Equations

Dimensional form

In this project, we consider the conservation laws of the mass, the momentum, and the internal energy, which are governed by

\[\pder{\tilde{u}_i}{\tilde{x}_i} = 0,\]

\[\pder{\tilde{u}_i}{\tilde{t}} + \tilde{u}_j \pder{\tilde{u}_i}{\tilde{x}_j} = - \frac{1}{\tilde{\rho}} \pder{\tilde{p}}{\tilde{x}_i} + \tilde{\nu} \pder{}{\tilde{x}_j} \pder{\tilde{u}_i}{\tilde{x}_j} + \tilde{g}_i,\]

\[\pder{\tilde{T}}{\tilde{t}} + \tilde{u}_j \pder{\tilde{T}}{\tilde{x}_j} = \tilde{\kappa} \pder{}{\tilde{x}_j} \pder{\tilde{T}}{\tilde{x}_j},\]

respectively, where the summation rule is applied.

Note

• \(\tilde{q}\) implies that the quantity \(q\) is dimensional (i.e., before normalised).

• We assume the physical properties (e.g., the density \(\tilde{\rho}\), the dynamic and kinematic viscosities \(\tilde{\mu}, \tilde{\nu}\), the thermal diffusivity \(\tilde{\kappa}\)) to be constant.

Non-dimensional form

In this project, we focus on Rayleigh-Bénard convection, which is an excellent model problem to shed light on the conservation properties. By adopting Boussinesq approximation and normalise the equations with proper scales, we obtain the following non-dimensional equations which play the central role in this project.

\[\pder{u_i}{x_i} = 0.\]

\[\pder{u_i}{t} + u_j \pder{u_i}{x_j} = - \pder{p}{x_i} + \frac{\sqrt{Pr}}{\sqrt{Ra}} \pder{}{x_j} \pder{u_i}{x_j} + T \delta_{ix}.\]

\[\pder{T}{t} + u_j \pder{T}{x_j} = \frac{1}{\sqrt{Pr} \sqrt{Ra}} \pder{}{x_j} \pder{T}{x_j}.\]

Here Rayleigh number \(Ra\) and Prandtl number \(Pr\) are dimensionless parameters given by

\[\begin{split}Ra & = \frac{\tilde{\beta} \tilde{g} {\tilde{l_x}}^3 \left( \Delta \tilde{T} \right)}{\tilde{\nu} \tilde{\kappa}}, \\ Pr & = \frac{\tilde{\nu}}{\tilde{\kappa}},\end{split}\]

where \(\tilde{\beta}\), \(\tilde{g}\), \(\tilde{l_x}\), and \(\Delta \tilde{T} = \tilde{T}_{H} - \tilde{T}_{L}\) are the thermal expansion coefficient \(\left[ K^{-1} \right]\), the gravitational acceleration \(\left[ L T^{-2} \right]\), the distance between the walls \(\left[ L \right]\), and the temperature difference \(\left[ K \right]\), respectively.

Also, by taking the inner product of the momentum balance and the velocity vector, a relation with respect to the squared velocity is obtained:

\[\pder{k}{t} + u_j \pder{k}{x_j} = - u_j \pder{p}{x_j} + \frac{\sqrt{Pr}}{\sqrt{Ra}} \pder{}{x_j} \left( u_i \pder{u_i}{x_j} \right) - \frac{\sqrt{Pr}}{\sqrt{Ra}} \pder{u_i}{x_j} \pder{u_i}{x_j} + u_i T \delta_{ix},\]

where

\[\tilde{k} \equiv \frac{1}{2} \tilde{u}_i \tilde{u}_i.\]

Similarly, by multiplying the temperature with the internal energy balance, we obtain the relation with respect to the squared temperature:

\[\pder{h}{t} + u_j \pder{h}{x_j} = \frac{1}{\sqrt{Pr} \sqrt{Ra}} \pder{}{x_j} \left( T \pder{T}{x_j} \right) - \frac{1}{\sqrt{Pr} \sqrt{Ra}} \pder{T}{x_j} \pder{T}{x_j},\]

where

\[\tilde{h} \equiv \frac{1}{2} \tilde{T} \tilde{T}.\]

Periodic boundary conditions are imposed in the homogeneous directions \(y\) and \(z\). The boundary conditions in the \(x\) (wall-normal) direction are listed here:

• \(\ux = 0\): Dirichlet condition, impermeable walls.

• \(\uy = \uz = 0\): Dirichlet condition, no-slip and stationary walls.

• \(\partial p / \partial x = 0\): Neumann condition.

• \(\vat{T}{x = 0}, \vat{T}{x = l_x \equiv 1}\): Dirichlet condition, fixed temperature satisfying \(\vat{T}{x = 0} - \vat{T}{x = 1} = 1\).

Note

• Without loss of generality, \(\tilde{\beta}\), \(\tilde{g}\), \(\tilde{l_x}\), and \(\Delta \tilde{T}\) are fixed to unity.

• The reference velocity scale \(\tilde{U} \left[ L T^{-1} \right]\) is defined by the other parameters \(\tilde{U} = \sqrt{\tilde{\beta} \tilde{g} \tilde{l_x} \left( \Delta \tilde{T} \right)} \left( = 1 \right)\), which is often called as the free-fall velocity.

