An approximate theory for the streaming motion past axisymmetric bodies at Reynolds numbers of from 1 to approximately 100

1977 ◽  
Vol 82 (3) ◽  
pp. 583-604 ◽  
Author(s):  
Michael S. Kolansky ◽  
Sheldon Weinbaum ◽  
Robert Pfeffer
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Pressure Gradient ◽  
Reynolds Numbers ◽  
Bluff Bodies ◽  
Approximate Theory ◽  
Viscous Term ◽  
Number Range ◽  
Curvature Effects ◽  
Flow Past

In Weinbaum et al. (1976) a simple new pressure hypothesis is derived which enables one to take account of the displacement interaction, the geometrical change in streamline radius of curvature and centrifugal effects in the thick viscous layers surrounding two-dimensional bluff bodies in the intermediate Reynolds number range O(1) < Re < O(102) using conventional Prandtl boundary-layer equations. The new pressure hypothesis states that the streamwise pressure gradient as a function of distance from the forward stagnation point on the displacement body is equal to the wall pressure gradient as a function of distance along the original body. This hypothesis is shown to be equivalent to stretching the streamwise body co-ordinate in conventional first-order boundary-layer theory. The present investigation shows that the same pressure hypothesis applies for the intermediate Reynolds number flow past axisymmetric bluff bodies except that the viscous term in the conventional axisymmetric boundary-layer equation must also be modified for transverse curvature effects O(δ) in the divergence of the stress tensor. The approximate solutions presented for the location of separation and the detailed surface pressure and vorticity distribution for the flow past spheres, spheroids and paraboloids of revolution at various Reynolds numbers in the range O(1) < Re < O(102) are in good agreement with available numerical Navier–Stokes solutions.

An approximate theory for incompressible viscous flow past two-dimensional bluff bodies in the intermediate Reynolds number regime O(1) < Re < O(102)

1976 ◽  
Vol 77 (1) ◽  
pp. 129-152 ◽  
Author(s):  
Sheldon Weinbaum ◽  
Michael S. Kolansky ◽  
Michael J. Gluckman ◽  
Robert Pfeffer
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Pressure Gradient ◽  
Boundary Layer Theory ◽  
Navier Stokes ◽  
Bluff Bodies ◽  
Approximate Theory ◽  
Two Dimensional ◽  
First Order ◽  
Flow Past

A new approximate theory is proposed for treating the flow past smoothly contoured two-dimensional bluff bodies in the intermediate Reynolds number rangeO(1) <Re< 0(102), where the displacement effect of the thick viscous layer near the surface of the body is large and a steady laminar wake is present. The theory is based on a new pressure hypothesis which enables one to take account of the displacement interaction and centrifugal effects in thick viscous layers using conventional first-order boundary-layer equations. The basic question asked is how the wall pressure gradient in ordinary boundary -layer theory must be modified if the pressure gradient along the displacement surface using the Prandtl pressure hypothesis is to be equal to the pressure gradient along this surface using a higher-order approximation to the Navier-Stokes equation in which centrifugal forces are considered. The inclusion of the normal pressure field with displacement interaction is shown to be equivalent to stretching the streamwise body co-ordinate in first-order boundary-layer theory such that the streamwise pressure gradient as a function of distance along the original and displacement body surfaces are equal.While the new theory is of a non-rigorous nature, it yields results for the location of separation and detailed surface pressure and vorticity distribution which are in remarkably good agreement with the large body of available numerical Navier-Stokes solutions. A novel feature of the new boundary-value problem is the development of a simple but accurate approximate method for determining the inviscid flow past an arbitrary two-dimensional displacement body with its wake.

