Instability and transitions of flow in a curved square duct: the development of two pairs of Dean vortices

1996 ◽  
Vol 314 ◽  
pp. 227-246 ◽  
Author(s):  
Philip A. J. Mees ◽  
K. Nandakumar ◽  
J. H. Masliyah
Keyword(s):  
Secondary Flow ◽  
Flow Pattern ◽  
Stokes Equations ◽  
Velocity Profiles ◽  
Navier Stokes ◽  
Flow State ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Measured Velocity ◽  
Dean Vortices

Steady developing flow of an incompressible Newtonian fluid in a curved duct of square cross-section (the Dean problem) is investigated both experimentally and numerically. This study is a continuation of the work by Bara, Nandakumar & Masliyah (1992) and is focused on flow rates between Dn = 200 and Dn = 600 (Dn = Re/(R/a)1/2, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension; the curvature ratio, R/a, is 15.1).Numerical simulations based on the steady three-dimensional Navier – Stokes equations predict the development of a 6-cell secondary flow pattern above a Dean number of 350. The 6-cell state consists of two large Ekman vortices and two pairs of small Dean vortices near the outer wall that result from the primary instability that is of centrifugal nature. The 6-cell flow state develops near θ = 80° and breaks down symmetrically into a 2-cell flow pattern.The apparatus used to verify the simulations had a duct dimension of 1.27 cm and a streamwise length of 270°. At a Dean number of 453, different velocity profiles of the 6-cell flow state at θ = 90° and spanwise profiles of the streamwise velocity at every 20° were measured using a laser-Doppler anemometer. All measured velocity profiles, as well as flow visualization of secondary flow patterns, are in very good agreement with the simulations, indicating that the parabolized Navier – Stokes equations give an accurate description of the flow.Based on the similarity with boundary layer flow over a concave wall (the Görtler problem), it is suggested that the transition to the 6-cell flow state is the result of a decreasing spanwise wavelength of the Dean vortices with increasing flow rate. A numerical stability analysis shows that the 6-cell flow state is unconditionally unstable. This is the first time that detailed experiments and simulations of the development of a 6-cell flow state are reported.

Dean vortices in finite-aspect-ratio ducts

2013 ◽  
Vol 716 ◽  
Author(s):  
Philip E. Haines ◽  
James P. Denier ◽  
Andrew P. Bassom
Keyword(s):  
Aspect Ratio ◽  
Stokes Equations ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Curved Channel ◽  
Dean Number ◽  
Element Analysis ◽  
Navier Stokes Equations ◽  
New Class ◽  
Dean Vortices

AbstractWe consider the development of Dean vortices in a curved channel of finite aspect ratio. Solutions to the axisymmetric Navier–Stokes equations are obtained through a finite-element analysis, allowing us to explore the complex and rich bifurcation pattern of the flow as the aspect ratio and Dean number vary. We demonstrate a new class of finite-amplitude vortices and discuss their relationship to similar structures seen in finite-length Taylor–Couette flow.

On laminar flow in curved semicircular ducts

1980 ◽  
Vol 99 (3) ◽  
pp. 469-479 ◽  
Author(s):  
Jacob H. Masliyah
Keyword(s):  
Secondary Flow ◽  
Stokes Equations ◽  
Outer Wall ◽  
Initial Guess ◽  
The Other ◽  
Navier Stokes ◽  
Dean Number ◽  
Solution Flow ◽  
Navier Stokes Equations ◽  
Semicircular Ducts

Calculations of the flow field under laminar conditions in a helical semicircular duct have been made by numerically solving the Navier–Stokes equations. With the flat wall of the duct being the outer wall, the solution of the momentum equations for Dean numbers below 105 gave, for the secondary flow, twin counter-rotating vortices of Taylor–Goertler type. However, above a Dean number of Dn = 105, two solutions were possible. One solution was similar to that obtained for Dn < 105. The other solution revealed four vortices for the secondary flow. For Dn > 105, convergence to either flow pattern depended on the initial guess used in the numerical solution. Flow visualization confirmed the possibility of the presence of both types of secondary flow patterns.

scholarly journals Off-Design Performance Prediction for Radial-Flow Impellers

1988 ◽  
Author(s):  
Shen C. Lee ◽  
Daying Chen
Keyword(s):  
Stokes Equations ◽  
Mechanical Energy ◽  
Navier Stokes ◽  
Shear Stresses ◽  
Pump Impeller ◽  
Navier Stokes Equations ◽  
Measured Velocity ◽  
Production Water ◽  
Performance Predictions ◽  
Stream Functions

A numerical method was developed to consider the two-dimensional flowfield between impeller blades of a given geometry. Solution of the laminar Navier-Stokes equations in geometry-oriented coordinates was obtained for stream functions and vorticities. Velocities and pressures were calculated to determine the output fluid-energy head. The circumferential components of the normal and shear stresses along the blade were evaluated to give the input mechanical-energy head. Performance predictions were obtained for different load conditions. Comparisons were made with the measured velocity vectors of the flowfield of an air-pump impeller and with the measured performance of a production water pump, good agreements were reached.

