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Dimensionless Numbers—Similarity Parameters: A Look at the Name Holders

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Mechanics of Flow Similarities

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Abstract

In Chaps. 2, 3 and 4 we presented various methods with which a count of dimensional numbers, depending on different methods, can be derived. Obviously, Buckingham’s \(\varPi \)theorem has, there is no doubt, the capacity with the greatest possible extent. Historically many of the similarity quantities for the first time were formulated as a single event, in particular those known from the 18th and the first part of the 19th century. Therefore, the appearance of the dimensionless numbers obviously was an evolutionary process. Most of the power products, later noted as dimensionless numbers or similarity parameters, were established before the mathematical calculus of the analysis of dimension, Buckingham’s \(\varPi \)theorem, was developed.

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Notes

  1. 1.

    In the Chaps. 2, 3 and 4 we distinguish the quantities for dimensional reference values (quantities with a bar \( \Longrightarrow \bar{x}\)), dimensional variables (clean quantities \(\Longrightarrow x\)) and dimensionless variables (quantities with a prime \(\Longrightarrow x^\prime \)). In this chapter we mainly use dimensional reference values (\(\bar{x}\)). For convenience the bar is always dropped.

  2. 2.

    Contributions to mathematics: geometry, trigonometry, graph theory, number theory, topology, infinitesimal calculus, algebra. Contributions to physics: continuum mechanics (fluid dynamics), optics, lunar theory, stability theory, ballistics, equation of gyroscope.

  3. 3.

    Note: The wave length is that of the cross wave. For definition see [8, 9].

  4. 4.

    Society of German Engineers.

  5. 5.

    Sometimes also described as the ratio of lift force to friction force.

  6. 6.

    The transport coefficient of the momentum is replaced by the transport coefficient of the energy.

  7. 7.

    There exists an analogue Péclet number with respect to mass transport, namely \(\displaystyle Pe^\prime = \frac{lu}{D_{ij}}\) with \(D_{ij}\) the binary diffusivity.

  8. 8.

    This equation can be derived from the energy equation, where only the heat conduction and the dissipation play a role (by integration of \(\displaystyle \frac{\partial }{\partial y} \lambda \frac{\partial T}{\partial y}\)), and when the boundary conditions of the Couette flow are applied, see Fig. 6.14.

  9. 9.

    Due to tragic circumstances he obviously took his life.

  10. 10.

    The major field of application of the Damköhler numbers consists in the discipline of chemical engineering (chemical reactor technology).

  11. 11.

    \(\rho \) the density in front of the bow shock, normally the freestream value and \(\hat{\rho }\) the density behind the bow shock.

  12. 12.

    When the fluid elements pass the bow shock, the chemical reactions are initiated due to the strong jump of the thermodynamic variables pressure, density and temperature inside the shock.

  13. 13.

    The temperatures forming the difference \(\varDelta T\) are defined with respect to the flow case (heat transfer case) considered, e.g. for \(T_{ref}\) often the recovery temperature \(T_r\) is applied, which seems to be problematic, in particular when the wall temperature tends towards the recovery temperature.

  14. 14.

    The second category can be found on Sect. 6.9.1.

  15. 15.

    Some Prandtl numbers:

    \(Pr_{\,liquid \,metals}<< 1\), \(Pr_{gases} \thickapprox 0.7\), \(Pr_{fluids} \thickapprox 7\), \(Pr_{\, tough \, fluids, \, oils} \thickapprox 70\).

  16. 16.

    We distinguish between an averaged Nusselt number, called global Nusselt number \(Nu_{glob} \equiv Nu_L\), and a local Nusselt number \(Nu_{loc} \equiv Nu_x\), which depends on a x-coordinate representing the main stream direction.

  17. 17.

    The specific heat is denoted by \(c_p\) for gases at constant pressure and c for fluids.

  18. 18.

    See Eqs. (2.12), (2.39), (2.46), (3.9).

  19. 19.

    These were the famous Antibes Workshops, [41, 46].

  20. 20.

    This configuration was designed by the Space Department of Messerschmitt—Bölkow—Blohm, Ottobrunn, Germany.

  21. 21.

