Influence of Surface Roughness on Shear Flow

2004 ◽  
Vol 71 (4) ◽  
pp. 459-464 ◽  
Author(s):  
S. Bhattacharyya ◽  
S. Mahapatra ◽  
F. T. Smith
Keyword(s):  
Cross Section ◽  
Numerical Solutions ◽  
Flat Surface ◽  
Stokes Equations ◽  
Circular Cross Section ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Maximum Pressure ◽  
Navier Stokes Equations ◽  
Uniform Shear

The local planar flow of incompressible fluid past an obstacle of semi-circular cross section is considered, the obstacle being mounted on a long flat surface. The far-field motion is one of uniform shear. Direct numerical solutions of the Navier-Stokes equations are described over a range of Reynolds numbers. The downstream eddy length and upstream position of maximum pressure gradient are found to agree with increased Reynolds number theory; in particular the agreement for the former quantity is close for Reynolds numbers above about 50.

scholarly journals Asymptotically based self-similarity solution of the Navier–Stokes equations for a porous tube with a non-circular cross-section

2017 ◽  
Vol 826 ◽  
pp. 396-420 ◽  
Author(s):  
M. Bouyges ◽  
F. Chedevergne ◽  
G. Casalis ◽  
J. Majdalani
Keyword(s):  
Cross Section ◽  
Similarity Solution ◽  
Stokes Equations ◽  
Circular Cross Section ◽  
Internal Flow ◽  
Navier Stokes ◽  
Solid Rocket Motor ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
Radial Deviation

This work introduces a similarity solution to the problem of a viscous, incompressible and rotational fluid in a right-cylindrical chamber with uniformly porous walls and a non-circular cross-section. The attendant idealization may be used to model the non-reactive internal flow field of a solid rocket motor with a star-shaped grain configuration. By mapping the radial domain to a circular pipe flow, the Navier–Stokes equations are converted to a fourth-order differential equation that is reminiscent of Berman’s classic expression. Then assuming a small radial deviation from a fixed chamber radius, asymptotic expansions of the three-component velocity and pressure fields are systematically pursued to the second order in the radial deviation amplitude. This enables us to derive a set of ordinary differential relations that can be readily solved for the mean flow variables. In the process of characterizing the ensuing flow motion, the axial, radial and tangential velocities are compared and shown to agree favourably with the simulation results of a finite-volume Navier–Stokes solver at different cross-flow Reynolds numbers, deviation amplitudes and circular wavenumbers.

Navier–Stokes solutions of unsteady separation induced by a vortex

2002 ◽  
Vol 465 ◽  
pp. 99-130 ◽  
Author(s):  
A. V. OBABKO ◽  
K. W. CASSEL
Keyword(s):  
Boundary Layer ◽  
Large Scale ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Recirculation Region ◽  
Small Scale ◽  
Navier Stokes Equations ◽  
Scale Interaction

Numerical solutions of the unsteady Navier–Stokes equations are considered for the flow induced by a thick-core vortex convecting along a surface in a two-dimensional incompressible flow. The presence of the vortex induces an adverse streamwise pressure gradient along the surface that leads to the formation of a secondary recirculation region followed by a narrow eruption of near-wall fluid in solutions of the unsteady boundary-layer equations. The locally thickening boundary layer in the vicinity of the eruption provokes an interaction between the viscous boundary layer and the outer inviscid flow. Numerical solutions of the Navier–Stokes equations show that the interaction occurs on two distinct streamwise length scales depending upon which of three Reynolds-number regimes is being considered. At high Reynolds numbers, the spike leads to a small-scale interaction; at moderate Reynolds numbers, the flow experiences a large-scale interaction followed by the small-scale interaction due to the spike; at low Reynolds numbers, large-scale interaction occurs, but there is no spike or subsequent small-scale interaction. The large-scale interaction is found to play an essential role in determining the overall evolution of unsteady separation in the moderate-Reynolds-number regime; it accelerates the spike formation process and leads to formation of secondary recirculation regions, splitting of the primary recirculation region into multiple corotating eddies and ejections of near-wall vorticity. These eddies later merge prior to being lifted away from the surface and causing detachment of the thick-core vortex.

