# Direct Numerical Simulations of Natural Convection in Near Shore Regions

5160 words (21 pages) Essay in Environmental Studies

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## 1.     Introduction

### 1.1    Problem description

In littoral regions, mixing and natural circulations of water driven by differential of temperature play a fundamental rule in many physical systems, such as biological and environmental fields across shore. It has a key role in characterizing, forecasting and maintaining the quality of water through the transport of pollutants, nutrients and dissolved and non-dissolved chemicals in reservoirs, lakes and coastal areas (Naghib, et al., 2018). Therefore, mixing and natural convection problems detain a large area of recent research attention.

Solar radiation absorption at various depths is generally following Beer’s Law:

I = Ie-ηd,

where I is radiation intensity at the local water depth d, Iis the radiation intensity (W/ m2) at the water surface when d = 0 and η is a bulk attenuation coefficient (m−1). For constant radiation intensity; I (t) = I= constant whilst for ramp radiation; the intensity increased according to the ramp function I(t)= (dI/dt)t where dI/dt = constant. Therefore, the amount of solar radiation at the water surface is assumed to be approximately uniform in space, but will vary in time(Naghib, et al., 2018).

According to the law, the amount of incoming solar radiation absorbed at the water surface in daytime is depth-dependent, its intensity decays exponentially with depth at a rate depending on the vertical attenuation coefficient of water. The latter depends on the intensity and wavelength of radiation and the turbidity and color of water (Hattori, et al., 2015b). Moreover, the depth of water between near shore and off-shore regions varies clearly, leading to a significant differential heating of the water column between shallow and deep regions.

As a result, a horizontal temperature gradient forms which causes the flow to drive and form a horizontal convection motion between the near shore areas and deeper water body, which is the reason of stable thermal stratification (Abolghasem, 2016). Whilst, the residual radiation penetrates into the water in near shore regions will be absorbed by the bottom layer are re-emitted back to the water body as a boundary heat flux. Thus, the water layer next to the bed heated up and forms a thermal boundary layer. As a result, an adverse temperature gradient forms causing the flow to drive and forms vertical convection circulations in the form of rising plumes, causing unstable thermal stratification which is the source of Rayleigh–Benard instability (Lei & Patterson, 2002; Hattori, et al., 2015b).

#### 1.2   Governing equations

With Boussinesq approximation, the flow in the littoral regions is mainly governed by the three-dimensional Navier-Stokes equations coupled with the temperature evolution and energy equation; the set of the dimensional governing equations are listed below:

$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=–\frac{1}{{\rho }_{0}}\frac{\partial P}{\partial x}+\nu {\mathrm{}}^{2}u$

$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}=–\frac{1}{{\rho }_{0}}\frac{\partial P}{\partial y}+\mathrm{\nu }{\mathrm{}}^{2}\mathrm{v}+g\left(T–{T}_{0}\right)$

$\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}=–\frac{1}{{\rho }_{0}}\frac{\partial P}{\partial x}+\nu {\mathrm{}}^{2}w\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}={\mathrm{\kappa }\mathrm{}}^{2}T+S\left(x,y,z\right)$

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$

where u, v and w represent the velocity components in x, y and z directions respectively; where x and z are the horizontal axes and y is the vertical axis; g is the gravitational acceleration (in the negative y direction); d is the water depth; t is time; T is the temperature; T0 is initial temperature;  is the Laplacian operator; P, ρ0,, ν, κ are the pressure, density, thermal expansion coefficient, kinematic viscosity and thermal diffusivity, respectively. The water temperature is assumed to be at a constant value thus the thermal water properties will be constant.

According to Beer’s law, the last term in equation 4 represents the internal heating source due to absorption of solar radiation, which is given by (Lei & Patterson, 2002):

where I0 represents the radiation intensity at the water surface, Cp is the specific heat of water at the reference temperature, and η is the attenuation coefficient which is considered constant over the water body (Farrow & Patterson, 1994).

Moreover, previous studies indicate that the flow in littoral regions is also governed by a number of non-dimensional parameters such as Rayleigh number (Ra) and Prandtl number (Pr). Cengel and Ghajar in their book Heat and heat transfer (2015) explained the physical meaning of Rayleigh number as “the product of the Grashof number, which describes the relationship between buoyancy and viscosity within the fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity. Hence the Rayleigh number itself may also be viewed as the ratio of buoyancy forces and (the products of) thermal and momentum diffusivities”. (page541) Should I include the page number when refer to book??