    The negative values of the Stanton number in Fig. 6.23 (below) are due to another definition of St compared to Eqs. (6.22)–(6.24), namely \(\displaystyle St = \frac{- \lambda \frac{\partial T}{\partial n}}{\rho _\infty u_\infty (h_0 - h_w)}\), which generates a negative sign.

  22. 22.

    These are some of the most critical aspects of hypersonic flight as the Columbia disaster has shown in 2003.

  23. 23.

    The fraying of the color filaments on the rear part of the upper side (leeward side) of the combat aircraft is due to the vortex break down and the subsequent turbulence onset.

  24. 24.

    For the flow along a flat plate with \(T_w = const\), \(\delta _m\) and \(\delta _T\) are identical for \(Pr = 1\).

  25. 25.

    The heat transfer coefficient between fluid and plate goes to infinity, from which follows, that the temperature of the fluid is equal to the temperature of the wall!.

  26. 26.

    The Reynolds number built with these values is \(Re = 3 \cdot 10^6\).

  27. 27.

    We have used here the Euler number in the form \(\displaystyle Eu = \frac{\varDelta p}{\rho u^2}\).

  28. 28.

    Also known as Laser-Doppler-Anemometry.

  29. 29.

    We use here the Greek letter \(\lambda \) to denote the free mean path, since it is common in the literature, knowing that \(\lambda \) is also used for the heat conduction coefficient.

  30. 30.

    National Aeronautics and Space Administration.

  31. 31.

    European Space Agency.

  32. 32.

    J. Ackeret, 1898–1981, obviously was the first, who in his habilitation treatise has named this ratio Mach number, [100].

  33. 33.

    In [98] is reported that E. Mach has believed, when he saw the photographs at the first time, that air mass was compressed in the front part (tip) of the projectile. In contrary to that P. Salcher shall have argued, that it looks more than a bow wave of a ship, which was obviously the right interpretation.

  34. 34.

    Archimedes, the famous Greek universal genius, about 287–212 B.C.

  35. 35.

    E. C. Bingham, an US chemist (1878–1945).

  36. 36.

    J.-B. Biot, a French physicist (1774–1862).

  37. 37.

    The reference length is mostly defined by the ratio between the volume of the body to the surface of the body \(l = V_{body} / S_{body}\).

  38. 38.

    The definition of the Nusselt number, Eq. (2.39), is formally the same as the Biot number, but in the Nusselt number definition the thermal conductivity is that of the fluid and in the Biot number definition that of the solid.

  39. 39.

    M. E. A. Bodenstein, a German physicist (1871–1942).

  40. 40.

    H. C. Brinkman, a Dutch physicist (1908–1961).

  41. 41.

    G. Galilei, the famous Italian astronomer (1564–1642).

  42. 42.

    P.-S. Laplace, the famous French mathematician and physicist (1749–1827).

  43. 43.

    W. Ohnesorge, a German engineer (1901–1976).

  44. 44.

    L. F. Richardson, a British mathematician and meteorologist (1881–1953).

  45. 45.

    In meteorology some variants of the Richardson number are used, which are denoted by flux Richardson number, gradient Richardson number, bulk Richardson number. For example, the gradient Richardson number characterizes the dynamic stability of a meteorological flow and reads \(\displaystyle Ri_g = \frac{g}{T_v} \cdot \frac{\partial T_v}{\partial z}/\left( \left( \frac{\partial U}{\partial z}\right) ^2 + \left( \frac{\partial V}{\partial z}\right) ^2\right) \).

  46. 46.

    A. Roshko, an US physicist and engineer (1923–2017).

  47. 47.

    T. K. Sherwood, an US engineer of chemistry (1903–1976).

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Weiland, C. (2020). Dimensionless Numbers—Similarity Parameters: A Look at the Name Holders. In: Mechanics of Flow Similarities. Springer, Cham. https://doi.org/10.1007/978-3-030-42930-0_6

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  • DOI: https://doi.org/10.1007/978-3-030-42930-0_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-42929-4

  • Online ISBN: 978-3-030-42930-0

  • eBook Packages: EngineeringEngineering (R0)

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