Laminar Flow Over Wavy Walls

1991 ◽  
Vol 113 (4) ◽  
pp. 574-578 ◽  
Author(s):  
V. C. Patel ◽  
J. Tyndall Chon ◽  
J. Y. Yoon
Keyword(s):  
Boundary Layer ◽  
Laminar Flow ◽  
Steady Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Flow Regimes ◽  
Navier Stokes ◽  
Wavy Wall ◽  
Fully Developed Flow ◽  
Navier Stokes Equations

The boundary layer over a wavy wall and fully-developed flow in a duct with a wavy wall are considered. Numerical solutions of the Navier-Stokes equations have been obtained to provide insights into the various steady flow regimes that are possible, and to illustrate the nuances of predicting flows containing multiple separation and reattachment points.

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.

The structure of two-dimensional separation

1990 ◽  
Vol 220 ◽  
pp. 397-411 ◽  
Author(s):  
Laura L. Pauley ◽  
Parviz Moin ◽  
William C. Reynolds
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Adverse Pressure Gradient ◽  
Navier Stokes ◽  
Stream Velocity ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Shedding Frequency

The separation of a two-dimensional laminar boundary layer under the influence of a suddenly imposed external adverse pressure gradient was studied by time-accurate numerical solutions of the Navier–Stokes equations. It was found that a strong adverse pressure gradient created periodic vortex shedding from the separation. The general features of the time-averaged results were similar to experimental results for laminar separation bubbles. Comparisons were made with the ‘steady’ separation experiments of Gaster (1966). It was found that his ‘bursting’ occurs under the same conditions as our periodic shedding, suggesting that bursting is actually periodic shedding which has been time-averaged. The Strouhal number based on the shedding frequency, local free-stream velocity, and boundary-layer momentum thickness at separation was independent of the Reynolds number and the pressure gradient. A criterion for onset of shedding was established. The shedding frequency was the same as that predicted for the most amplified linear inviscid instability of the separated shear layer.

scholarly journals The numerical solution of the asymptotic equations of trailing edge flow

1974 ◽  
Vol 340 (1620) ◽  
pp. 91-111 ◽  
Keyword(s):  
Boundary Layer ◽  
Outer Layer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Trailing Edge ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Asymptotic Equations ◽  
Displacement Thickness

According to Stewartson (1969, 1974) and to Messiter (1970), the flow near the trailing edge of a flat plate has a limit structure for Reynolds number Re →∞ consisting of three layers over a distance O (Re -3/8 ) from the trailing edge: the inner layer of thickness O ( Re -5/8 ) in which the usual boundary layer equations apply; an intermediate layer of thickness O ( Re -1/2 ) in which simplified inviscid equations hold, and the outer layer of thickness O ( Re -3/8 ) in which the full inviscid equations hold. These asymptotic equations have been solved numerically by means of a Cauchy-integral algorithm for the outer layer and a modified Crank-Nicholson boundary layer program for the displacement-thickness interaction between the layers. Results of the computation compare well with experimental data of Janour and with numerical solutions of the Navier-Stokes equations by Dennis & Chang (1969) and Dennis & Dunwoody (1966).

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.

Boundary layer problems for the two-dimensional compressible Navier–Stokes equations

2016 ◽  
Vol 14 (01) ◽  
pp. 1-37 ◽  
Author(s):  
Shengbo Gong ◽  
Yan Guo ◽  
Ya-Guang Wang
Keyword(s):  
Boundary Layer ◽  
Laminar Flow ◽  
Stokes Equations ◽  
Degenerate Parabolic Equations ◽  
Navier Stokes ◽  
Large Space ◽  
Comparison Theory ◽  
Navier Stokes Equations ◽  
Boundary Layer Problems ◽  
Space Interval

We study the well-posedness of the boundary layer problems for compressible Navier–Stokes equations. Under the non-negative assumption on the laminar flow, we investigate the local spatial existence of solution for the steady equations. Meanwhile, we also obtain the solution for the unsteady case with monotonic laminar flow, which exists for either long time small space interval or short time large space interval. Moreover, the limit of these solutions with vanishing Mach number is considered. Our proof is based on the comparison theory for the degenerate parabolic equations obtained by the Crocco transformation or von Mises transformation.

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 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.

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.

Hybrid Solution for Developing Laminar Flow in Wavy-Wall Channels via Integral Transforms

2007 ◽  
Author(s):  
Roseane L. Silva ◽  
Carlos A. C. Santos ◽  
Joa˜o N. N. Quaresma ◽  
Renato M. Cotta
Keyword(s):  
Laminar Flow ◽  
Stokes Equations ◽  
Integral Transform ◽  
Integral Transforms ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Wavy Wall ◽  
Navier Stokes Equations ◽  
Convergence Behavior ◽  
Hybrid Solution

The analysis of two-dimensional laminar flow in the entrance region of arbitrarily shaped ducts is undertaken by application of the Generalized Integral Transform Technique (GITT) in the solution of the steady Navier-Stokes equations for incompressible flow. The streamfunction-only formulation is adopted, and a general filtering solution that adapts to the irregular contour is proposed to enhance the convergence behavior of the eigenfunction expansion. The case of a wavy-wall channel is then considered more closely in order to report some numerical results illustrating the expansions convergence behavior. In addition to reporting results of streamfunction, the product of friction factor-Reynolds number is also calculated and compared against results from discrete methods available in the literature for different Reynolds numbers and amplitudes of the wavy channel.

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