Unsteady flow past a rotating circular cylinder at Reynolds numbers 103 and 104

1990 ◽  
Vol 220 ◽  
pp. 459-484 ◽  
Author(s):  
H. M. Badr ◽  
M. Coutanceau ◽  
S. C. R. Dennis ◽  
C. Ménard
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Unsteady Flow ◽  
Rotation Rate ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Number Range ◽  
Flow Past

The unsteady flow past a circular cylinder which starts translating and rotating impulsively from rest in a viscous fluid is investigated both theoretically and experimentally in the Reynolds number range 103 [les ] R [les ] 104 and for rotational to translational surface speed ratios between 0.5 and 3. The theoretical study is based on numerical solutions of the two-dimensional unsteady Navier–Stokes equations while the experimental investigation is based on visualization of the flow using very fine suspended particles. The object of the study is to examine the effect of increase of rotation on the flow structure. There is excellent agreement between the numerical and experimental results for all speed ratios considered, except in the case of the highest rotation rate. Here three-dimensional effects become more pronounced in the experiments and the laminar flow breaks down, while the calculated flow starts to approach a steady state. For lower rotation rates a periodic structure of vortex evolution and shedding develops in the calculations which is repeated exactly as time advances. Another feature of the calculations is the discrepancy in the lift and drag forces at high Reynolds numbers resulting from solving the boundary-layer limit of the equations of motion rather than the full Navier–Stokes equations. Typical results are given for selected values of the Reynolds number and rotation rate.

Pressure-driven flow past spheres moving in a circular tube

2007 ◽  
Vol 592 ◽  
pp. 233-262 ◽  
Author(s):  
G. J. SHEARD ◽  
K. RYAN
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Axial Force ◽  
Flow Rate ◽  
Circular Tube ◽  
Reynolds Numbers ◽  
Diameter Ratio ◽  
Tube Diameter ◽  
Flow Past ◽  
Pressure Driven Flow

A computational investigation, supported by a theoretical analysis, is performed to investigate a pressure-driven flow around a line of equispaced spheres moving at a prescribed velocity along the axis of a circular tube. This fundamental study underpins a range of applications including physiological circulation research. A spectral-element formulation in cylindrical coordinates is employed to solve for the incompressible fluid flow past the spheres, and the flows are computed in the reference frame of the translating spheres.Both the volume flow rate relative to the spheres and the forces acting on each sphere are computed for specific sphere-to-tube diameter ratios and sphere spacing ratios. Conditions at which zero axial force on the spheres are identified, and a region of unsteady flow is detected at higher Reynolds numbers (based on tube diameter and sphere velocity). A regular perturbation analysis and the reciprocal theorem are employed to predict flow rate and drag coefficient trends at low Reynolds numbers. Importantly, the zero drag condition is well-described by theory, and states that at this condition, the sphere velocity is proportional to the applied pressure gradient. This result was verified for a range of spacing and diameter ratios. Theoretical approximations agree with computational results for Reynolds numbers up toO(100).The geometry dependence of the zero axial force condition is examined, and for a particular choice of the applied dimensionless pressure gradient, it is found that this condition occurs at increasing Reynolds numbers with increasing diameter ratio, and decreasing Reynolds number with increasing sphere spacing.Three-dimensional simulations and predictions of a Floquet linear stability analysis independently elucidate the bifurcation scenario with increasing Reynolds number for a specific diameter ratio and sphere spacing. The steady axisymmetric flow first experiences a small region of time-dependent non-axisymmetric instability, before undergoing a regular bifurcation to a steady non-axisymmetric state with azimuthal symmetrym= 1. Landau modelling verifies that both the regular non-axisymmetric mode and the axisymmetric Hopf transition occur through a supercritical (non-hysteretic) bifurcation.

An Analysis of Axisymmetric Turbulent Flow Past a Long Cylinder

1972 ◽  
Vol 94 (1) ◽  
pp. 200-204 ◽  
Author(s):  
F. M. White
Keyword(s):  
Boundary Layer ◽  
Shape Factor ◽  
Boundary Layer Thickness ◽  
Axial Flow ◽  
Reynolds Numbers ◽  
Small Radius ◽  
Law Of The Wall ◽  
Curvature Effects ◽  
Flow Past ◽  
Long Cylinder

An analysis is presented which predicts the properties of an arbitrarily thick turbulent boundary layer for incompressible axial flow past a long cylinder. The approach makes use of a modified form of the law-of-the-wall, deduced by G. N. V. Rao, which properly accounts for transverse curvature effects. Using this law, the theory which follows is equivalent to an exact solution to the axisymmetric equations of continuity and momentum for zero pressure gradient. Numerical results show that curvature increases skin friction and overall drag and decreases the boundary layer thickness and the integral thicknesses. The velocity profile is flattened and the shape factor approaches unity at large curvature. Comparison with several sources of friction data show better overall agreement than previous theories, except for an unexplained discrepancy with data for moving nylon fibers at very small radius Reynolds numbers.