Optimized Accelerating Parameters of Convergence for the Navier-Stokes Equations

1992 ◽  
Author(s):  
Bashar S. AbdulNour
Keyword(s):  
Stream Function ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Relaxation Method ◽  
Reynolds Numbers ◽  
Computer Time ◽  
Navier Stokes ◽  
Computational Time ◽  
Navier Stokes Equations ◽  
Successive Over Relaxation

Abstract An over-relaxation procedure, that includes weighing factors, is applied to the steady, two-dimensional Navier-Stokes equations in order to reduce the computational time. The benefits obtained from this strategy are illustrated by the problem of viscous flow in the entrance region of an unconstricted and a constricted channel. The describing equations are expressed in terms of the stream function and vorticity. The convergence domain for the Successive Over-Relaxation method and the optimum values of the accelerating parameters, which consist of the over-relaxation and weighting factors for both the stream function and vorticity, are discussed. Numerical solutions are obtained for Reynolds numbers ranging from 20 to 2000. The computer time is reduced by as much as a factor of six using the optimum values of the accelerating parameters.

Numerical and asymptotic methods for certain viscous free-surface flows

1992 ◽  
Vol 340 (1656) ◽  
pp. 1-45 ◽  
Keyword(s):  
Reynolds Number ◽  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Free Surface Flows ◽  
Navier Stokes ◽  
Lubrication Approximation ◽  
Navier Stokes Equations ◽  
Flow Rates

This paper concerns the two-dimensional motion of a viscous liquid down a perturbed inclined plane under the influence of gravity, and the main goal is the prediction of the surface height as the fluid flows over the perturbations. The specific perturbations chosen for the present study were two humps stretching laterally across an otherwise uniform plane, with the flow being confined in the lateral direction by the walls of a channel. Theoretical predictions of the flow have been obtained by finite-element approximations to the Navier-Stokes equations and also by a variety of lubrication approximations. The predictions from the various models are compared with experimental measurements of the free-surface profiles. The principal aim of this study is the establishment and assessment of certain numerical and asymptotic models for the description of a class of free-surface flows, exemplified by the particular case of flow over a perturbed inclined plane. The laboratory experiments were made over a range of flow rates such that the Reynolds number, based on the volume flux per unit width and the kinematical viscosity of the fluid, ranged between 0.369 and 36.6. It was found that, at the smaller Reynolds numbers, a standard lubrication approximation provided a very good representation of the experimental measurements but, as the flow rate was increased, the standard model did not capture several important features of the flow. On the other hand, a lubrication approximation allowing for surface tension and inertial effects expanded the range of applicability of the basic theory by almost an order of magnitude, up to Reynolds numbers approaching 10. At larger flow rates, numerical solutions to the full equations of motion provided a description of the experimental results to within about 4% , up to a Reynolds number of 25, beyond which we were unable to obtain numerical solutions. It is not known why numerical solutions were not possible at larger flow rates, but it is possible that there is a bifurcation of the Navier-Stokes equations to a branch of unsteady motions near a Reynolds number of 25.

Complex solutions of the Dean equations and non-uniqueness at all Reynolds numbers

2017 ◽  
Vol 818 ◽  
pp. 241-259 ◽  
Author(s):  
F. A. T. Boshier ◽  
A. J. Mestel
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Vortex Solution ◽  
Vortex Solutions ◽  
Complex Solutions ◽  
Unknown Solution

Steady incompressible flow down a slowly curving circular pipe is considered, analytically and numerically. Both real and complex solutions are investigated. Using high-order Hermite–Padé approximants, the Dean series solution is analytically continued outside its circle of convergence, where it predicts a complex solution branch for real positive Dean number, $K$. This is confirmed by numerical solution. It is shown that other previously unknown solution branches exist for all $K>0$, which are related to an unforced complex eigensolution. This non-uniqueness is believed to be generic to the Navier–Stokes equations in most geometries. By means of path continuation, numerical solutions are followed around the complex $K$-plane. The standard Dean two-vortex solution is shown to lie on the same hypersurface as the eigensolution and the four-vortex solutions found in the literature. Elliptic pipes are considered and shown to exhibit similar behaviour to the circular case. There is an imaginary singularity limiting convergence of the Dean series, an unforced solution at $K=0$ and non-uniqueness for $K>0$, culminating in a real bifurcation.

scholarly journals Transonic flow over localised heating elements in boundary layers

2018 ◽  
Vol 844 ◽  
pp. 746-765 ◽  
Author(s):  
A. F. Aljohani ◽  
J. S. B. Gajjar
Keyword(s):  
Supersonic Flow ◽  
Transonic Flow ◽  
Energy Equation ◽  
Flat Surface ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Upper Deck ◽  
Lower Deck

The problem of transonic flow past an array of micro-electro-mechanical-type (MEMS-type) heating elements placed on a flat surface is investigated using the triple-deck theory. The compressible Navier–Stokes equations supplemented by the energy equation are considered for large Reynolds numbers. The triple-deck problem is formulated with the aid of the method of matched expansions. The resulting nonlinear viscous lower deck problem, coupled with the upper deck problem governed by the nonlinear Kármán–Guderley equation, is solved using a numerical method based on Chebyshev collocation and finite differences. Our results show the differences in subsonic and supersonic flow behaviour over heated elements. The results indicate the possibility of using the elements to favourably control the transonic flow field.

