Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-01T00:01:01.512Z Has data issue: false hasContentIssue false
  • English
  • Français

Transonic flows of single-phase supercritical fluids over thin airfoils

Published online by Cambridge University Press:  17 March 2021

Zvi Rusak Open the ORCID record for Zvi Rusak [Opens in a new window]  and
Akashdeep Singh Virk Open the ORCID record for Akashdeep Singh Virk [Opens in a new window]
Zvi Rusak
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY12180USA
Akashdeep Singh Virk*
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY12180USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A small-disturbance model for a steady, two-dimensional, inviscid and transonic flow of a single-phase real gas around a thin airfoil is presented. The approach explores the nonlinear interactions among near-sonic speed of the flow, small thickness ratio of the airfoil and upstream properties of the fluid. The gas thermodynamic properties are related by a general equation of state. Information about thermodynamic modelling of the gas is lumped into one similarity parameter, $K_G$, related to the fundamental derivative of gas dynamics. The flow field is described by a modified transonic small-disturbance problem. The theory applies to any working fluid of interest. Model problems are derived for steam flows described by the perfect, van der Waals, virial and Redlich–Kwong gas equations of state. Predictions are compared according to the various gas models under various free-stream operating conditions from low subcritical to high supercritical thermodynamic states to gain insights into the sensitivity of the small-disturbance problem solution to thermodynamic modelling of the gas. Results show that transonic flows are independent of gas modelling at low subcritical thermodynamic conditions. However, at near-critical and supercritical thermodynamic conditions, transonic flow behaviour is significantly sensitive to gas modelling and variations of $K_G$. The upstream flow critical Mach number increases as the flow approaches thermodynamic critical state and a wider range of upstream Mach numbers can be found where pressure drag is zero. However, at supercritical conditions, $K_G$ increases, resulting in lower critical Mach numbers and higher pressure drags.

JFM classification

Compressible Flows: Shock waves Compressible Flows: Gas dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A61
Check for updates
Copyright
© The Author(s), 2021. Published by 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.)