Experiments on the flow past spheres at very high Reynolds numbers

1972 ◽  
Vol 54 (3) ◽  
pp. 565-575 ◽  
Author(s):  
Elmar Achenbach
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layer Separation ◽  
Boundary Layer Transition ◽  
Layer Transition ◽  
Local Flow ◽  
Friction Forces ◽  
Local Skin ◽  
Number Range ◽  
Flow Past

The present work is concerned with the flow past spheres in the Reynolds number range 5 × 104 [les ] Re [les ] 6 × 106. Results are reported for the case of a smooth surface. The total drag, the local static pressure and the local skin friction distribution were measured at a turbulence level of about 0·45%. The present results are compared with other available data as far as possible. Information is obtained from the local flow parameters on the positions of boundary-layer transition from laminar to turbulent flow and of boundary-layer separation. Finally the dependence of friction forces on Reynolds number is pointed out.

Calculation of incompressible flow past a circular cylinder at moderate Reynolds numbers

1969 ◽  
Vol 37 (1) ◽  
pp. 95-114 ◽  
Author(s):  
Robert Leigh Underwood
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Differential Equations ◽  
Incompressible Flow ◽  
Equations Of Motion ◽  
Pressure Coefficient ◽  
Reynolds Numbers ◽  
Number Range ◽  
Flow Parameters ◽  
Flow Past

The steady, two-dimensional, incompressible flow past a circular cylinder is calculated for Reynolds numbers up to ten. An accurate description of the flow field is found by employing the semi-analytical method of series truncation to reduce the governing partial differential equations of motion to a system of ordinary differential equations which can be integrated numerically. Results are given for Reynolds numbers between 0.4 and 10.0 (based on diameter). The Reynolds number at which separation first occurs behind the cylinder is found to be 5.75. Over the entire Reynolds number range investigated, characteristic flow parameters such as the drag coefficient, pressure coefficient, standing eddy length, and streamline pattern compare favourably with available experimental data and numerical solution results.

Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer

2009 ◽  
Vol 630 ◽  
pp. 5-41 ◽  
Author(s):  
XIAOHUA WU ◽  
PARVIZ MOIN
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Shear Stress ◽  
Pressure Gradient ◽  
Flat Plate ◽  
Plate Boundary ◽  
Transitional Region ◽  
Bypass Transition ◽  
Number Range ◽  
Flat Plate Boundary Layer

A nominally-zero-pressure-gradient incompressible boundary layer over a smooth flat plate was simulated for a continuous momentum thickness Reynolds number range of 80 ≤ Reθ ≤ 940. Transition which is completed at approximately Reθ = 750 was triggered by intermittent localized disturbances arising from patches of isotropic turbulence introduced periodically from the free stream at Reθ = 80. Streamwise pressure gradient is quantified with several measures and is demonstrated to be weak. Blasius boundary layer is maintained in the early transitional region of 80 < Reθ < 180 within which the maximum deviation of skin friction from the theoretical solution is less than 1%. Mean and second-order turbulence statistics are compared with classic experimental data, and they constitute a rare DNS dataset for the spatially developing zero-pressure-gradient turbulent flat-plate boundary layer. Our calculations indicate that in the present spatially developing low-Reynolds-number turbulent flat-plate boundary layer, total shear stress mildly overshoots the wall shear stress in the near-wall region of 2–20 wall units with vanishing normal gradient at the wall. Overshoots as high as 10% across a wider percentage of the boundary layer thickness exist in the late transitional region. The former is a residual effect of the latter. The instantaneous flow fields are vividly populated by hairpin vortices. This is the first time that direct evidence (in the form of a solution of the Navier–Stokes equations, obeying the statistical measurements, as opposed to synthetic superposition of the structures) shows such dominance of these structures. Hairpin packets arising from upstream fragmented Λ structures are found to be instrumental in the breakdown of the present boundary layer bypass transition.

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