The importance of the understanding of the heat mechanisms is the comprehension of the thermal boundary layer. In CFD simulations the mesh generation is directly related with the size of the thermal boundary layer. Not only the knowledge about the flow is important, but the appropriate techniques for thermal boundary layer definition.

Thermal boundary layer

In the forced convection, to conceive a thermal boundary layer is not difficult, because there is a thermal boundary layer and a kinematic one. For natural convection regime there is still a thermal boundary layer, but even without a velocity field, there is also a kinematic boundary layer. Considering a problem with two walls, hot and cold ones, due to the temperature difference, there are plums (Figure 1) that rejected from the bottom wall and go towards the upper wall. Hopefully, it is known that these plums shapes, as the ones seen in Figure 1, are due to the hot fluid that is rising. This one becomes cold near the cold wall and go back down. Hence, there is a structure which is known to be very stable. This means that even it is turbulent, the structure high time is very bad. Hence, to beat the wall, these plums or cluster of plums, have a rise time that is very absolute and the presence of the boundary layer with the known time scales. Actually, it is possible to have very big structures inducing the motion, that are long in time, which means that also a natural convection can have a kinematic boundary layer.

Figure 2 illustrates the structure captured in live time, this is responsible for the so-called mean wind. The velocity field induced by the structure is very hot in time, it creates the boundary layer. Hence, depending on the Prandtl number, there are two effectively possible scenarios, as seen in Figure 3.

These are the effects of the mean wind, due to the plums, along the life time of the structure, these create a boundary layer at the wall. The height of this one with respect to the height of the thermal boundary layer is given by λ_θ/λ_u. This is also depends on Pr, if Pr < 1, then the thermal boundary layer thickness (λ_θ) is larger than the kinematic boundary layer thickness (λ_u). The opposite is true, thus the kinematic boundary layer thickness is larger than the thermal boundary layer thickness.

λ_θ > λ_u ; Pr < 1

λ_u > λ_θ ; Pr > 1

It is known that the kinematic boundary layer plays a relevant role on the calculation of the heat exchange. In particular, if the thermal boundary layer becomes turbulent in its zone, the entire flow is turbulent, but the boundary layer induced by mean wind can be laminar, the heat exchange increase a lot. For this reason it is important to address the behavior of this boundary layer by using its own Reynolds number, which is calculated using the mean wind free stream velocity.

Nu = f(Pr,Ra)

Re_MW = U_MW∙L/ν = f(Pr,Ra)

This approach is taken, because there is a theory that allows to have a measure of the scales of this problem. The first result of this theory is that the thickness of the thermal boundary layer is simply given by the Nusselt number.

δ_θ = L/(2Nu) ; δ = 0.25L/√Re ; δ = 1.38∙log(K²∙Re)/(K²∙Re)

The higher is the heat exchange, higher is Nu, thus lower is the thermal boundary layer thickness. Actually, how much heat exchange are performed depends on the temperature gradient at the wall. In the quiescent or laminar regime, the temperature is linear and the heat exchange depends on the temperature gradient at the wall.

Figure 4 illustrates that in case of a turbulent flow, the temperature profile has this shape and the slope is large, with respect to the laminar case. Hence, the heat exchange increases a lot, thus larger the heat exchange, larger Nu. Therefore, it is a bit intuitive that the thickness of the boundary layer depends on Nu, thus also depends on the heat exchange. Finally, the heat exchange depends on the slope of these profiles, everything is connected. Bottom line, the larger the heat exchange, large the slope and smaller will be the thermal boundary layer. The other results are respective to the thickness of the kinematic boundary layer, these depends on Re of the mean wind free stream velocity, U_MW. This can not be known, a priori, because this is based in U_MW and this requires to follow the Grossman and Lohse theory that shared a table which provides the height of the kinematic boundary layer (Figure 5).

From this theory it is possible to derive some useful quantities for mesh generation. For this, it is important to start the analysis knowing, for instance, the result as a function of Ra, it is possible to calculate the height of the thermal boundary layer for different Pr.

