Draft:Scale analysis of internal natural convection

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Scale Analysis of Internal natural convection

Internal Natural Convection

Internal natural convection is the movement of fluid caused by temperature differences within it. Warmer fluid becomes lighter and rises, while cooler fluid, being heavier, sinks. This creates a natural circulation that helps transfer heat within the fluid.

This process is important in many areas, such as heating systems, cooling devices, and environmental studies. Understanding how internal natural convection works helps engineers and scientists study and predict fluid movement in different situations.

Introduction

Fluid-filled enclosures are important in many engineering and natural systems. The way fluid moves inside an enclosure, such as the air between the panes of a double-pane window, can differ greatly from how it moves outside, in the natural convection boundary layer.

Governing Equations

The behaviour of fluid in internal natural convection is described by equations that follow the principles of mass, momentum, and energy conservation. To simplify these equations, the Boussinesq approximation is used. This means the fluid's density (ρ) is considered constant everywhere except in the vertical momentum equation, where it varies slightly to account for temperature changes. In this term, the density is expressed as ρ[1−β(T−T0​)], where β is the coefficient of thermal expansion, and T0​ is the reference temperature.

This approach helps predict how the fluid will flow and transfer heat within the enclosure under varying conditions. These are the governing equations used in the scale analysis.

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0$ (1)

${\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}=-{\frac {1}{\rho }}{\frac {\partial P}{\partial x}}+v\left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right)$ (2)

${\frac {\partial v}{\partial t}}+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}=-{\frac {1}{\rho }}{\frac {\partial P}{\partial y}}+v\left({\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {\partial ^{2}v}{\partial y^{2}}}\right)-g\left[1-\beta \left(T-T_{0}\right)\right]$ (3)

${\frac {\partial T}{\partial t}}+u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}=\alpha \left({\frac {\partial ^{2}T}{\partial x^{2}}}+{\frac {\partial ^{2}T}{\partial y^{2}}}\right)$ (4)

Scale Analysis

Consider a rectangular enclosure with a height H and a length L. This space is filled with a fluid, like air or water, that behaves according to Newtonian fluid principles. If the side walls of this enclosure are suddenly heated and cooled to temperatures of +ΔT/2 and −ΔT/2, respectively, while the top and bottom walls remain insulated, the fluid inside will start to move. Initially, the fluid is at an even temperature (T=0) and is not moving (u=0 and v=0).

Instead of solving the governing equations through numerical methods, scale analysis is used to predict the flow and heat transfer patterns within the enclosure. Scale analysis identifies the key factors influencing the system and provides an understanding of the flow regimes and thermal interactions that may occur.

Thermal Boundary Layer

Right after t=0, the fluid near the sidewalls is almost stationary, with velocities u→0 and v→0. During this time, the thickness of the thermal boundary layer (𝛿 _T​) is much smaller than the height (H) of the enclosure (𝛿 _T≪H). Near the sidewalls, the energy equation (4) shows a balance between thermal inertia and heat conduction perpendicular to the wall.

${\frac {\Delta T}{t}}\sim \alpha {\frac {\Delta T}{\delta _{T}^{2}}}$ (5)

In the governing equations, we assume that the fluid velocities 𝑢 and 𝑣 are initially zero, as the fluid is at rest. This simplification is valid because, near 𝑡 = 0 ⁺, the thermal boundary layer thickness 𝛿 _T is much smaller than the height of the enclosure. In other words, 𝑦 ∼ 𝐻 and 𝑥 ∼ 𝛿 _T, where 𝐻 is the height of the enclosure, leading to the conclusion that temperature variations along the vertical axis are much more gradual than along the horizontal axis. From here, we get

$\delta _{T}\sim (\alpha t)^{1/2}$ (6)

This means that boundary layer thickness rises along with the heated wall.

Velocity Scale

To calculate this, the first step is to eliminate 𝑃 from the equations. To do this, we first take the partial derivative of equation (2) with respect to 𝑦 , and then the partial derivative of equation (3) with respect to 𝑥. By subtracting these two resulting equations, we effectively eliminate the pressure term, simplifying the system for further analysis.

${\frac {\partial }{\partial x}}\left({\frac {\partial v}{\partial t}}+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}\right)-{\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}\right)=v\left[{\frac {\partial }{\partial x}}\left({\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {\partial ^{2}v}{\partial y^{2}}}\right)-{\frac {\partial }{\partial y}}\left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right)\right]+g\beta {\frac {\partial T}{\partial x}}$ (7)

The resulting equation (7) consists of three fundamental groups of terms: inertia terms on the left-hand side, along with four viscous diffusion terms and a buoyancy term on the right-hand side. It can be easily demonstrated that three dominant terms emerge from each of these fundamental groups.

${\text{Inertia: }}{\frac {\partial ^{2}v}{\partial x\partial t}},\quad {\text{Friction: }}\nu {\frac {\partial ^{3}v}{\partial x^{3}}},\quad {\text{Buoyancy: }}g\beta {\frac {\partial T}{\partial x}}$ (8)

Now performing the scale analysis, we get

${\frac {v}{\delta _{T}t}},\quad v{\frac {v}{\delta _{T}^{3}}}\sim {\frac {g\beta \Delta T}{\delta _{T}}}$ (9)

The driving force in the natural convection process within the enclosure is the buoyancy effect.A critical aspect of the analysis is determining whether this buoyancy effect is counteracted primarily by viscous friction or by inertial forces. To simplify it more , multiply by (𝛿 _T)³ on each side we get-

${\text{Inertia: }}{\frac {1}{\operatorname {Pr} }},\quad {\text{Friction: }}1,\quad {\text{Buoyancy: }}\sim {\frac {g\beta \Delta T\delta _{T}^{2}}{\nu v}}$ (10)

Thus, for fluids with a Prandtl number of approximately 1 or higher (Pr>=1), the appropriate momentum balance immediately after 𝑡 = 0 ⁺ is between the buoyancy force and frictional forces.

$1\sim {\frac {g\beta \Delta T\delta _{T}^{2}}{\nu v}}$ (11)

Hence , we conclude that the initial vertical velocity scale is,

$v\sim {\frac {g\beta \Delta T\alpha t}{\nu }}$ (12)

Hence , we can conclude that the vertical velocity scale is applicable to fluids such as water and oils with Prandtl number >1 and is only partially applicable to gases which have Prandtl number ~ 1.

References