📂Fluid Mechanics

Boussinesq Approximation

Model¹²

Introducing the heat equation into the Navier–Stokes equations to account for buoyancy arising from heat transfer in a fluid and changes in density is called the Boussinesq approximation. $$ \mathbf{u} = \mathbf{u} \left( t ; \mathbf{x} \right) = \left( u_{1} \left( t ; \mathbf{x} \right) , u_{2} \left( t ; \mathbf{x} \right) , u_{3} \left( t ; \mathbf{x} \right) \right) $$ Specifically, suppose that in three-dimensional space the velocity field at time $t$ and spatial coordinates $\mathbf{x} = \left( x_{1} , x_{2} , x_{3} \right)$ is represented by the velocity vector above. Similarly, $p : \mathbb{R}^{3} \to \mathbb{R}$ denotes the pressure $p = p \left( \mathbf{x} \right)$ applied at each coordinate.

Assume that the fluid is incompressible, so that $\nabla \cdot \mathbf{u} = 0$, and that the density $\rho$ is given as the following linear function of the constant reference density $\rho_{0}$, the reference temperature $\theta_{0}$, and the temperature $\theta : \mathbb{R}^{3} \to \mathbb{R}$. $$ \rho = \rho_{0} \left( 1 - \beta \left( \theta - \theta_{0} \right) \right) $$ Here, $\beta$ is the coefficient of thermal expansion, assumed to be constant. With the gravitational acceleration $\mathbf{g} = \left( 0 , 0 , g \right)$, the governing equations are as follows. $$ \begin{align*} {\frac{ D \mathbf{u} }{ D t }} =& - {\frac{ 1 }{ \rho_{0} }} \nabla p + \nu \nabla^{2} \mathbf{u} + {\frac{ \rho }{ \rho_{0} }} \mathbf{g} \\ {\frac{ D \theta }{ D t }} =& \alpha \nabla^{2} \theta \end{align*} $$ Here, $\nabla \cdot$ is the divergence, $\nu = \mu / \rho$ is the ratio of the kinematic viscosity coefficient $\mu$ to the density $\rho$, and $\alpha$ is the thermal diffusivity.

Explanation

$$ {\frac{ D }{ D t }} = {\frac{ \partial }{ \partial t }} + {\frac{ \partial }{ \partial x_{1} }} + {\frac{ \partial }{ \partial x_{2} }} + {\frac{ \partial }{ \partial x_{3} }} $$ The material derivative $D$ is defined as above with respect to the three dimensions, excluding $\theta$. Writing the governing equations with $u = u_{1}$, $v = u_{2}$, $w = u_{3}$ and $x = x_{1}$, $y = x_{2}$, $z = x_{3}$, we obtain the following.

$$ \begin{align*} {\frac{\partial u}{\partial t}} + u {\frac{\partial u}{\partial x}} + v {\frac{\partial u}{\partial y}} + w {\frac{\partial u}{\partial z}} =& - {\frac{ 1 }{ \rho_{0} }} {\frac{\partial p}{\partial x}} + \nu \nabla^{2} u \\ {\frac{\partial v}{\partial t}} + u {\frac{\partial v}{\partial x}} + v {\frac{\partial v}{\partial y}} + w {\frac{\partial v}{\partial z}} =& - {\frac{ 1 }{ \rho_{0} }} {\frac{\partial p}{\partial y}} + \nu \nabla^{2} v \\ {\frac{\partial w}{\partial t}} + u {\frac{\partial w}{\partial x}} + v {\frac{\partial w}{\partial y}} + w {\frac{\partial w}{\partial z}} =& - {\frac{ 1 }{ \rho_{0} }} {\frac{\partial p}{\partial z}} + \nu \nabla^{2} w - {\frac{ \rho g }{ \rho_{0} }} \\ {\frac{\partial \theta}{\partial t}} + u {\frac{\partial \theta}{\partial x}} + v {\frac{\partial \theta}{\partial y}} + w {\frac{\partial \theta}{\partial z}} =& \alpha \nabla^{2} \theta \\ \rho =& \rho_{0} \left( 1 - \beta \left( \theta - \theta_{0} \right) \right) \end{align*} $$

Looking at the equations, one can see that the buoyancy term ${ \rho g } / { \rho_{0} }$, which depends on density and gravity, appears only along the $z$-axis. The fluid moves according to $(u, v, w)$ and transports the heat-related quantity $\theta$ along with this flow; the resulting change in temperature alters the density and affects $w$, which in turn ultimately affects $(u, v, w)$.