**Research and Numerical Algorithms in Computational Fluid Dynamics Simulations**

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Sharaf F. Al-Sharif

*Saudi Arabia*

**Reynolds Stress Transport Modelling**

*Center of Excellence in Desalination Technology, King Abdulaziz University, Jeddah*

The Reynolds–averaged Navier–Stokes (RANS) approach is the most commonly employed approach in CFD for industrial applications, and is likely to continue to be so for the foreseeable future. The need to handle complex wall-bounded flows, and the need to evaluate large numbers of design variations usually prohibits high-fidelity approaches such as direct numerical simulation (DNS), and large-eddy simulation (LES). The application of Reynolds–averaging to the equations of motion introduces a set of unclosed terms, the *Reynolds Stresses*, into the mean flow momentum equations, and turbulence models are needed to provide closure of these terms before the set of equations can be solved. Within the framework of RANS approaches, a hierarchy of modelling schemes exists based on the level of sophistication in which these unclosed terms are modelled. In Reynolds stress transport (RST) modelling, rather than assuming a direct (linear or non–linear) link between the Reynolds stresses and mean strain, a separate transport equation for each of the stress components is solved. This in principle provides a number of advantages over other RANS models, which

This chapter aims to provide a general introduction and overview of Reynolds Stress transport modelling. The first section will provide a brief historical background on the development of this class of models. Next, the theoretical background and rationale underlying the most common modelling practises within this framework are presented. This is followed by a discussion of some numerical implementation issues specific to RST modelling within the context of the finite volume method. Finally the chapter is closed with some concluding

Early work leading to the development of Reynold-stress transport (RST) modelling was mainly theoretical, due to the relative complexity of this level of modelling compared to the available computational capabilities of the time. Chou (1945) constructed a formal solution to the fluctuating pressure Poisson equation that is the basis for current models of the pressure–strain-rate correlation. Later Rotta (1951), laid the foundation for Reynolds stress transport modelling by being the first to develop a closed model of all the terms in the exact equation (Speziale, 1991). Because of limited computational capability at the time, successful computations were not carried out until several decades later (Speziale, 1991). Another important development came when the continuum mechanics community speculated on the potential similarity between turbulent flow and the flow of non-Newtonian fluids (Gatski,

**1. Introduction**

will be reviewed here.

**2. Development of RST modelling**

remarks.


**9. The light entrainment of the circadian clock in zebrafish would reflect a cellular response to photooxidative stress**

It is conceivable that the development of circadian clocks is one way to segregate daytime from nighttime processes with light-dark cycles acting as selective forces

*Oncogenes and Carcinogenesis*

**7. Shared regulatory pathway for circadian clock and DNA repair in**

*zPHR. The induced zPHR repairs UV-damaged DNA in a light-dependent manner.*

Although solar light has several beneficial uses, including the regulation of circadian clocks, the UV component of solar light is harmful to living cells because it produces cytotoxic and mutagenic lesions in DNA called cyclobutane pyrimidine dimers (CPDs) and pyrimidine (6-4) pyrimidone photoproducts [(6-4) photoproducts] (**Figure 2**) [51, 52]. Natural selective pressure has forced the development of a self-defense system mediated by photoreactivation. Photoreactivation is the light-dependent DNA repair mechanism mediated by DNA photolyases (PHRs), which bind to and repair the UV-induced DNA damage using visible light as an energy source [53]. Two classes of PHRs have been identified, one specific for CPDs and the other specific for (6-4) photoproducts. Importantly, both the induction of PHRs in response to light and subsequent light-dependent repair of DNA by PHRs are essential for a successful photoreactivation in zebrafish cells [42, 54, 55]. Notably, the expression of the zebrafish *Phr* repairing (6-4) photoproduct (*z64Phr*) is regulated by the same light-induced MAPK cascades as those controlling the expression of the clock genes *zCry1a* and *zPer2* (**Figure 2**) [49]. The light-induced ERK activation triggers the expression of *z64Phr*, whereas the light-induced p38 activation inhibits it. Thus, the light-dependent DNA repair and regulation of the circadian clock are governed by shared regulatory pathways. Both CRYs and PHRs belong to the DNA photolyase/cryptochrome protein family and have highly similar amino acid sequences [42, 55, 56]. Evolutionary studies have shown that the animal CRY proteins functionally diverged first from the CPD photolyase and then further

*A proposal model of light signaling pathways involved in shared control of the circadian clock and DNA repair in zebrafish. In a variety of organisms, light induces ROS production. In zebrafish cells, the light-induced ROS stimulate intracellular MAPK/ERK signaling pathway, which transduces photic signal to zCry1a expression. The light-induced zCRY1a interacts directly with the zCLOCK:zBMAL complex and modifies its transcriptional capacity. Notably, the zCLOCK:zBMAL complex regulates the transcription of a variety of genes involved in cellular stress responses and DDR. UV component of sunlight induces DNA damage. Light-induced ROS and activation of MAPK/ERK pathway also induce expression of a DNA repair enzyme,*

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**zebrafish**

**Figure 2.**

[69, 70]. In this scenario, increasing levels of oxygen-free radicals during the daytime may be a decisive factor in relegating the anabolic processes of mitosis, growth, and consolidation to the dark hours. Thus, it is reasonable that cellular signaling cascade mediated by ROS is utilized in the regulation of the circadian clocks and that common regulatory pathways mediate both cellular responses to photooxidative stress and the light-dependent regulation of the circadian clocks (**Figure 2**).

In addition to the photooxidative stress derived from sunlight, the UV component of it is major source of harm to organisms [51, 52]. In zebrafish, the light induces expression of PHRs which repair UV-damaged DNA in a light-dependent manner (**Figure 2**) [49, 71]. Importantly, this light induction of *DNA Phr* expression appears to be mediated by photooxidative stress [68, 72]. These observations are consistent with the idea that photooxidative stress may be utilized as a signal to activate DNA repair enzymes that can protect the organism's DNA from UV-induced damage. The fact that ROS, a well-known inducer of oxidative stress, can activate *zCry1a* transcription in zebrafish cells [66], together with the finding that *zCry1a* and *DNA Phr* are governed by shared light-induced signaling pathways [49], strongly suggests that, at least in zebrafish, the light entrainment of the circadian clock reflects a long-standing cellular response to photooxidative stress (**Figure 2**). The zCRY1a protein interacts directly with the CLOCK (NPAS2):BMAL complexes and regulates its transcriptional capacity [67, 73]. The complexes regulate a variety of key genes involved in cellular stress responses, DNA repair, and cell cycle regulation [14, 74]. Thus, the circadian clock protein zCRY1a may be the key integrator of oxidative stress that controls the core circadian machinery to regulate the transcription of genes responsible for DDR and cell cycle adjustments.