scholarly journals Propagation of Solitary Waves over a Submerged Slotted Barrier

2020 ◽  
Vol 8 (6) ◽  
pp. 419 ◽  
Author(s):  
Yun-Ta Wu ◽  
Shih-Chun Hsiao
Keyword(s):  
Flow Pattern ◽  
Solitary Waves ◽  
Vortex Shedding ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Model Data ◽  
Free Surface Elevation ◽  
Complex Flow ◽  
Laboratory Observation ◽  
Navier Stokes Equations

In this article, the interaction of solitary waves and a submerged slotted barrier is investigated in which the slotted barrier consists of three impermeable elements and its porosity can be determined by the distance between the two neighboring elements. A new experiment is conducted to measure free surface elevation, velocity, and turbulent kinetic energy. Numerical simulation is performed using a two-dimensional model based on the Reynolds-Averaged Navier-Stokes equations and the non-linear k-ɛ turbulence model. A detailed flow pattern is illustrated by a flow visualization technique. A laboratory observation indicates that flow separations occur at each element of the slotted barrier and the vortex shedding process is then triggered due to the complicated interaction of those induced vortices that further create a complex flow pattern. During the vortex shedding process, seeding particles that are initially accumulated near the seafloor are suspended by an upward jet formed by vortices interacting. Model-data comparisons are carried out to examine the accuracy of the model. Overall model-data comparisons are in satisfactory agreement, but modeled results sometimes fail to predict the positions of the induced vortices. Since the measured data is unique in terms of velocity and turbulence, the dataset can be used for further improvement of numerical modeling.

scholarly journals Effects of Hydrostatic Pocket Shape on the Flow Pattern and Pressure Distribution

1995 ◽  
Vol 1 (3-4) ◽  
pp. 225-235 ◽  
Author(s):  
M. J. Braun ◽  
M. Dzodzo
Keyword(s):  
Numerical Simulation ◽  
Pressure Distribution ◽  
Flow Pattern ◽  
Stokes Equations ◽  
The Other ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Hydrostatic Bearing ◽  
Collocated Grid ◽  
Dimensionless Formulation

The flow in a hydrostatic pocket is numerically simulated using a dimensionless formulation of the 2-D Navier-Stokes equations written in primitive variables, for a body fitted coordinates system, and applied through a collocated grid. In essence, we continue the work of Braun et al. 1993a, 1993b] and extend it to the study of the effects of the pocket geometric format on the flow pattern and pressure distribution. The model includes the coupling between the pocket flow and a finite length feedline flow, on one hand, and the pocket and its adjacent lands on the other hand. In this context we shall present, on a comparative basis, the flow and the pressure patterns at the runner surface for square, ramped-Rayleigh step, and arc of circle pockets. Geometrically all pockets have the same footprint, same lands length, and same capillary feedline. The numerical simulation uses the Reynolds number based on the lid(runner) velocity and the inlet jet strengthFas the dynamic similarity parameters. The study aims at establishing criteria for the optimization of the pocket geometry in the larger context of the performance of a hydrostatic bearing.

Flow Pattern on the Surface of Airfoil With Lower Reynolds Number

2009 ◽  
Author(s):  
Xu Sun ◽  
Jia-Zhong Zhang
Keyword(s):  
Reynolds Number ◽  
Flow Pattern ◽  
Stokes Equations ◽  
Separation Bubble ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Naca0012 Airfoil ◽  
Lower Reynolds Number ◽  
Characteristic Based Split

In this paper, aerodynamic performance of the NACA0012 airfoil in the incompressible flow with a lower Reynolds number (Re) is investigated numerically from the viewpoints of flow pattern and nonlinear dynamics. First, the characteristic-based split (CBS) finite element method is introduced for the approximation of the incompressible Navier-Stokes equations, and then the lid-driven cavity flow and flow around a circular cylinder are calculated for varification. Then, at Re = 1000, flow fields around the NACA0012 airfoil at a series of angles of attack are simulated. With the increase of the attack angle, great change of the flow pattern appears, and the flow structures such as trailing edge vortex, separation bubble and shedding vortex are observed. Moreover, it is found that the separation bubble plays an important role in the deterioration of the flow stability at higher attack angles, and the vortex shedding can be taken as the result of a Hopf bifurcation while the bifurcation parameter is the angle of attack.

