Natural Convection in a Cavity

This is a standard benchmark with many data available.


The goal of this project is to simulate the fluid flow under natural convection: the heated fluid circulates towards the low temperature under the action of density and gravity differences. The phenomenon is important, in the sense it models evacuation of heat, generated by friction forces for example, with a cooling fluid.

We shall put in place a simple convection problem in order to study the phenomenon without having to handle the difficulties of more complex domaines. We describe then some necessary transformations to the equations, then we define quantities of interest to be able to compare the simulations with different parameter values.


Natural Convection Cavity
Figure 1 : Geometry of natural convection benchmark.

To study the convection, we use a model problem: it consists in a rectangular tank of height 1 and width W, in which the fluid is enclosed, see figure Geometry of natural convection benchmark.. We wish to know the fluid velocity \mathbf{u}, the fluid pressure p and fluid temperature \theta.

We introduce the adimensionalized Navier-Stokes and heat equations parametrized by the Grashof and Prandtl numbers. These parameters allow to describe the various regimes of the fluid flow and heat transfer in the tank when varying them.

The adimensionalized steady incompressible Navier-Stokes equations reads:

\begin{split} \mathbf{u}\cdot\nabla \mathbf{u} + \nabla p - \frac{1}{\sqrt{\text{Gr}}} \Delta \mathbf{u} &= \theta \mathbf{e}_2 \\ \nabla \cdot \mathbf{u} &= 0\ \text{sur}\ \Omega\\ \mathbf{u} &= \mathbf{0}\ \text{sur}\ \partial \Omega \end{split}

where \mathrm{Gr} is the Grashof number, \mathbf{u} the adimensionalized velocity and p adimensionalized pressure and \theta the adimensionalized temperature. The temperature is in fact the difference between the temperature in the tank and the temperature T_0 on boundary \Gamma_1.

The heat equation reads:

\begin{split} \mathbf{u} \cdot \nabla \theta -\frac{1}{\sqrt{\text{Gr}}{\mathrm{Pr}}} \Delta \theta &= 0\\ \theta &= 0\ \text{sur}\ \Gamma_1\\ \frac{\partial \theta}{\partial n} &= 0\ \text{sur}\ \Gamma_{2,4}\\ \frac{\partial \theta}{\partial n} &= 1\ \text{sur}\ \Gamma_3 \end{split}

where \mathrm{Pr} is the Prandtl number.

Influence of parameters

what are the effects of the Grashof and Prandtl numbers ? We remark that both terms with these parameters appear in front of the \Delta parameter, they thus act on the diffusive terms. If we increase the Grashof number or the Prandtl number the coefficients multiplying the diffusive terms decrease, and this the convection, that is to say the transport of the heat via the fluid, becomes dominant. This leads also to a more difficult and complex flows to simulate, see figure [fig:heatns:2]. The influence of the Grashof and Prandtl numbers are different but they generate similar difficulties and flow configurations. Thus we look only here at the influence of the Grashof number which shall vary in [1, 1e7].

flow grashof
Figure 2Velocity norm with respect to Grashof

Quantities of interest

We would like to compare the results of many simulations with respect to the Grashof defined in the previous section. We introduce two quantities which will allow us to observe the behavior of the flow and heat transfer.

Mean temperature

We consider first the mean temperature on boundary \Gamma_3

T_3 = \int_{\Gamma_3} \theta

This quantity should decrease with increasing Grashof because the fluid flows faster and will transport more heat which will cool down the heated boundary \Gamma_3. We observe this behavior on the figure [fig:heatns:3].

Mean temperature with respect to the Grashof number

Flow rate

Another quantity of interest is the flow rate through the middle of the tank. We define a segment \Gamma_f as being the vertical top semi-segment located at W/2 with height 1/2, see figure [fig:heatns:1]. The flow rate, denoted \mathrm{D}_f, reads \mathrm{D}_f = \int_{\Gamma_f} \mathbf{u} \cdot \mathbf{e}_1

where \mathbf{e}_1=(1,0). Note that the flow rate can be negative or positive depending on the direction in which the fluid flows.

As a function of the Grashof, we shall see a increase in the flow rate. This is true for small Grashof, but starting at 1e3 the flow rate decreases. The fluid is contained in a boundary layer which is becoming smaller as the Grashof increases.

image::debit_grashof.png[Behavior of the flow rate with respect to the Grashof number; h = 0.02, \mathbb{P}_3 for the velocity, \mathbb{P}_2 for the pressure and \mathbb{P}_1 for the temperature.]

Running the model

$ mpirun -np 4 /usr/local/bin/feelpp_toolbox_fluid_2d --config-file cfd2d.cfg