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Three-dimensional viscous flows with large secondary velocity

Published online by Cambridge University Press:  20 April 2006

W. R. Briley  and
H. Mcdonald
W. R. Briley
Affiliation:
Scientific Research Associates, Inc., Glastonbury, CT 06033, U.S.A.
H. Mcdonald
Affiliation:
Scientific Research Associates, Inc., Glastonbury, CT 06033, U.S.A.
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Abstract

A new system of approximating equations is derived for three-dimensional steady viscous compressible flows in which a primary flow direction is present, but in which both transverse velocity components can be large. If the transverse velocity vector which corrects a given potential flow is first decomposed into ‘potential’ and ‘rotational’ vector components, then a re-examination of three-dimensional boundary-layer theory shows that both components (vϕ, wϕ) of the potential-velocity vector may be assumed small, whereas both components (vψ, wψ) of the rotational-velocity vector and hence of the composite secondary flow (v, w) can remain of order unity. An assumption of small scalar potential then leads to a system of governing equations whose characteristic polynomial has a non-elliptic form for arbitrary Mach number, without introducing any direct approximation of either streamwise or transverse pressure gradient terms. These non-elliptic equations can be solved very economically as a well-posed initial/boundary-value problem. Computed results for laminar subsonic flow in a curved square duct confirm the small scalar-potential approximation for both large (R/d = 100) and small (R/d = 2) radius of curvature. Other computations for R/d = 2.3 are in good agreement with the measurements of Taylor, Whitelaw & Yianneskis (1980).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 144 , July 1984 , pp. 47 - 77
Copyright
© 1984 Cambridge University Press

