Skip to main content Accessibility help
Hostname: page-component-59b7f5684b-8dvf2 Total loading time: 0.316 Render date: 2022-09-28T06:52:24.550Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

A steady separated viscous corner flow

Published online by Cambridge University Press:  26 April 2006

Robert McLachlan
Applied Mathematics, California Institute of Technology, Pasadena, CA 91125. USA Current address: Program in Applied Mathematics, Campus Box 526, University of Colorado at Boulder, CO 80309–0526, USA.


An example is presented of a separated flow in an unbounded domain in which, as the Reynolds number becomes large, the separated region remains of size 0(1) and tends to a non-trivial Prandtl-Batchelor flow. The multigrid method is used to obtain rapid convergence to the solution of the discretized Navier-Stokes equations at Reynolds numbers of up to 5000. Extremely fine grids and tests of an integral property of the flow ensure accuracy. The flow exhibits the separation of a boundary layer with ensuing formation of a downstream eddy and reattachment of a free shear layer. The asymptotic (’triple deck’) theory of laminar separation from a leading edge, due to Sychev (1979), is clarified and compared to the numerical solutions. Much better qualitative agreement is obtained than has been reported previously. Together with a plausible choice of two free parameters, the data can be extrapolated to infinite Reynolds number, giving quantitative agreement with triple-deck theory with errors of 20% or less. The development of a region of constant vorticity is observed in the downstream eddy, and the global infinite-Reynolds-number limit is a Prandtl-Batchelor flow; however, when the plate is stationary, the occurrence of secondary separation suggests that the limiting flow contains an infinite sequence of eddies behind the separation point. Secondary separation can be averted by driving the plate, and in this case the limit is a single-vortex Prandtl-Batchelor flow of the type found by Moore, Saffman & Tanveer (1988); detailed, encouraging comparisons are made to the vortex-sheet strength and position. Altering the boundary condition on the plate gives viscous eddies that approximate different members of the family of inviscid solutions.