References

REFERENCES

Bakhtar, F., Ebrahimi, M. & Webb, R.A. 1995 On the performance of a cascade of turbine rotor tip section blading in nucleating steam: part 1: surface pressure distributions. Proc. Inst. Mech. Engrs 209 (2), 115–124.Google Scholar
Clifford, A.A. & Williams, J. R. 2000 Introduction to supercritical fluids and their applications. In Supercritical fluid methods and Protocols, pp. 1–16. Humana Press.CrossRefGoogle Scholar
Cole, J.D. & Cook, L.P. 1986 Transonic Aerodynamics, 1st edn, North-Holland Series in Applied Mathematics and Mechanics, vol. 30. Elsevier Science Publishers B.V.Google Scholar
Colonna, P., Nannan, N.R., Guardone, A.L. & Van der Stelt, T.P. 2009 On the computation of the fundamental derivative of gas dynamics using equations of state. Fluid Phase Equilib. 286 (1), 43–54.CrossRefGoogle Scholar
Cramer, M.S. 1996 Transonic flows of arbitrary gases. In Theroretical Fluid Mechanics Conference, p. 2116. AIAA, Inc.CrossRefGoogle Scholar
Cramer, M.S., Whitlock, S.T. & Tarkenton, G.M. 1996 Transonic and boundary layer similarity laws in dense gases. Trans. ASME: J. Fluids Engng 118, 481–485.Google Scholar
Fomin, N.A. 2010 110 years of experiments on shock tubes. J. Engng Phys. Thermophys. 83 (6), 1118–1135.CrossRefGoogle Scholar
Guderley, K.G. 1947 Considerations on the structure of mixed subsonic-supersonic flow patterns. Tech. Rep. F-TR-2168-ND. HQ Air Materiel Command, Wright Field, Dayton, Ohio.Google Scholar
Hawthorne, W.R. (Ed.) 2017 Aerodynamics of Turbines and Compressors. (HSA-1), vol. 1. Princeton University Press.Google Scholar
Kluwick, A. 1993 Transonic nozzle flow of dense gases. J. Fluid Mech. 247, 661–668.CrossRefGoogle Scholar
Kluwick, A. & Cox, E.A. 2018 Steady small-disturbance transonic dense gas flow past two-dimensional compression/expansion ramps. J. Fluid Mech. 848, 756–787.CrossRefGoogle Scholar
Kluwick, A. & Cox, E.A. 2019 Weak shock reflection in channel flows for dense gases. J. Fluid Mech. 874, 131–157.CrossRefGoogle Scholar
Krupp, J.A. & Murman, E.M. 1972 Computation of transonic flows past lifting airfoils and slender bodies. AIAA J. 10 (7), 880–886.CrossRefGoogle Scholar
Kuethe, A.M. & Chow, C.-Y. 1976 Foundations of Aerodynamics. John Wiley and Sons, Inc.Google Scholar
Lee, J.-C. & Rusak, Z. 2000 Parametric investigation of nonadiabatic compressible flow around airfoils. Phys. Fluids 13 (1), 315–323.CrossRefGoogle Scholar
Moran, M.J., Shapiro, H.N, Boettner, D.D. & Bailey, M.B. 2014 Fundamentals of Engineering Thermodynamics, 8th edn. John Wiley and Sons, Inc.Google Scholar
Murman, E.M. & Cole, J.D. 1971 Calculation of plane steady transonic flows. AIAA J. 9 (1), 114–121.CrossRefGoogle Scholar
Nixon, D. (Ed.) 1982 Transonic Aerodynamics. American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar
Nixon, D. (Ed.) 1989 Unsteady Transonic Aerodynamics. American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar
Oswatitsch, K. & Zierep, J. 1960 Das problem des senkrechten stoles an der gekriimmten wand. J. Appl. Math. Mech. 40, 143–144.Google Scholar
Rusak, Z. 1993 Transonic flow around the leading edge of a thin airfoil with a parabolic nose. J. Fluid Mech. 248, 1–26.CrossRefGoogle Scholar
Rusak, Z. & Lee, J.-C. 2000 a Transonic flow of moist air around a thin airfoil with non-equilibrium and homogeneous condensation. J. Fluid Mech. 403, 173–199.CrossRefGoogle Scholar
Rusak, Z. & Lee, J.-C. 2000 b Transonic small-disturbance theory-a tool for aerodynamic analysis and design. Can. Aeronaut. Space J. 46 (2), 74–86.Google Scholar
Rusak, Z. & Wang, C.W. 1997 Transonic flow of dense gases around an airfoil with a parabolic nose. J. Fluid Mech. 346, 1–21.CrossRefGoogle Scholar
Sattelmayer, T. 1997 Combustion chamber for gas turbine engine. U.S. Patent 5,626,017.Google Scholar
Schnerr, G.H. & Dohrmann, U. 1990 Transonic flow around airfoils with relaxation and energy supply by homogeneous condensation. AIAA J. 28 (7), 1187–1193.CrossRefGoogle Scholar
Schwendeman, D.W., Kropinski, M.C.A. & Cole, J.D. 1993 On the construction and calculation of optimal nonlifting critical airfoils. Z. Angew. Math. Phys. 44 (3), 556–571.CrossRefGoogle Scholar
Thompson, P.A. 1971 A fundamental derivative in gasdynamics. Phys. Fluids 14 (9), 1843–1849.CrossRefGoogle Scholar
Virk, A.S. & Rusak, Z. 2019 A small disturbance model for transonic flow of pure steam with condensation. Trans. ASME: J. Fluid Engng 141 (3), 031204.Google Scholar
Virk, A.S. & Rusak, Z. 2020 Near-sonic pure steam flow with real-gas effects and non-equilibrium and homogeneous condensation around thin airfoils. J. Fluid Mech. 884, A30.CrossRefGoogle Scholar
Von Kármán, T. 1947 The similarity law of transonic flow. J. Math. Phys. 26 (1–4), 182–190.CrossRefGoogle Scholar
Wagner, W. & Kretzschmar, H.J. 2007 International Steam Tables-Properties of Water and Steam based on the Industrial Formulation IAPWS-IF97: Tables, Algorithms, Diagrams, and CD-ROM Electronic Steam Tables-All of the Equations of IAPWS-IF97 Including a Complete Set of Supplementary Backward Equations for Fast Calculations of Heat Cycles, Boilers, and Steam Turbines. 2nd edn. Springer Science & Business Media.Google Scholar