Accordingly, Rayleigh number, which classify the type of the natural convection boundary layer as laminar, transition or turbulent, can be stated in many different ways. For temperature difference, Ra is defined as (where L is the length):

Where, Prandtl number can be given by Pr = ν/κ, and it is constant for water at 7.0 (Hattori, et al., 2015a). While for radiation intensity, Ra can be written as:

## 2.     Literature review

Numerous studies have been motivated to investigate the behavior of the radiation-induced natural convection in a fluid layer and the resulting convection transport using different approaches; including analytical (Farrow & Patterson, 1993a), experimental (Naghib, et al., 2018), numerical (Hattori, et al., 2015a) and scaling analysis (Mao, et al., 2009) under various circumstances and boundary conditions (Amber & O’Donovan, 2018).  The motivation is to get better understanding of the process mechanism, prediction of the flow pattern and fluid temperature and to figure out the effect of changing the controlled parameters and boundary conditions on convection circulations. (references in red to be converted to numbers once bibliography finalized??)

Most of the recently published research on radiation-induced natural convection in near shore regions focused on complex geometries such as triangular or trapezoidal cavities where the bed layer is inclined near the shore and horizontal in deeper regions offshore; in order to reflect real field conditions of shallow littoral water in lakes or reservoirs. Whilst earliest studies conducted their investigations on simple geometries where a flat horizontal bed layer being used.

Correspondingly, earlier models of convective circulation were reported by (Cormack, et al., 1974), (Patterson & Imberger, 1980) & (Patterson, 1984), where investigations of steady laminar flow considered in an enclosed rectangular cavity with differential heating at vertical walls or internally heated source; the results shows a single Rayleigh-Benard cell (vortex) driven by internal buoyancy sources. The latter found that the overall flow state in the domain can be classified into three distinct regimes: conductive, transitional or convective, based on the value of the Grashof number at different combinations of the Prandtl number and the cavity aspect ratio. This finding was also concluded by Patterson & Imberger (1980).

Many other researches continued in analysing the flow in cavities with a horizontal flat bottom, some will be discussed in the forthcoming sections. However, a more representative geometry, such as a triangular cavity, is desired to reflect actual field situation in lakes and reservoirs; where the depth of water is decreasing towards the shore. One of which examined by (Farrow & Patterson, 1993a) and (1993b) with a relatively small bottom slope. By solving the one-dimensional conduction equation, they examined the onset of instability in a two-dimensional triangular wedge imposed to a diurnal heating and cooling radiation uniformly distributed on a top horizontal open surface. The bottom surface of the domain was heated by absorption of the residual radiation which is re-emitted back as heat flux. These investigations have been modified by Farrow & Patterson (1994) with a more physically realistic model, where they included the exponential decay rate of radiation with depth and the effect of depth dependant absorptions of radiation. As a result, the residual radiation absorbed at the bottom layer decreases off-shore causing a temperature gradient which drive the flow up the sloped bottom, which is the main source for Rayleigh–Benard instability. As a result, the flow in the domain forms two distinct regions: conduction near shore and convection off-shore. See Figure1.

Figure 1: Isotherms at steady state (Mao, et al., 2013).

Following, further discussions of previous studies are categorised according to the adopted procedure.

### 2.1  Experimental studies on radiation induced natural convection

A large body of experimental studies have been conducted to ensure the importance of buoyancy driven flow. One of the earlier field observations of radiation induced natural convection in a triangular tank was reported by Lei and Patterson (2002). This experiment aimed at demonstrating the volumetric heating in a triangular tank directly illuminated by a theatre spot light source at the top surface. Values of temperatures and fluid velocities were measured at different times and locations along the sloped bottom.

The results revealed that the convective circulation of flow is classified into three distinct stages: (i) an initial growth stage; where conduction dominates the flow and a distinct thermal boundary layer developed along the sloping bottom, (ii) a transitional stage; which is marked by the onset on instability in a form of plumes arising up from the boundary layer, and (iii) a quasi-steady stage; in which the average temperature of flow in the domain grows steadily with time. The Authors stated that these findings have been previously reported by Farrow and Patterson (1993a). However, despite these significant results, the model reflects simplified case of the real conditions in reservoirs.

Many other field observations focused on the transport mechanisms have been conducted to report its importance in maintaining the quality of water. Some of which examined by Adams and Wells (1984), Monismith et al. (1990), MacIntyre and Melack (1995), Schladow et al. (2002), Macintyre (2012), Xing et al. (2014) and most recently by Naghib et al. (2018). The latter conducted laboratory scale experiments to study the characteristics of the boundary layer formed due to convection circulation in the near shore of lakes and reservoirs.

