Once it is understood the three main heat transference mechanism, which, in automotive field, the convection is main one, it is important to develop the theories and methods to correctly evaluate the convection into CFD environment.

Oberbeck-Boussinesq Approximation

The idea behind the Oberbeck-Boussinesq Approximation is, that in the most of applications, the temperature fluctuation occurs internally. This means that the temperature variation does not occurs with a huge ΔT. Actually, a small ΔT are most common. Hence, the density fluctuation will be small. This means that it is possible to assume that there are points of the flow that the density variation is small. If this is the case, the only density variation accounted is due to the buoyancy terms. The parts of the flow that has a low temperature variation are considered incompressible. The buoyancy term can be written in function of Taylor expansion for temperature, which can be given by:

ρ = ρ₀ + (∂ρ/∂T)∙(T – T₀)

Since the temperature difference is small, it is possible to update this equation by a truncated version:

ρ ≈ ρ₀[1 – β(T – T₀)]

Where β is the volume expansion coefficient, which is also the derivative of density ∂ρ/∂T. Hence, it is possible to account the variation of density ρ only due to the buoyancy term.

∇∙u = 0

(∂u/∂t) + u∙∇u = – (1/ρ₀)∇p* + ν∇²u + βgθz

(∂θ/∂t) + u∙∇θ = α∇²θ + (q_g“‘/ρc)

p* = p + ρ₀∙g∙z

θ = T – T₀

After defined ρ due to the buoyancy term, it is possible to conclude this one at the mass conservation. Since in this case there is no density variation, what will occur is the pressure variation, which is given by p*. The buoyancy term will be accounted by βgθz, where θ is the temperature variation. In this equation, this is the only term that induces the motion. θ is used instead of T to account the difference about the reference temperature T₀. In this way, five equations are used to solve this kind of problem. The Oberbeck-Boussinesq Approximation (OBA) is a simpler way to solve this kind of problems, which is even more simple if it was considered a fully compressible problem.

Determination of the regime

Understanding that convection is a heat transfer that occurs in a flow motion, another good strategy is to work non-dimensional forms.

∇∙u = 0

(∂u/∂t) + u∙∇u = – ∇p* + (1/Re)∇²u + Ri∙θz

(∂θ/∂t) + u∙∇θ = [1/(Pr∙Re)]∇²θ

Re = U∙L/ν ; Pr = ν/α ; Ri = gβΔTL/U²

Uff = √gβΔTL

For this, u, L and T are the reference velocity, length and temperature. Hence, three new parameters will arise, the classical Reynolds number (Re), the Prandtl (Pr) and the Richardson (Ri) number. This last one multiply the buoyancy term, which allows to characterize if the motion due to buoyancy is important or not. In addition, it is possible to notice that Ri is gβΔTL, the buoyancy term, divided by U², which accounts the forced velocity. Actually, the term gβΔTL is also a velocity term, but it is squared. Hence Uff is the so called free fall velocity, as described above. This is a rough measure of the motion induced by buoyancy. The velocity induced by buoyancy can be measured using the free fall velocity definition. The Prandtl number suggests how is diffused the temperature field with respect to the velocity field. This diffuses in the kinematic viscosity µ, the temperature field diffuses into λ, which is the thermal conductivity. Hence, by the non-dimensional form of OBA it is possible to notice, that the diffusivity of the temperature field is equal the diffusivity of the velocity field (1/Re) multiplied by 1/Pr. Actually this last one is the ratio between diffusivity of the velocity with respect to the diffusivity of the temperature field. When Pr > 1, this means that the diffusivity of the velocity field is higher than the one of the temperature field. In cases of Pr < 1, occurs the opposite. Regarding liquids, Pr is larger than 1. For instance, water has Pr = 7, which means that the diffusivity of the velocity field is seven times higher than the diffusivity of the temperature field.

Richardson number

The convection regime can be defined by the Richardson number, it measures how intense is the motion induced by buoyancy with respect to the motion imposed by an external agent, which can be a free stream or an imposed flux in pipe. The Richardson number is given by:

Ri = gβΔTL/U²

Ri allows to distinguish between the three main regime of convection. In forced convection the regime is simply driven by an external agent as a free stream, a pump or a fan. In the natural convection there is no imposed motion, instead the motion is driven by the temperature field. Hence, there are different values for each kind of regime since Ri measures the velocity induced by buoyancy respective to the one induced by an external agent. This means that if Ri is very smaller than 1, the motion is induced by buoyancy can be neglected. Hence, in this case, there is a flow which is a forced one. On the opposite case, when Ri ⋙ 1, this means that the motion is induced by the temperature difference and, therefore, the density difference (buoyancy) are extremely more intense than the imposed motion. In this case it is said that the flow is under natural convection. In cases which the regime is between these two previous cases, it is said that the flow is under a mix of convection regime.

Forced convection

Considering the forced convection, Ri < 1 , thus the buoyancy does not play a hole. Hence, the buoyancy term that is accounted on the following equation becomes zero.