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References

Anderson, O. L. 1980 Calculation of internal viscous flows in axisymmetric ducts at moderate to high Reynolds numbers. Comp. Fluids 8, 391–441.Google Scholar
Anderson, O. L. & Hankins, G. B. 1981 Development of a parabolic finite difference method for 3-D high Reynolds number viscous internal flows. In Computers in Flow Predictions and Fluid Dynamics Experiments, 119–128. ASME.
Baker, A. J. & Orzechowski, J. A. 1983 An interaction algorithm for three-dimensional turbulent subsonic aerodynamic juncture region flow. AIAA J. 21, 524–533.Google Scholar
Blottner, F. G. 1977 Numerical solution of slender channel laminar flows. Comp. Meth. Appl. Mech. & Engng 11, 319–339.Google Scholar
Briley, W. R. 1974 Numerical method for predicting three-dimensional steady viscous flow in ducts. J. Comp. Phys. 14, 8–28.Google Scholar
Briley, W. R. & McDonald, H. 1977 Solution of the multidimensional compressible Navier—Stokes equations by a generalized implicit method. J. Comp. Phys. 23, 372–397.Google Scholar
Briley, W. R. & McDonald, H. 1980 On the structure and use of linearized block ADI and related schemes. J. Comp. Phys. 34, 54–73.Google Scholar
Briley, W. R., McDonald, H. & Shamroth, S. J. 1983 A low Mach number Euler formulation and application to time-iterative LBI schemes, AIAA J. 21, 1467–1469.Google Scholar
Caretto, L. S., Curr, R. M. & Spalding, D. B. 1973 Two numerical methods for three-dimensional boundary layers. Comp. Meth. Appl. Mech. & Engng 1, 39–57.Google Scholar
Courant, R. & Hilbert, D. 1966 Methods of Mathematical Physics, vol. II. Interscience.
Davis, R. T. & Rubin, S. G. 1980 Non-Navier—Stokes viscous flow computations. Comp. Fluids 8, 103–131.Google Scholar
Garabedian, P. R. 1964 Partial Differential Equations. Wiley.
Ghia, K. N., Ghia, U., Shin, C. T. & Reddy, D. R. 1981 Multigrid simulation of asymptotic curved-duct flows using a semi-implicit numerical technique. In Computers in Flow Predictions and Fluid Dynamics Experiments, pp. 11–25. ASME.
Ghia, K. N. & Sokhey, J. S. 1977 Laminar incompressible viscous flow in curved ducts of regular cross-sections. Trans ASME I: J. Fluids Engng 99, 640–648.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. 34, 813–818.Google Scholar
Hah, C. 1983 A Navier—Stokes analysis of three-dimensional tubulent flows inside turbine blade rows at design and off-design conditions. ASME Paper 83-GT-40.Google Scholar
Hawthorne, W. R. 1951 Secondary circulation in fluid flow. Proc. R. Soc. Lond. A 206, 374–387.Google Scholar
Horlock, J. H. & Lakshminarayana, B. 1973 Secondary flows: theory, experiment, and application in turbomachinery aerodynamics. Ann. Rev. Fluid Mech. 5, 247–280.Google Scholar
Humphrey, J. A. C., Taylor, A. M. K. & Whitelaw, J. H. 1977 Laminar flow in a square duct of strong curvature. J. Fluid Mech. 83, 509–527.Google Scholar
Johnston, J. P. 1960 On the three-dimensional turbulent boundary layer generated by secondary flow. Trans. ASMED: J. Basic Engng 82, 233–248.Google Scholar
Khosla, P. K. & Rubin, S. G. 1983 A composite velocity procedure for the compressible Navier—Stokes equations. AIAA J. 21, 1546–1551.Google Scholar
Kreskovsky, J. P., Briley, W. R. & McDonald, H. 1981 Analysis and Computation of Three-Dimensional Flow in Strongly Curved Ducts. In Computers in Flow Predictions and Fluid Dynamics Experiments, pp. 129–140. ASME.
Lakshminarayana, B. & Horlock, J. H. 1973 Generalized expressions for secondary vorticity using intrinsic coordinates. J. Fluid Mech. 59, 97–115.Google Scholar
Levy, R., Briley, W. R. & McDonald, H. 1983 Viscous primary/secondary flow analysis for use with nonorthogonal coordinate systems. AIAA Paper 83–0556.Google Scholar
Moore, J. & Moore, J. G. 1979 A calculation procedure for three-dimensional viscous compressible duct flow. Parts I and II. Trans. ASME I: J. Fluids Engng 101, 415–428.Google Scholar
Patankar, S. V. & Spalding, D. B. 1972 A calculation procedure for heat, mass, and momentum transfer in three-dimensional parabolic flows. Intl J. Heat Mass Transfer 15, 1787–1805.Google Scholar
Pratap, V. S. & Spalding, D. B. 1976 Fluid flow and heat transfer in three-dimensional duct flows. Intl J. Heat Mass Transfer 19, 1183–1187.Google Scholar
Pulliam, T. H. & Steger, J. L. 1980 Implicit finite-difference simulations of three-dimensional compressible flow. AIAA J. 18, 159–167.Google Scholar
Ralston, A. 1965 A First Course in Numerical Analysis, McGraw-Hill.
Roberts, G. O. 1971 Computational meshes for boundary layer problems. In Proc. 2nd Intl. Conf. Num. Meth. Fluid Dyn. (ed. M. Holt). Lecture Notes in Physics, vol. 8, pp. 171–177. Springer.
Rubin, S. G., Khosla, P. K. & Saari, S. 1977 Laminar flow in rectangular channels. Parts I and II. Comp. Fluids 5, 151–173.Google Scholar
Rubin, S. G. & Reddy, D. R. 1983 Global PNS solutions for laminar and turbulent flow. AIAA Paper 83–1911.Google Scholar
Schiff, L. B. & Steger, J. L. 1980 Numerical simulation of steady supersonic viscous flow. AIAA J. 18, 1421–1430.Google Scholar
Stanitz, J. D., Osborn, W. M. & Mizisin, J. 1953 An experimental investigation of secondary flow in an accelerating rectangular elbow with 90 degrees of turning. NACA TN 3015.Google Scholar
Taylor, A. M. K. P., Whitelaw, J. H. & Yianneskis, M. 1982 Curved ducts with strong secondary motion; velocity measurements of developing laminar and Turbulent flows. Trans ASME I: J. Fluids Engng 104, 350–358.Google Scholar
Vigneron, Y. C., Rakich, J. V. & Tannehill, J. C. 1978 Calculation of supersonic viscous flow over delta wings with sharp subsonic leading edges. AIAA Paper 78–1137.Google Scholar
Williams, G. P. 1969 Numerical integration of the three-dimensional Navier—Stokes equations for incompressible flow. J. Fluid Mech. 37, 727–750.Google Scholar