Abstract
In this chapter we (re-)familiarize the reader with concepts from mathematics, thermodynamics, and aerodynamics that are fundamental to the methods presented in the remainder of the book. This includes a short review of vector algebra and partial-differential equations to provide the mathematical insight into the governing equations of fluid flow. Examples include the one-dimensional wave equation and the one-dimensional heat equation. A basic review of thermodynamics is given including the equation of state, the first law and the second law of thermodynamics. Also the isentropic relations between pressure, density and temperature are derived. In the aerodynamics section, the Navier-Stokes equations are derived in integral and derivative form. To simulate transonic flow, often simplifications of the Navier-Stokes equations are used. Therefore, it is shown how the Reynolds-Averaged Navier-Stokes (RANS) equations can be obtained and how the k-epsilon turbulence model can be used to close the RANS equations. For inviscid flow the Euler equations, and the full-potential equation are derived. Using examples from the literature, it is shown how well each of these models can predict the outcome of aerodynamic experiments in transonic conditions. This chapter contains 11 examples and concludes with 29 practice problems.
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Notes
- 1.
The speed of sound will be derived in Sect. 2.5.
- 2.
The fundamental theorem of calculus specifies the relationship between the two central operations of calculus: differentiation and integration [11].
- 3.
This theorem applies to any vector field, not only \({\varvec{V}}\) but also mass flow or force fields.
- 4.
The Kronecker delta is defined as follows: \(\delta _{\textit{ij}}={\left\{ \begin{array}{ll} 1\quad \text {for}~i=j\\ 0\quad \text {for}~i\ne j\end{array}\right. }\).
References
Anderson, J.: Computational Fluid Dynamics: The Basics and Applications, 1st edn. McGraw Hill, New York (1995)
Anderson, J.: Modern Compressible Flow with Historic Perspective, 3rd edn. McGraw Hill, New York (2003)
Anderson, J.: Fundamentals of Aerodynamics, 5th edn. McGraw Hill, New York (2010)
Anon.: Direct numerical simulation (DNS) wikipedia. http://en.wikipedia.org/wiki/Direct_numerical_simulation (2012)
Blackwell, J.A.: Scale effects on supercritical airfoils. In: Proceedings of ICAS, pp. 370–283 (1978)
Cebeci, T.: Analysis of Turbulent Flows. Elsevier, Amsterdam (2004)
Cook, P.H., McDonald, M.A., Firmin, M.C.P.: Aerofoil RAE 2822—pressure distributions, and boundary layer and wake measurements. In: AGARD-138 (1979)
Drela, M., Giles, M.B.: Viscous-inviscid analysis of transonic and low Reynolds number airfoils. AIAA J. 25(10), 1347–1355 (1987). doi:10.2514/3.9789
Farokhi, S.: Aircraft Propulsion, 2nd edn. Wiley, Chichester, UK (2014)
Ferrari, C., Tricomi, F.G.: Transonic Aerodynamics. Academic Press, New York (1968)
Keener, J.P.: Principles of Applied Mathematics. Westview Press, Boulder (1999)
Launder, B.E., Spalding, D.B.: Lectures in Mathematical Models of Turbulence. Academic Press, London (1972)
Mohammadi, B., Pironneau, O.: Analysis of the K-epsilon Turbulence Model. Wiley, Chichester, UK (1994)
Prandtl, L.: Über die ausgebildete turbulenz. In: Proceedings of the Second International Congress of Applied Mechanics, pp. 62–75. Orell Füssli Verlag, Zurich (1927)
Reynolds, O.: On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philos. Trans. R. Soc. Lond. 186, 123–164 (1895)
Ruijgrok, G.J.J.: Elements of Airplane Acoustics. Eburon, Delft (2003)
Schlichting, H., Gestern, K.: Boundary Layer Theory, 8th edn. Springer, Berlin (1999)
Strauss, W.A.: Partial Differential Equations: An Introduction. Wiley, Chichester (1992)
Tannehill, J.C., Anderson, D.A., Pletcher, R.H.: Computational Fluid Mechanics and Heat Transfer, 2nd edn. Taylor & Francis, Philadelphia (1997)
Van Muijden, J., Broekhuizen, A.J., Van der Wees, A.J., Van der Vooren, J.: Flow analysis and drag prediction for transonic transport wing/body configurations using a viscous-inviscid interaction type method. In: Proceedings of the 19th ICAS Congress. Anaheim, California (1994)
Van Wylen, G., Sonntag, R.: Fundamentals of Classical Thermodynamics. Wiley, Chichester, UK (1973)
Wilcox, D.: Turbulence Modeling for CFD, 2nd edn. DCW Industries, Canada (1998)
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Vos, R., Farokhi, S. (2015). Review of Fundamental Equations. In: Introduction to Transonic Aerodynamics. Fluid Mechanics and Its Applications, vol 110. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9747-4_2
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