The Steady Horizontal Flow of a Wall Jet Into a Large-Width Cavity

1998 ◽  
Vol 120 (1) ◽  
pp. 70-75 ◽  
Author(s):  
K. O. Homan ◽  
S. L. Soo
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Finite Domain ◽  
Wall Jet ◽  
Navier Stokes Equations ◽  
Self Similar ◽  
Large Width

This paper treats the steady flow of a wall jet into a large-width cavity for which the primary axis is normal to the direction of the jet inflow. Numerical solutions of the two-dimensional Navier-Stokes equations are computed for inlet Reynolds numbers of 10 to 50 and tank width to inlet height ratios of 16 to 128. The length and velocity scales of the wall jet boundary layer exhibit close agreement with the classic wall jet similarity solution and published experimental data but the width of the region for which the comparison proves to be favorable has a limited extent. This departure from a self-similar evolution of the wall jet is shown to result from the finite domain width and its influence on the large recirculation cell located immediately above the wall jet boundary layer.

Oscillatory flow states in an enclosed cylinder with a rotating endwall

1999 ◽  
Vol 389 ◽  
pp. 101-118 ◽  
Author(s):  
J. L. STEVENS ◽  
J. M. LOPEZ ◽  
B. J. CANTWELL
Keyword(s):  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Numerical Study ◽  
Modulation Frequency ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Multiple Time ◽  
Navier Stokes Equations ◽  
Flow States

A combined experimental and numerical investigation is presented of the multiple oscillatory states that exist in the flows produced in a completely filled, enclosed, circular cylinder driven by the constant rotation of one of its endwalls. The flow in a cylinder of height to radius ratio 2.5 is interrogated experimentally using flow visualization and digitized images to extract quantitative temporal information. Numerical solutions of the axisymmetric Navier–Stokes equations are used to study the same flow over a range of Reynolds numbers where the flow is observed to remain axisymmetric. Three oscillatory states have been identified, two of them are periodic and the third is quasi-periodic with a modulation frequency much smaller than the base frequency. The range of Reynolds numbers for which the quasi-periodic flow exists brackets the switch between the two periodic states. The results from the combined experimental and numerical study agree both qualitatively and quantitatively, providing unambiguous evidence of the existence and robustness of these multiple time-dependent states.

Capsular flow in pipelines

1972 ◽  
Vol 56 (1) ◽  
pp. 49-59 ◽  
Author(s):  
A. E. Vardy ◽  
M. I. G. Bloor ◽  
J. A. Fox
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Steady Motion ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Hydraulic Pressure ◽  
Central Difference ◽  
Navier Stokes Equations

The problem considered is that of the steady motion of a series of neutrally buoyant, flat-faced, rigid, cylindrical capsules along the axis of a pipeline under the influence of a hydraulic pressure gradient. The Navier-Stokes equations are non-dimensionalized and expressed in central-difference form. Numerical solutions are found by the method of relaxation for Reynolds numbers up to 20 000 and a close agreement is obtained with readings from a laboratory apparatus for Reynolds numbers up to 2200.The flow is examined in detail and the existence of toroidal vortices between successive capsules is demonstrated. Their shape is shown to be increasingly influenced by inertial forces as the Reynolds number increases, but the overall pressure gradient is not greatly dependent on the Reynolds number.

Heat Transfer of Coupled Fluid Flow Within a Channel With a Permeable Base

2009 ◽  
Vol 131 (11) ◽  
Author(s):  
Rosemarie Mohais ◽  
Balswaroop Bhatt
Keyword(s):  
Heat Transfer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Effective Means ◽  
Reynolds Numbers ◽  
Model Systems ◽  
Surface Velocity ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Small Reynolds Numbers

We examine the heat transfer in a Newtonian fluid confined within a channel with a lower permeable wall. The upper wall of the channel is impermeable and driven by an accelerating surface velocity. Through a similarity solution, the Navier–Stokes equations are reduced to a fourth-order differential equation; the analytical solutions of which determined for small Reynolds numbers show dependence of the temperature and heat transfer profiles on the slip parameter based on the properties of the porous channel base. For larger Reynolds numbers, numerical solutions for three main groups of solutions show that the Reynolds number strongly influences the heat transfer profile. However, the slip conditions associated with the porous base of the channel can be used to alter these heat transfer profiles for large Reynolds numbers. The presence of a porous base in a channel can thus serve as an effective means of reducing or enhancing heat transfer performance in model systems.

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