Abstract
In the last two Chaps. 10 and 11, some of the elements of the foundations of turbulence theory and turbulent closure procedures were laid down. It was shown, how the Reynolds averaged field equations could be derived from the full balance laws of physics by applying to them a filter operation 〈•〉. The properties of the filter were so assigned to include the ergodic hypothesis, which, in the averaging process, means that multiple averaging is an invariant operation2, i.e., does not lead to new results, viz. 〈f〉 = 〈〈• • 〈f〉 • •〉〉 for any physical quantity f. As for the physical balance laws of mass, momenta and energy this procedure led to the so-called Reynolds averaged balance equations (of Navier-Stokes and Fourier).
The text for this chapter was thorougly read and criticised by Amsini Sadiki.
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The text for this chapter was thorougly read and criticised by AMSINI SADIKI.
In mathematics operators with such properties are also called projections.
The REYNOLDS stress deviator has only five independent components.
The motivation of the nonlinear REYNOLDS stress models is very similar to that used in phenomenological continuum mechanics: The REYNOLDS stress tensor and other closure variables are assumed to depend on a certain set of variables, and then thermodynamic arguments (second law) are used to reduce these dependences. In the other approach the balance law for the REYNOLDS stresses is simplified; it leads to an implicit relation for the REYNOLDS stress tensor which may equally be viewed as a phenomenological relation of a certain constitutive class. They simply express the dependencies more directly.
The term “anisotropy” in connection with the closure functional for the REYNOLDS stress tensor is differently used in the turbulence literature than in the continuum mechanics literature. When turbulence modellers speak of the anisotropic REYNOLDS stress, then RD, the deviator of the REYNOLDS stress tensor, is meant. Any deviation of RD from RD = pvt(D) is then called to contribute to the anisotropy of RD. This is different from the classical notion of anisotropy which is a property usually expressed in the reference configuration. The reader should be aware of the difference of the jargons.
We shall give further reasons based on significant physical results, as we proceed in our developments, when they are better understood than here.
Papers dealing with such realizability conditions are e.g. by Du VACHAT [247], RUNG et al. [200], SHIN et al. [214].
The reader should be aware of the difference in notation: € is used for internal energy, e for the turbulent dissipation rate.
The radiative source is assumed not to have any fluctuation. This makes it a true source.
In rational thermodynamics in which the entropy principle is formulated with the CLAUSIUS- DUHEM inequality, entropy flux equals heat flux divided by absolute temperature. Here no distinction is made between empirical and absolute temperature and coldness is set equal to the inverse temperature 19 = 1/0 by definition. Generally, the coldness is not necessarily regarded to be equal to the inverse temperature, but as a derived quantity that could possibly satisfy such a relationship, particularly in equilibrium.
Very close to absolute zero on the molecular level, quantum mechanical effects play a role; to these we have no analogue in the turbulent case.
We are taking this view of open systems thermodynamics here mainly in order to simplify the computations which otherwise become rather complicated. Research in thermodynamics of turbulence is at a rather early stage. With knowledge accumulating it may well become necessary to repeat the analysis for closed systems thermodynamics.
Inspite of this rule, authors proposing higher order turbulent closure conditions have occasionally not followed this rule and nevertheless proposed turbulent closure relations which do obey the rule of material frame indifferance. This is, of course, permissible, but it is not compelling.
These arguments anticipate that K a grad,ÛT and kE a grade or at least that K and Ice will contain such contributions.
See LUMLEY [140] and SAFFMAN [206]. This statement is not exactly correct, because the variable list (12.4.4)2 contains (W) which is not usually an independent constitutive variable, because non-NEwToNian behaviour is always postulated to be materially objective, i.e., indifferent. Such non-NEwToNian type closure relations have been proposed by many authors, among others by SPEZIALE [220], [222], [226], SPEZIALE et al. [221], [223], SPEZIALE & GATSKI [225], SPEZIALE & Xu [224], WANG [253], CRAFT et al. [55], [54] and POPE [183], [184].
The independent variables in (12.4.11)2 are the same so those listed in (12.4.9), but are written differently. The form listed in (12.4.11)2 facilitates the explicit computations.
We use, for instance, the fact that tr((D°) (D)) = tr((D)°(D)) which can easily be proven with the aid of the definitions (12.4.7)1.
Notice that ßT contains as elements the terms involving é and grad é.
See for instance CHOWDHURY & AHMADI [48], CABON & SCOTT [41], CRAFT, LAUNDER & SUGA [53], [55], [54].
The representation (12.4.33) is a slight generalisation of that presented by SADIKI and HUTTER [204]. It differs from that presented in [204] by the term involving ßl which is stated as ß1(D). Both are possible forms because in a density preserving material the difference can be absorbed in the pressure.
To be more precise, the classical BOUSSINESQ type closure RD = vt (D) generates normal stresses RD = RD = RE, which are all equal. A closure scheme that is more general than this produces, in general, diagonal components of RD of which not all are equal to one another. This fact alludes to “anisotropy”. In the continuum mechanical literature differences in the normal stresses are referred to as normal stress effects.
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Hutter, K., Jöhnk, K. (2004). Thermodynamic Formulation of Turbulent Closure Relations of First Order Level — Algebraic Reynolds Stress Models. In: Continuum Methods of Physical Modeling. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06402-3_13
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DOI: https://doi.org/10.1007/978-3-662-06402-3_13
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