There are the laws for the thermal boundary layer thickness (Figure 5) formulas as a function of Ra and these can be written as function Nu. All these allow to have the thickness of the thermal boundary layer. By using this theory, it is possible to notice that the formulas that are functions of Ra, can also be a function of Gr, thus a function of Pr. This makes possible to recognize different regimes (Figure 6). By these, it is possible to verify if the thermal boundary layer is turbulent or not, thus these results are useful to characterize the problem before to perform a simulation. This means, building properly the mesh. This all passages suggest that solving CFD analysis for race cars is no an easy task, because it is important to know the coupling between temperature and velocity, that results in many different scenarios. Consequently these require different approaches.

Range of Oberbeck-Boussinesq approximation validity

The Oberbeck-Boussinesq Approximation (OBA) assumes that the temperature difference is small, the associated density fluctuation is small, thus this is considered only in the buoyancy term. All the rest is considered as compressible. Actually, there is another reference, that provides some a table to verify if the respective application accepts or not OBA. This is given by Gray & Giorgini. It is another tool that help to decide if the solver will be compressible or incompressible. This one is OBA, while the compressible solver means the entire approximations on Figure 7.

To decide between these two methods, the application of Gray & Giorgini is advisable. This is the linearization of the density, heat capacity, viscosity, expansion coefficient and conductivity variations with respect to the temperature difference and the pressure field.

The next step is given by the non-dimensional coefficients given by the derivatives (Figure 8) multiplied by ΔP and ΔT regarding the information on how much deformation are being lost by considering incompressible. Hence, by using the terms described in Figure 8, the main non-dimensional one is ε_n. When this one is less than 0.1, it can be considered that is a good approximation, their effect can be negligible. Hence, those terms define how good is OBA. If these terms are describing that a huge error is being generated, it is advisable to consider the compressible solver. The reason why OBA is used is due to the fast calculation, but it must be verified if its results are reliable.

Although this evaluation can appear difficult, there is a support graph as the one illustrated by Figure 9. This describes a relation between θ, which is the temperature difference, while L is the size of the volume. Considering that this graph is for air, it is possible to derive a region which OBA error is about or less than 10%, then it is possible to use the incompressible flow. In addition, it is possible to notice that increasing the temperature difference between the two walls, also the distance between these can be increased to maintain OBA. Considering a temperature difference at distance L given by 10⁴ cm, this means that there is 100 m of possibilities to use OBA. It is possible to notice that in terms of air, the lowest L is 1 cm, this means that for automotive applications, the size L is not a problem. The problem is ΔT. As can be seen, above 20 °C of temperature difference, this assumption can not be made.

Figure 10 illustrates the same graph as the one in Figure 9, but for water. It can be seen that temperature difference range is completely different, but the size of the problem still is not an issue.

Conclusion

Therefore, when dealing with a thermal process some decision should be taken. First, recognize if the analysis is about a forced, natural or mixed convection, because by using Ri, it is possible to verify if the regime is forced or natural. In the first case, the problem is more simple, because it uses the classical incompressible and an additional equation for temperature since buoyancy does not play a role. However, the only critical part for forced convection is which type of fluid is being considered, because for high Pr the small scales are given by temperature, while for low Pr the small scales are given by velocity. For cases which the regime is natural convection, the type of the solvers is also important, compressible or incompressible. It is possible to use Gray & Giorigini map to decide to run with OBA or not. Once decided, it is necessary to move to other aspects, which is the mesh. In general, for the large Pr the Bachelor scale is bigger than the Kolmogorov scale, thus the mesh should be built to resolve the temperature field, instead of the velocity field. When adding the wall layer to resolve the boundary layer, it is interesting to apply the Grossman and Lohse theory to know which of the boundary layer, thermal or kinematic ones, is the smallest.

References

• This article is based on the lecture notes written during the Industrial Aerodynamic lectures attended at Dallara Academy.