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Optimum plane diffusers in laminar flow

Published online by Cambridge University Press:  26 April 2006

Hayri Çlabuk  and
Vijay Modi
Hayri Çlabuk
Affiliation:
Department of Mechanical Engineering, Columbia University, New York, NY 10027, USA
Vijay Modi
Affiliation:
Department of Mechanical Engineering, Columbia University, New York, NY 10027, USA
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Abstract

The problem of determining the profile of a plane diffuser (of given upstream width and length) that provides the maximum static pressure rise is solved. Two-dimensional, incompressible, laminar flow governed by the steady-state Navier-Stokes equations is assumed through the diffuser. Recent advances in computational resources and algorithms have made it possible to solve the ‘direct’ problem of determining such a flow through a body of known geometry. In this paper, a set of ‘adjoint’ equations is obtained, the solution to which permits the calculation of the direction and relative magnitude of change in the diffuser profile that leads to a higher pressure rise. The direct as well as the adjoint set of partial differential equations are obtained for Dirichlet-type inflow and outflow conditions. Repeatedly modifying the diffuser geometry with each solution to these two sets of equations with the above boundary conditions would in principle lead to a diffuser with the maximum static pressure rise, also called the optimum diffuser. The optimality condition, that the shear stress all along the wall must vanish for the optimum diffuser, is also recovered from the analysis. It is postulated that the adjoint set of equations continues to hold even if the computationally inconvenient Dirichlet-type outflow boundary condition is replaced by Neumann-type conditions. It is shown that numerical solutions obtained in this fashion do satisfy the optimality condition.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 237 , April 1992 , pp. 373 - 393
Copyright
© 1992 Cambridge University Press

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References

Beavers, G. S., Sparrow, E M. & Magnuson, R. A. 1970 Experiments on hydrodynamically developing flow in rectangular ducts of arbitrary aspect ratio. Intl J. Heat Mass Transfer 13, 689–701.Google Scholar
cLabuk, H. 1991 Optimum design in fluid mechanics. Ph.D. thesis, Columbia University Department of Mechanical Engineering.
cLabuk, H. & Modi, V. 1990 Shape optimization analysis: first- and second-order necessary conditions. Optimal Control Applics Meth. 11, 173–190.Google Scholar
Chang, P. K. 1976 Control of Flow Separation. McGraw Hill.
Chorin, A. J. 1967 A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26.Google Scholar
Garabedian, P. R. & Korn, D. G. 1971 Numerical design of transonic airfoils. In Proc. SYNSPADE, 1970 (ed. B. Hubbard), pp. 253–271. Academic.
Garabedian, P. & McFadden, G. B. 1982 Computational fluid dynamics of airfoils and wings. In Proc. Symp. on Transonic, Shock, and Multidimensional Flows, Madison, 1981 (ed. R. Meyer), pp. 1–16. Academic.
Giles, M., Drela, M. & Thompkins, W. T. 1985 Newton solution of direct and inverse transonic Euler equations. AIAA Paper 85–1530; Proc. AIAA 7th Computational Fluid Dynamics Conf., Cincinnati, pp. 394–402.Google Scholar
Glowinski, R. & Pironneau, O. 1975 On the numerical computation of the minimum drag profile in laminar flow. J. Fluid Mech. 72, 385–389.Google Scholar
Goldstein, R. J. & Kreid, D. K. 1967 Measurement of laminar flow development in a square duct using a laser-Doppler flowmeter. Trans. ASME E: J. Appl. Mech. 89, 813–818.Google Scholar
Henne, P. A. 1980 An inverse transonic wing design method. AIAA paper 80–0330.Google Scholar
Hicks, R. M. & Henne, P. A. 1979 Wing design by numerical optimization. AIAA paper 79-0080.Google Scholar
Hirsch, C. 1990 Numerical Computation of Internal and External Flows, vol. 2, p. 603. John Wiley.
Jameson, A. 1988 Aerodynamic design via control theory. NASA ICASE Rep. 88–64.Google Scholar
Jameson, A., Schmidt, W. & Turkel, E. 1981 Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA paper 81–1259.Google Scholar
Kreiss, H.-O. & Lorenz, J. 1989 Initial-Boundary Value Problems and the Navier-Stokes Equations, pp. 347–349. Academic.
Lighthill, M. J. 1945 A new method of two dimensional aerodynamic design. Aero. Res. Counc. R and M 2112.Google Scholar
Lions, J. L. 1968 ContoCle Optimal de SysteGmes GoverneAs par des Equations aux DeriveAes Partielles. Paris:Dunod.
Lions, J. L. & Magenes, E. 1967 ProbleAmes aux Limites Non-homogeGnes, vol. 1.Paris:Dunod.
McFadden, G. B. 1979 An artificial viscosity method for the design of supercritical airfoils. New York University Rep. C00-3077-158.Google Scholar
Pironneau, O. 1973 On optimum profiles in Stokes flow. J. Fluid Mech. 59, 117–128.Google Scholar
Pironneau, O. 1974 On optimum design in fluid mechanics. J. Fluid Mech. 64, 97–110.Google Scholar
Rizzi, A. & Eriksson, L.-E. 1984 Computation of flow around wings based on the Euler equations. J. Fluid Mech. 148, 45–71.Google Scholar
Rizzi, A. & Eriksson, L.-E. 1985 Computation of inviscid incompressible flow with rotation. J. Fluid Mech. 153, 275–312.Google Scholar
Sung, C.-H. 1987 An explicit Runge-Kutta method for 3D turbulent incompressible flows. David W. Taylor Naval Ship Research and Development Center, Ship Hydromechanics Department Rep. DTNSRDC/GHD-1244-01.Google Scholar
Swanson, R. C. & Turkel, E. 1985 A multistage time-stepping scheme for the Navier-Stokes equations. AIAA paper 85–35.Google Scholar
Thompson, J. F., Thames, F. C. & Mastin, C. W. 1974 Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies. J. Comput. Phys. 15, 299–319.Google Scholar
Tranen, J. L. 1974 A rapid computer aided transonic airfoil design method. AIAA paper 74–501.Google Scholar
Tuck, E. O. 1968 Proc. Conf. Hydraul. Fluid Mech., p. 29. Australia: Institute of Engineers.
Vatsa, V. N. 1986 Accurate solutions for transonic viscous flow over finite wings. AIAA paper 86–1052.Google Scholar
Volpe, G. & Melnik, R. E. 1986 The design of transonic aerofoils by a well posed inverse method. Intl J. Numer. Methods Engng 22, 341–361.Google Scholar
Watson, S. R. 1971 Towards the minimum drag on a body of given volume in slow viscous flow. J. Inst. Maths Applics. 7, 367–376.Google Scholar