Research Article
© 1991 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Batchelor, G. K. 1956 A proposal concerning laminar wakes behind bluff bodies at large Reynolds number. J. Fluid Mech. 1, 380398.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Mechanics. Cambridge University Press.
Brandt, A. 1984 Multigrid methods: 1984 Guide with Applications to Fluid Dynamics. Rehovot, Israel: Weizmann Institute of Science.
Brodetsky, S. 1923 Discontinuous fluid motion past circular and elliptic cylinders. Proc. R. Soc. Lond, A. 102, 542553.Google Scholar
Carpenter, M. & Homsy, G. M. 1990 High Marangoni number convection in a square cavity: Part II. Phys. Fluids A 2, 137149.Google Scholar
Carrier, G. F. & Lin, C. O. 1948 On the nature of the boundary layer near the leading edge of a flat plate. Q. Appl. Maths 6, 6368.Google Scholar
Cheng, H. K. 1984 Laminar separation from airfoil beyond trailing edge stall. AIAA Paper 84–1612.Google Scholar
Cheng, J. K. & Lee, C. J. 1985 Laminar separation studied as an airfoil problem. In Proc. Symp. Numer. Phys. Aspects Aerodyn. Flows III (ed. T. Cebeci). Springer.
Cheng, J. K. & Smith, F. T. 1982 The influence of airfoil thickness and Reynolds number on separation. J. Appl. Math. Phys. 33, 151180.Google Scholar
Daniels, P. G. 1979 Laminar boundary layer reattachment in supersonic flow. J. Fluid Mech. 90, 289303.Google Scholar
Dennis, S. C. R. & Smith, F. T. 1980 Steady flow through a channel with a symmetrical constriction in the form of a step. Proc. R. Soc. Lond. A 372, 393414.Google Scholar
Dommelen, L. L. van & Shen, S. F. 1984 Interactive separation from a fixed wall. In Proc. Symp. Numer. Phys. Aspects Aerodyn. Flows, 2nd Long Beach Conf. (ed. T. Cebeci). Springer.
Elliott, J. W., Smith, F. T. & Cowley, S. J. 1983 Breakdown of boundary layers: (i) on moving surfaces; (ii) in semi-similar unsteady flow; (iii) in fully unsteady flow, Geophys. Astrophys. Fluid Dyn. 25, 77138.Google Scholar
Fornberg, B, 1980 A numerical study of steady viscous flow past a circular cylinder. J. Fluid Mech. 98, 819855.Google Scholar
Fornberg, B. 1985 Steady viscous flow past a circular cylinder up to Reynolds number 600. J. Comput. Phys. 61, 297320.Google Scholar
Korolev, C. P. 1980 TsAGI, Uch. Zap. 11, 716.
Leal, L. G. 1973 Steady separated flow in a linearly decelerated free stream. J. Fluid Mech. 59, 513535.Google Scholar
McLachlan, R. I. 1990 Separated viscous coiner flows via multigrid. Ph.D. thesis, California Institute of Technology.
McLachlan, R. I. 1991 The boundary layer on a finite flat plate. Phys. Fluids A 3, 341348.Google Scholar
Messiter, A. F. 1975 Laminar separation - a local asymptotic flow description for constant pressure downstream. AGARD Symp. on Separated Flows.
Milos, F. S., Acrivos, A. & Kim, J. 1987 Steady flow past sudden expansions at large Reynolds number II. Navier—Stokes solutions for the cascade expansion. Phys. Fluids 30, 718.Google Scholar
Moore, D. W., Saffman, P. G. & Tanveer, S. 1988 The calculation of some Batchelor flows: the Sadovskii vortex and rotational corner flow. Phys. Fluids 31, 978990 (referred to herein as MST).Google Scholar
Pépin, F. M. 1990 Simulation of the flow past an impulsively started cylinder using a discrete vortex method. Ph.D. thesis, GALCIT, California Institute of Technology.
Prandtl, L. 1905 Motion of fluids with very little viscosity. Göttingen; transl. in NACA TM-452 (1928).Google Scholar
Schreiber, J. & Keller, H. B. 1983 Driven cavity flows by efficient numerical techniques. J. Comput. Phys. 49, 310333.Google Scholar
Smith, F. T. 1977 The laminar separation of an incompressible fluid streaming past a smooth surface. Proc. R. Soc. Lond. A 356, 443463.Google Scholar
Smith, F. T. 1979 Laminar flow of an incompressible fluid past a bluff body: the separation, reattachment, eddy properties and drag. J. Fluid Mech. 92, 171205.Google Scholar
Smith, F. T. 1981 Comparisons and comments concerning recent calculations for flow past a circular cylinder. J. Fluid Mech. 113, 407110.Google Scholar
Smith, F. T. 1986 Steady and unsteady boundary-layer separation. Ann. Rev. Fluid Mech. 18, 197220.Google Scholar
Smith, F. T. 1987 Theory of high-Reynolds-number flow past a blunt body. In Studies of Vortex Dominated Flows (ed. M. Y. Hussaini & X. Salas). Springer.
Smith, J. H. B. 1982 The representation of planar separated flow by regions of uniform vorticity. In Vortex Motion, pp. 157172. Vieweg.
Smith, J. H. B. 1986 Vortex flows in aerodynamics. Ann. Rev. Fluid Mech. 18, 221242.Google Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14, 145.Google Scholar
Suh, Y. K. & Liu, C. S. 1990 Study on the flow structure around a flat plate in a stagnation flow field. J. Fluid Mech. 214, 469487.Google Scholar
Sychev, V. V. 1972 Concerning laminar separation. Izv. Akad. Nauk. SSSR, Mekh. Zhid. Gaza 3, 4759.Google Scholar
Sychev, V. V. 1979 Boundary layer separation from a plane surface. TsAGI, Uch. Zap. 9, 2029; (transl. in NASA TM-75828).Google Scholar
Velde, E. F. van De & Keller, H. B. 1987 The parallel solution of nonlinear elliptic equations. In Parallel Computations and Their Impact on Mechanics fed. A. K. Noor), pp. 127153.
Woods, L. C. 1954 Aeronaut. Q. 5, 176184.
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A steady separated viscous corner flow
Available formats

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

A steady separated viscous corner flow
Available formats

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

A steady separated viscous corner flow
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *