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.

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.

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.

scholarly journals Natural convection in a shallow cavity with differentially heated end walls. Part 2. Numerical solutions

1974 ◽  
Vol 65 (2) ◽  
pp. 231-246 ◽  
Author(s):  
D. E. Cormack ◽  
L. G. Leal ◽  
J. H. Seinfeld
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Aspect Ratio ◽  
Excellent Agreement ◽  
Asymptotic Theory ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Shallow Cavity

Numerical solutions of the full Navier-Stokes equations are obtained for the problem of natural convection in closed cavities of small aspect ratio with differentially heated end walls. These solutions cover the parameter range Pr = 6·983, 10 ≤ Gr 2 × 104 and 0·05 [les ] A [les ] 1. A comparison with the asymptotic theory of part 1 shows excellent agreement between the analytical and numerical solutions provided that A [lsim ] 0·1 and Gr2A3Pr2 [lsim ] 105. In addition, the numerical solutions demonstrate the transition between the shallow-cavity limit of part 1 and the boundary-layer limit; A fixed, Gr → ∞.

A Finite Element Analysis of the Navier-Stokes Equations Applied to High Speed Thin Film Lubrication

1987 ◽  
Vol 109 (1) ◽  
pp. 71-76 ◽  
Author(s):  
J. O. Medwell ◽  
D. T. Gethin ◽  
C. Taylor
Keyword(s):  
Finite Element ◽  
High Speed ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
The Finite Element Method ◽  
Thin Film Lubrication ◽  
Element Analysis ◽  
Navier Stokes Equations ◽  
Film Lubrication

The performance of a cylindrical bore bearing fed by two axial grooves orthogonal to the load line is analyzed by solving the Navier-Stokes equations using the finite element method. This produces detailed information about the three-dimensional velocity and pressure field within the hydrodynamic film. It is also shown that the method may be applied to long bearing geometries where recirculatory flows occur and in which the governing equations are elliptic. As expected the analysis confirms that lubricant inertia does not affect bearing performance significantly.

A finite element analysis of steady viscous flow in triangular cavities

1999 ◽  
Vol 213 (3) ◽  
pp. 263-276 ◽  
Author(s):  
P. H. Gaskell ◽  
H. M. Thompson ◽  
M. D. Savage
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Published Data ◽  
Uniform Motion ◽  
Element Formulation ◽  
Computational Results ◽  
Element Analysis ◽  
Navier Stokes Equations ◽  
Side Walls

A finite element formulation of the Navier—Stokes equations, written in terms of the stream function, ψ, and vorticity, ω, for a Newtonian fluid in the absence of body forces, is used to solve the problem of flow in a triangular cavity, driven by the uniform motion of one of its side walls. A key feature of the numerical method is that the difficulties associated with specifying ω at the corners are addressed and overcome by applying analytical boundary conditions on ω near these singularities. The computational results are found to agree well with previously published data and, for small stagnant corner angles, reveal the existence of a sequence of secondary recirculations whose relative sizes and strengths are in accord with Moffatt's classical theory. It is shown that, as the stagnant corner angle is increased beyond approximately 40°, the secondary recirculations diminish in size rapidly.

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