The experiment conducted in a water-filled triangular cavity with a sloping bottom and an open top subjected to radiative heating from a halogen lamp to simulate solar radiation. The study extended the previous work of Lei and Patterson (2002) and investigated a wider range of Rayleigh numbers and different parameters. Results showed that the development of the flow can fall into three different categories based on their temporal state; initiation, transitional and quasi steady which shows consistence with the outcomes of previous experimental observations of Lei and Patterson (2002). On the other hand, the flow in the domain spatially developed three distinct sub-regions; conductive, stable convection and unstable convective, showing a good agreement and consistent with previous scaling (Lei & Patterson, 2003) and numerical results (Lei & Patterson, 2003).

Literature stated that results reported by experimental studies can provide a good understanding of the physical mechanism of natural convection; since many field observations provide reliable results that showed good agreement and consistency with different approaches. Nevertheless, the limitation of such approach is that the model used a simplified form of real conditions, for instant, the impact of winds, waves, tides and heat exchanged between water and the surrounding were neglected. Furthermore, constant radiation from spot light or theatre were used while solar radiation in reality changed with time and climate, hence, the velocity of the rising plumes, as well as, the rate of mixing and transport would be higher than the experimental values. Therefore, the value of Rayleigh number in lakes and reservoirs is higher than the values obtained in the laboratories, which in turns encouraged scientist and researchers to use more accurate methods to examine the flow in littoral regions (Amber & O’Donovan, 2018).

### 2.2  Analytical and scaling analysis on radiation induced natural convection

A large number of Benard type stability problems have been conducted using analytical scaling analysis. Lei and Patterson (2003) introduced scales to study the instability of flow in a two-dimensional triangular shallow wedge imposed to direct solar radiation at the top surface. The results of scaling analysis revealed that the flow in the domain, based on Rayleigh number, eventually forms two distinct regions classified as; a conductive region nearshore which becomes isothermal and stationary, and a convective/transitional region offshore that has a flow velocity associated with large-scale circulation. The introduced scales were also used to figure out the properties of flow in the aforementioned regions.

Detailed scaling analysis of flow at high Rayleigh numbers in a water-filled triangular enclosure with relatively high bottom slope have been reported by Mao et al. (2009). The aim of this investigation was to extend what previously initiated and include more detailed properties of flow development. From the scaling analysis, two critical functions of Rayleigh number have been derived with respect to the horizontal position and used to identify the onset of instability and the range of the unstable region at different offshore distances, as well as, to illuminate the characteristics of the boundary layer.

The study revealed the possibility of four different flow scenarios, which depends on the bottom slope and maximum water depth. The flow in the domain for each scenario forms three possible flow regimes depending on Rayleigh number; each flow regime can be characterized by distinct subregions based on the dominant mode of heat transfer into: conduction near shore, stable convection and unstable convection off-shore. The dividing point between each successive subregion can be determined by comparing the values of the critical function of Rayleigh number with the global Rayleigh number.

In addition, Mao, et al., (2009) in this study established a set of steady state scales to illuminate the flow thermal features and variation with the horizontal position in different subregions. They aimed at the use of detailed scaling analysis to determine the onset time for instability and the range of the unstable region at different offshore positions.

However, despite the importance of previously cited analytical and scaling analysis investigation and many others, the limitation of such analysis that it was not capable to provide more realistic details on stability properties of the unstable region or to considered the transient growth of surface thermal layer (Hattori, et al., 2015a).

### 2.3  Numerical Analysis on radiation induced natural convection

Numerical simulation have been widley implemented and reported to validate the results of expiremental observations and scaling analysis. One of the earlier studies conducted by Lei & Patterson (2003) on a three-dimensional numerical simulation to examine radiation induced natural convection in a triangular wedge with a bottom slope of 0.1. The results illuminated the development of three distinct stages of the flow: (i) an initial stage; where conduction dominates at the bottom boundary and forms a thermal boundary layer along the sloping bottom. (ii) a transitional stage; which is characterised by instability due to rising plumes from the bottom boundary up the slope. (iii) a quasi-steady stage; in which the average temperature of the wedge raises steadily. These findings provided a strong confirmation of the experimental observations obtained by Lei & Patterson (2002).