Some numerical experiments on developing laminar flow in circular-sectioned bends

1985 ◽  
Vol 154 ◽  
pp. 357-375 ◽  
Author(s):  
J. A. C. Humphrey ◽  
H. Iacovides ◽  
B. E. Launder
Keyword(s):  
Shear Stress ◽  
Laminar Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Streamwise Velocity ◽  
Navier Stokes ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Numerical Predictions

The paper reports numerical solutions to a semi-elliptic truncation of the Navier–Stokes equations for the case of developing laminar flow in circular-sectioned bends over a range of Dean numbers. The ratios of bend radius to pipe radius are 7:1 and 20:1, corresponding with the configurations examined experimentally by Talbot and his co-workers in recent years. The semi-elliptic treatment facilitates a much finer grid than has been possible in earlier studies. Numerical accuracy has been further improved by assuming radial equilibrium over a thin sublayer immediately adjacent to the wall and by re-formulating the boundary conditions at the pipe centre.Streamwise velocity profiles at Dean numbers of 183 and 565 are in excellent agreement with laser-Doppler measurements by Agrawal, Talbot & Gong (1978). Good, albeit less complete, accord is found with the secondary velocities, though the differences that exist may be mainly due to the difficulty of making these measurements. The paper provides new information on the behaviour of the streamwise shear stress around the inner line of symmetry. Upstream of the point of minimum shear stress, our numerical predictions display a progressive shift towards the result of Stewartson, Cebici & Chang (1980) as the Dean number is successively raised. Downstream of the minimum, however, in contrast with the monotonic approach to an asymptotic level reported by Stewartson, the numerical solutions display a damped oscillatory behaviour reminiscent of those from Hawthorne's (1951) inviscid-flow calculations. The amplitude of the oscillation grows as the Dean number is raised.

Numerical Simulation of Electromagnetic Stirring Effect on Flow Pattern of Metal Melt

2011 ◽  
Vol 264-265 ◽  
pp. 1574-1579
Author(s):  
H. Namaki ◽  
S. Hossein Seyedein ◽  
M.R. Afshar Moghadam ◽  
R. Ghasemzadeh
Keyword(s):  
Flow Pattern ◽  
Maxwell Equations ◽  
Flow Model ◽  
Stokes Equations ◽  
Electromagnetic Stirring ◽  
Navier Stokes ◽  
Simultaneous Solution ◽  
Navier Stokes Equations ◽  
Stirring Effect ◽  
Good Agreement

In this study, a mathematical model was developed to simulate 2-D axisymmetric melt flow under magnetic field in a cylindrical container. The modeling of this process required the simultaneous solution of the turbulent Navier-Stokes equations together with Maxwell equations. The flow pattern in liquid bath was obtained using a two-equation κ-є turbulent flow model, which was further used to obtain the solute distribution. The governing differential equations were solved numerically using finite volume based finite difference method. The computed results, were found to be in good agreement with the measurements reported in the literature. The effect of stirring parameters on temperature homogeneity of the melt have been discussed and presented.

A Contribution to the Numerical Solution of Developing Laminar Flow in the Entrance Region of Concentric Annuli With Rotating Inner Walls

1974 ◽  
Vol 96 (4) ◽  
pp. 333-340 ◽  
Author(s):  
J. E. R. Coney ◽  
M. A. I. El-Shaarawi
Keyword(s):  
Boundary Layer ◽  
Laminar Flow ◽  
Stokes Equations ◽  
Complete Solution ◽  
Numerical Differentiation ◽  
Velocity Profiles ◽  
Entrance Region ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Special Case

The boundary layer simplification of the Navier-Stokes equations for hydrodynamically developing laminar flow with constant physical properties in the entrance region of concentric annuli with rotating inner walls have been numerically solved using a simple linearized finite-difference scheme. Additional results to those existing in the literature by Martin and Payne [1–2] will be presented here. An advantage of the analysis used in this paper is that it does not solve for the stream function and vorticity, but predicts the development of tangential, axial and radial velocity profiles directly, thus avoiding numerical differentiation. Results for the development of these velocity profiles, pressure drop and friction factor are presented for five annuli radii ratios (0.3, 0.5, 0.674, 0.727 and 0.90) at various values of the parameter Re2/Ta. The paper may be considered as a direct comparison between the boundary layer solution and the complete solution of the Navier-Stokes equations [1–2] for that special case.

scholarly journals A Geometrical Regularity Criterion in Terms of Velocity Profiles for the Three-Dimensional Navier–Stokes Equations

2019 ◽  
Vol 72 (4) ◽  
pp. 545-562 ◽  
Author(s):  
C V Tran ◽  
X Yu
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Regularity Criterion ◽  
Velocity Profiles ◽  
Regularity Criteria ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Large Velocity ◽  
Whole Space ◽  
Velocity Region

Summary In this article, we present a new kind of regularity criteria for the global well-posedness problem of the three-dimensional Navier–Stokes equations in the whole space. The novelty of the new results is that they involve only the profiles of the magnitude of the velocity. One particular consequence of our theorem is as follows. If for every fixed $t\in (0,T)$, the ‘large velocity’ region $\Omega:=\{(x,t)\mid |u(x,t)|&gt;C(q)\left|\mkern-2mu\left|{u}\right|\mkern-2mu\right|_{L^{3q-6}}\}$, for some $C(q)$ appropriately defined, shrinks fast enough as $q\nearrow \infty$, then the solution remains regular beyond $T$. We examine and discuss velocity profiles satisfying our criterion. It remains to be seen whether these profiles are typical of general Navier–Stokes flows.

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