# Problem Description

Fig. b.1 shows the cross-section of a rectangular channel filled by hot water. The channel walls have lower temperature and the liquid free surface does not exchange heat with the ambient air. The channel length is assumed to be large enough such that the problem can be modelled in two dimensions. Because there is no heat flux to the air, the surface can be represented by an adiabatic wall (q = 0) with a slip condition for the velocity.

Fig. b.1: Cross-section of the channel

The following values are chosen for the calcultation:
\begin{equation} T_{hot} = 80 ^{\circ} C, \ T_{cold} = 80 ^{\circ} C, \ L = 7 cm \end{equation} The temperature difference causes the buoyancy convection inside the channel. The cooled water sinks to the bottom of the channel along the channel walls and the hot water lifts up in the middle of the channel. The convection continues until reaching the Equilibrium temperature.

Density variation is very small and therefore it can be assumed that the fluid is incompressible. The influence of temperature on the viscosity is negligible, so the Boussinesq approximation can be used. The reference temperature for the Boussinesq approximation is given by the arithmetic mean. For this problem $\Theta_{ref} = 50 ^{\circ} C$. That is also the reference temperature for all properties of water, which are used for the simulation using OpenFOAM.

Two different cases will be considered:

# Case (1): Free slip condition for channel walls

The simulation is started for a $20 \times 20$ uniform mesh, as shown in Fig. b.2. For simulation, the "buoyantBoussinesqPimpleFoam"-solver is employed. This solver is mainly used for incompressible, transient and turbulent buoyancy-driven flows. However, instead of turbulence model, a laminar model is used. There are slip conditions for velocity along all walls.
Fig. b.2: $20 \times 20$ uniform mesh

Maximum timestep is set to 0.5 seconds, with a maximum Courant-number of 0.5. In total 2000 seconds are simulated.

Four simulations are done and the mesh is refined after each run by splitting the cells, so that the number of cells has increased by a factor of 4. All cells are quadratic and have the same size. For convergence control the temperature in the middle of the domain was sampled for each 100 seconds. The results are shown in Fig. b.3. It seems that the temperature has converged by the fourth run. Total run time of the calculation, as shown in Fig. b.4, has significantly increased after each refinement.

Fig. b.3: Convergence study for different mesh sizes, case (1)

Fig. b.4: Calculation time for different mesh sizes, case (1)

The results for the finest mesh (25600 cells) are shown in Fig. b.5, b.6 and b.7 for the time $t = 100 s$. Fig. b.5 shows streamlines colored by velocity magnitude. Along the cooled walls there is a thin layer of sinking water, in which the maximum velocity is reached. There are two eddies along the two vertical walls. The same result is represented by the vector-plot in Fig. b.6.

Fig. b.5: Flow streamlines for $t = 100 s$, case (1)

Fig. b.6: Velocity vectors for $t = 100 s$, case (1)

Figure 7 displays temperature field inside the channel. Because of circulation, the warm water assembles at the surface while the cool water sinks to the bottom. It’s also possible to see, that the surface is adiabatic. The flow as well as the temperature distribution are symmetric with respect to the vertical centerline.

Fig. b.7: Temperature distribution for $t = 100 s$, case (1)

# Case (2): No-slip condition for channel walls

The simulation setup for the case (2) is the same as case (1), with the exception, that there is a no-slip condition along the channel walls. The mesh sizes are exactly the same as in case (1).

The convergence study is shown in Fig. b.8. There is nearly no difference between the 6400-cells-mesh and the 25600-cells-mesh concerning to the same sample point for temperature. That means that the time for cooling down the water is already very good represented by the 6400-cells-mesh.

Fig. b.8: Convergence study for different mesh sizes, case (2)

Fig. b.9: Calculation time for different mesh sizes, case (2)

Results for case (2) are shown in Fig. b.10 and b.11 for $t = 100 s$. Fig. b.10 shows the streamlines, again colored by velocity magnitude. There is also a thin layer along the walls, in which the water sinks to the bottom. In the vector-plot in Fig. b.11, it is visible that the fluid flows toward the surface in the middle of the channel. The center of the eddies are in the upper half of the domain.
Fig. b.10: Flow streamlines for $t = 100 s$, case (2)

Fig. b.11: Velocity vectors for $t = 100 s$, case (2)

# Comparison

Both cases reproduce the expected flow behavior. The slip condition of case (1) causes higher maximum velocities and therefore the water cools down faster than in case (2). This behavior can be seen, comparing Fig. b.3 and b.8.

The higher velocity influences the Courant-number and thus the calculation-time is increased in case (2), as it can be seen comparing Fig. b.4 with Fig. b.9.

The eddies of case (2) are not as slim as in case (1), which is also an effect of the velocity difference.

The sourse-codes for both cases can be downloaded here:

Case (1): Free slip condition for channel walls
Case (2): No-slip condition for channel walls

Special thanks to Florian Schmidl for his contributions to prepare this post.