Direct numerical simulations (DNS) is one of the major tools that have been used recently to track turbulent flows behaviour and enhance the experience in the area of modelling research. This approach has been implemented in many natural convection studies associated with high Rayleigh number; to solve the coupled Navier-Stokes, temperature evolution and energy equations by solving a large range time and length scales of turbulence without using turbulent models. DNS are considered as the simplest and most accurate method in resolving small scales of turbulent motion; with a very fine grid up to the smallest scale quoted as the Kolmogorov length scale (i.e. η = (ν3/)1/4), where ν is the kinematic viscosity and ε is the energy dissipation rate per unit mass. Therefore, high-performance computing techniques with a relatively large memory and high-speed super machines are required to carry out the numerical computations of such large number of grids; thus, a high implementation cost of DNS is expected (Ferziger & Peric, 2002).

Recent studies have been conducted by Hattori et al. (2015a) followed by (2015b) using DNS to model natural convection in a three-dimensional parallelepiped cavity imposed to solar radiation. The chosen geometry represents a good consistency with real conditions of deep-water bodies with negligible varying topography and bottom slope. The governing equations were solved using SNS code, developed in Fortran90 language; which uses highly accurate numeric methods. Different schemes have been considered and compared by investigating the impact of various parameters such as the cavity aspect ratio, the water depth and the corresponding Rayleigh number on the instability of the thermal boundary layer (i.e. the critical time for the onset of stability and the growth rate of the boundary layer).

Results revealed the development of two distinct layers; Stable surface thermal layer which is produced by direct absorption of solar radiation and unstable bottom thermal layer which is developed due to the absorption of the residual solar radiation by the bottom layers and re-emission of plumes. The formation of the thermal layers and their stability properties were also investigated using scaling analysis. In addition, vertical temperature profiles which obtained from numerical results are compared with the theoretical solution used by Farrow & Patterson (1993a).

It is notable that the complexity of the investigated numerical research has been increased significantly. However, the related Rayleigh numbers are still limited to low values. This arise the importance of such research which is discussed in the following section.

## 3.     Expected research contribution:

As mentioned earlier, the significant implications of natural convection and mixing on water quality generates lots of research attention. Hence, a large scale of researches was conducted on Benard stability problems over the last years. Despite of these recent studies, previous research available were only limited on investigating laminar flow with relatively small Rayleigh number; the reason behind that is the complexity of turbulent flow nature. Therefore, knowledge of the turbulent flow behavior in nearshore regions is still limited.

However, this study will proceed with examining the behavior of turbulent flow, that has a relatively high Rayleigh number, in three-dimensional domain imposed to isothermal heating through solar radiation. The main aspect that will be addressed in this research is the turbulent flow features in littoral regions which will be achieved by investigating the behavior of natural convection induced by radiation in near shore regions and to study the characteristics of the associated turbulent flow in the domain.

The mode of the research will encompass modeling and simulation of flow in near shore regions. A series of unsteady three-dimensional Direct Numerical Simulations will be conducted using ANSYS Fluent Flow Modelling Software using the unsteady laminar model to solve the time-dependent Navier-Stokes equations, coupled with analytical approach and scaling analysis. In order to investigate the behavior of natural convection of turbulent flow in the domain of interest, a very fine grid should be used with a sufficient time interval to resolve the Kolmogorov scales (i.e. Length scale of turbulent motion) as mentioned earlier. Then, the effects of density variation, temperature gradient and Rayleigh number will be analysed in detail.

Consequently, understanding the behavior of turbulent flow will provide a better insight into thermodynamics, mixing, exchange and transport process, which in turns would be effective in reducing the cost of solar energy systems (Amber & O’Donovan, 2018).

## 4.      Bibliography

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• Adams, E. & Wells, S., 1984. Wells Field measurements on side arms of lake Anna, VA. Journal of Hydraulic Engineering, 110(6), pp. 773-793.
• Amber, I. & O’Donovan, T., 2018. Natural convection induced by the absorption of solar radiation: A review. Renewable and Sustainable Energy Reviews, Volume 82, pp. 3526-3545.
• Cengel, Y. A. & Ghajar, A. J., 2015. Heat and Mass Transfer, Fundamentals and Applications. fifth ed. s.l.:McGraw-Hill.
• Cormack, D. E., Leal, L. G. & Imberger, J., 1974. Natural convection in a shallow cavity with differentially heated end walls. Part 2. Numerical solutions. Journal of Fluid Mechanics, Volume 65, pp. 231-246.
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