Quadratic Quantities

We investigate the properties of the mentioned relations with respect to the quadratic quantities \(k\) and \(h\). In particular, we focus on how the net amount of them:

\[ \begin{align}\begin{aligned}& \int \int \int k dx dy dz,\\& \int \int \int h dx dy dz,\end{aligned}\end{align} \]

behave, which follow

\[ \begin{align}\begin{aligned}& \int \int \int \pder{k}{t} dx dy dz = \int \int \int \left\{ - u_j \pder{k}{x_j} - u_i \pder{p}{x_i} + \frac{\sqrt{Pr}}{\sqrt{Ra}} \pder{}{x_j} \left( u_i \pder{u_i}{x_j} \right) - \frac{\sqrt{Pr}}{\sqrt{Ra}} \pder{u_i}{x_j} \pder{u_i}{x_j} + u_i T \delta_{ix} \right\} dx dy dz,\\& \int \int \int \pder{h}{t} dx dy dz = \int \int \int \left\{ - u_j \pder{h}{x_j} + \frac{1}{\sqrt{Pr} \sqrt{Ra}} \pder{}{x_j} \left( T \pder{T}{x_j} \right) - \frac{1}{\sqrt{Pr} \sqrt{Ra}} \pder{T}{x_j} \pder{T}{x_j} \right\} dx dy dz,\end{aligned}\end{align} \]

giving

\[ \begin{align}\begin{aligned}& \int \int \int \pder{k}{t} dx dy dz = \int \int \int u_x T dx dy dz - \int \int \int \frac{\sqrt{Pr}}{\sqrt{Ra}} \pder{u_i}{x_j} \pder{u_i}{x_j} dx dy dz,\\& \int \int \int \pder{h}{t} dx dy dz = \vat{ T }{x = 1} \int \int \frac{1}{\sqrt{Pr} \sqrt{Ra}} \vat{ \pder{T}{x} }{x = 1} dy dz - \vat{ T }{x = 0} \int \int \frac{1}{\sqrt{Pr} \sqrt{Ra}} \vat{ \pder{T}{x} }{x = 0} dy dz - \int \int \int \frac{1}{\sqrt{Pr} \sqrt{Ra}} \pder{T}{x_j} \pder{T}{x_j} dx dy dz.\end{aligned}\end{align} \]

In this project, we aim at faithfully replicating these relations from a numerical standpoint.

Derivation is given below, focusing on the individual components.

Advective terms

We have

\[- \int \int \int u_j \pder{k}{x_j} dx dy dz = - \int \int \int \left( \pder{u_j k}{x_j} - k \pder{u_j}{x_j} \right) dx dy dz,\]

which is, due to the incompressibility:

\[- \int \int \int \pder{u_j k}{x_j} dx dy dz = - \int \int \vat{ \left( u_x k \right) }{x = 1} dy dz + \int \int \vat{ \left( u_x k \right) }{x = 0} dy dz,\]

where the divergence theorem and the periodic boundary conditions are utilized. Since the walls are impermeable, \(u_x \equiv 0\) on the walls and thus this leads to 0, indicating that the advective terms do not affect the total amount of \(k\).

Note that exactly the same statement holds for the advective terms of \(h\).

Pressure-gradient terms

We consider

\[- \int \int \int u_i \pder{p}{x_i} dx dy dz.\]

By following the identical algebra we adopted to investigate the advective terms, we find that the pressure-gradient terms do not affect the net amount of \(k\) either.

Diffusive terms - conduction

We have

\[\int \int \int \frac{\sqrt{Pr}}{\sqrt{Ra}} \pder{}{x_j} \left( u_i \pder{u_i}{x_j} \right) dx dy dz.\]

By utilizing the divergence theorem, this leads to

\[\int \int \frac{\sqrt{Pr}}{\sqrt{Ra}} \vat{ \left( u_i \pder{u_i}{x} \right) }{x = 1} dy dz - \int \int \frac{\sqrt{Pr}}{\sqrt{Ra}} \vat{ \left( u_i \pder{u_i}{x} \right) }{x = 0} dy dz,\]

where periodic boundary conditions are assumed in the \(y\) and \(z\) directions. Since we assume that the walls are impermeable and fixed (i.e., \(u_i \equiv 0_i\) on the walls), the integrands are all zero for all directions, indicating that the conductive terms do not alter the total amount of \(k\).

Regarding the conductive terms with respect to \(h\), we also have an analogous relation:

\[\int \int \frac{1}{\sqrt{Pr} \sqrt{Ra}} \vat{ \left( T \pder{T}{x} \right) }{x = 1} dy dz - \int \int \frac{1}{\sqrt{Pr} \sqrt{Ra}} \vat{ \left( T \pder{T}{x} \right) }{x = 0} dy dz.\]

Since they are non-zero in general, we find that the conduction plays a role in the budget of \(h\).

Diffusive terms - dissipation

We have

\[- \int \int \int \frac{\sqrt{Pr}}{\sqrt{Ra}} \pder{u_i}{x_j} \pder{u_i}{x_j} dx dy dz\]

and

\[- \int \int \int \frac{1}{\sqrt{Pr} \sqrt{Ra}} \pder{T}{x_j} \pder{T}{x_j} dx dy dz,\]

which are always non-positive and dissipate \(k\) and \(h\).

Body force term

We have

\[\int \int \int u_x T dx dy dz,\]

which works as a source term and alters the net amount of \(k\).