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Basic Fluid Mechanics

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Introduction to Wind Turbine Aerodynamics

Part of the book series: Green Energy and Technology ((GREEN))

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Abstract

Air regarded as an ideal gas may be described by its mass density \(\rho = dm/dV\).

Ich behaupte aber, daß in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist (Immanuel Kant, 1786) [1]. (However, I claim that in every special doctrine of nature, there can be only as much proper science as there is mathematics therein. (Ref Stanford Encyclopedia of Philosophy))

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Notes

  1. 1.

    A tensor may be represented by a n \(\times \) n matrix. However, like scalars and vectors, it is defined by its transformation rules under change of coordinates.

  2. 2.

    Heinrich Blasius, \(\star \) 09.08.1883, \(\dagger \) 24.04.1970, was one of L. Prandtl’s Prandtl, Ludwig (1875–1953)first PhD students. His work on forces and boundary layer flat plate theory is discussed in almost every textbook of fluid mechanics even today. Less known is that he taught over 50 years (1912–1970) at Ingenieurschule (Polytechnic—now University of Applied Sciences) Hamburg.

  3. 3.

    It may interest to note that here, the essential argument is to avoid singularities in terms of infinite velocities, whereas in other applications, this does not apply, for example, the infinite pressure at an inclined flat plate.

  4. 4.

    Here, we use Einstein’s summation convention, to sum over all dual-indexed variables: \(u_i \cdot u_i := \sum _{i=1}^N u_i \cdot u_i\).

  5. 5.

    Sometimes called chaos theory.

  6. 6.

    By random, we mean that there exist only probabilities for the field quantities, at least in the sense of classical statistical physics.

  7. 7.

    Many well-known physicists worked on turbulence: W. Heisenberg, L. Onsager, Carl-Friedrich von Weizsäcker, to name a few of them. An often repeated quote is that of Richard Feynman that turbulence is the most important unresolved problem in classical physics.

  8. 8.

    Especially on 2D turbulence for which he was awarded the Dirac Medal in 2003.

  9. 9.

    It is well known from the theory of linear partial differential equations that a number of problems are simplified if formulated by Fourier transform into wave number–frequency co-space. The NSE equation then reads as

    $$\begin{aligned}&k_{\beta } \cdot u_{\beta }(\mathbf{k},t) \end{aligned}$$
    (3.115)
    $$\begin{aligned}&\left( \frac{\partial }{\partial t} + \nu \cdot k^2 \right) u_{\alpha } (\mathbf{k},t) = M_{\alpha \beta \gamma } (\mathbf{k},t) \sum _{\mathbf{j}} u_{\beta } (\mathbf{j},t) \cdot u_{\gamma } (\mathbf{k} - \mathbf{j},t) \; \mathrm{and} \end{aligned}$$
    (3.116)
    $$\begin{aligned}&M_{\alpha \beta \gamma } (\mathbf{k},t) = (2 i)^{-1} \left( k_{\beta } D_{\alpha \gamma } (\mathbf{k}) + k_{\gamma } D_{\alpha \beta } (\mathbf{k}) \right) \end{aligned}$$
    (3.117)
  10. 10.

    Taylor’s frozen turbulence hypothesis has been used here. It states that time series may be used instead of spatially varying values.

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  1. Mechanical Engineering Department, University of Applied Sciences, Kiel, Germany

    A. P. Schaffarczyk

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Schaffarczyk, A.P. (2014). Basic Fluid Mechanics. In: Introduction to Wind Turbine Aerodynamics. Green Energy and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36409-9_3

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  • DOI: https://doi.org/10.1007/978-3-642-36409-9_3

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