573 ◽  
pp. 479-502 ◽  
Author(s):  
PEDRO C. FERNANDES ◽  
FRÉDÉRIC RISSO ◽  
PATRICIA ERN ◽  
JACQUES MAGNAUDET
Keyword(s):  
Reynolds Number ◽  
Aspect Ratio ◽  
Axial Velocity ◽  
Body Shape ◽  
Rigid Bodies ◽  
The Body ◽  
Body Motion ◽  
Wake Instability ◽  
Moving Bodies

This paper reports on an experimental study of the motion of freely rising axisym- metric rigid bodies in a low-viscosity fluid. We consider flat cylinders with height h smaller than the diameter d and density ρb close to the density ρf of the fluid. We have investigated the role of the Reynolds number based on the mean rise velocity um in the range 80 ≤ Re = umd/ν ≤ 330 and that of the aspect ratio in the range 1.5 ≤ χ = d/h ≤ 20. Beyond a critical Reynolds number, Rec, which depends on the aspect ratio, both the body velocity and the orientation start to oscillate periodically. The body motion is observed to be essentially two-dimensional. Its description is particularly simple in the coordinate system rotating with the body and having its origin fixed in the laboratory; the axial velocity is then found to be constant whereas the rotation and the lateral velocity are described well by two harmonic functions of time having the same angular frequency, ω. In parallel, direct numerical simulations of the flow around fixed bodies were carried out. They allowed us to determine (i) the threshold, Recf1(χ), of the primary regular bifurcation that causes the breaking of the axial symmetry of the wake as well as (ii) the threshold, Recf2(χ), and frequency, ωf, of the secondary Hopf bifurcation leading to wake oscillations. As χ increases, i.e. the body becomes thinner, the critical Reynolds numbers, Recf1 and Recf2, decrease. Introducing a Reynolds number Re* based on the velocity in the recirculating wake makes it possible to obtain thresholds $\hbox{\it Re}^*_{cf1}$ and $\hbox{\it Re}^*_{cf2}$ that are independent of χ. Comparison with fixed bodies allowed us to clarify the role of the body shape. The oscillations of thick moving bodies (χ < 6) are essentially triggered by the wake instability observed for a fixed body: Rec(χ) is equal to Recf1(χ) and ω is close to ωf. However, in the range 6 ≤ χ ≤ 10 the flow corrections induced by the translation and rotation of freely moving bodies are found to be able to delay the onset of wake oscillations, causing Rec to increase strongly with χ. An analysis of the evolution of the parameters characterizing the motion in the rotating frame reveals that the constant axial velocity scales with the gravitational velocity based on the body thickness, $\sqrt{((\rho_f-\rho_b)/\rho_f)\,gh}$, while the relevant length and velocity scales for the oscillations are the body diameter d and the gravitational velocity based on d, $\sqrt{((\rho_f-\rho_b)/\rho_f)\,gd}$, respectively. Using this scaling, the dimensionless amplitudes and frequency of the body's oscillations are found to depend only on the modified Reynolds number, Re*; they no longer depend on the body shape.

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