(∂u/∂t) + u∙∇u = – ∇p* + (1/Re)∇²u ; ∇∙u = 0 ; βgθz = 0

In the case which the buoyancy term is neglected, the velocity field behave as usual, thus the temperature does not influence the velocity field solution.

(∂θ/∂t) + u∙∇θ = [1/(Pr∙Re)]∇²θ ; Nu = f(Pr,Re)

These equations develop the behavior without influence of the temperature field. On the opposite case, the temperature field is influenced by the forced motion. Hence there is only a one way coupling, which means that the velocity field influences the temperature field, but not the opposite. This is the forced convection, which is characterized by a very small Ri and the neglected buoyancy term (βgθz = 0). Hence, this is a connection case, which is governed by Re and Pr. In addition, the heat exchange introduced by Nu, actually this is function of Re and Pr. This one is given by the fluid considered, thus to improve the heat exchange it is necessary to improve, roughly, the Reynolds number.

Effect of the Prandtl number

Also regarding the forced convection, there is the effects due to Pr. This is a measure of the temperature field, with the diffusivity of the velocity field. Considering a laminar boundary layer, the thickness of the boundary layer depends on the diffusivity. In a laminar boundary layer the thickness is the square root of Re. However, there are also the effects of the temperature field on the boundary layer. In the laminar conditions, there is also Δθ, which is the thickness of the thermal boundary layer, where the temperature reaches 99% of the free stream temperature. The thickness of the thermal boundary layer depends only on the diffusivity of the temperature. Hence, the ratio between the rate of the thermal boundary layer with respect to the rate of the velocity boundary layer is given by the ratio between two diffusivity coefficients, which is Pr.

δ_θ/δ ~ Pr^-1/2 for Pr < 1

δ_θ/δ ~ Pr^-1/3 for Pr > 1

After some calculations, for laminar boundary layers, this allows to do the concept of the thermal boundary layer, because with thermal processes it must be considered also the boundary layer for the temperature. However, this one does not match the boundary layer for the velocity field, because it depends of Pr.

δ/δ_θ < 1 for Pr < 1 → δ < δ_θ

δ/δ_θ > 1 for Pr > 1 → δ > δ_θ

Therefore, when Pr is less than 1, the thickness of the thermal boundary layer is larger than the thickness of the kinematic boundary layer. This means, that the temperature reach 99% of the free stream temperature further away from the wall with respect to the velocity field. The opposite occurs with the Prandtl number larger than 1, the power unit is different, is 3. This means that Pr is usually assigned for liquids, because Pr ⋙ 1 means liquids. Hence, for this case, the thickness of the thermal boundary layer is smaller than the kinetic one. The temperature field reaches 99% free stream temperature within the kinematic boundary layer.

Turbulent boundary layer

Regarding the turbulent boundary layer, there are two relevant aspects. First, when considering wall turbulence, everything is scaled in friction. This is valid for the velocity field. However, in the case of the temperature field, everything is in the logarithmic scale. Once this still is calculated in friction unit, and the friction of the temperature field is the same unit of the velocity field multiplied by Pr, it is possible to write:

(θ,θ_RMS) = f(Pr^0.6y⁺)

By using these units, the mean temperature field for near-wall scales has the same RMS, the standard deviation of the temperature field has a peak which is always at the same position where the scales are at the friction wall-distance, like the velocity field in the wall distance. Hence, the Prandtl number is considered at a re-scaled friction wall-distance.

η_θ/η ~ Pr^-3/4 for Pr < 1 → η < η_θ

η_θ/η ~ Pr^-1/2 for Pr > 1 → η > η_θ

Hence, if Pr > 1 means that the boundary layer is being re-scaled with respect to the velocity boundary layer. The second very important aspect for turbulence is related to scales. The velocity field in turbulent flows is a multiscale phenomenon. In addition, the smaller scales are called, Kolmogorov Scales (KS). When running a simulation, it is required a discretization in the order of Kolmogorov Scales. Otherwise, it should be used RANS or other methods. The temperature is also a multiscale phenomena, because the fluctuation of temperature field follows the fluctuation of the velocity field. Hence, there are fluctuation of temperatures at different sizes and intensity. Also in this case, it is possible to define a smaller scale than KS, the Bachelor Scales (BS), which smaller scales than temperature field occur. As the smaller scales of velocity field depends of the diffusivity of the temperature field, thus the smallest scales between these two fields depends on the diffusivity. The Kolmogorov Scales depends on the diffusivity of the velocity scales, while the Bachelor Scales depend on the diffusivity of temperature. The ratio between these two depends on the Prandtl number. Hence, if Pr < 1, η_θ/η > 1, which means that BS is larger KS. When solving this problem, it is necessary to be de-meshed the CAD model, to resolve KS of the velocity field assuring that the temperature field is also being resolved in a better way. The reason is that the smaller scales of the temperature field are larger than KS.

The opposite occurs in case that liquids are being considered, this ratio is less than 1, which means that the Bachelor scale is lower than the Kolmogorov scale. Hence, if the simulation is started considering KS, its results have a completely wrong temperature field, because when dealing with liquids, the temperature field requires a fine resolution of the mesh with respect to the velocity scale. Therefore, a thermal process simulation which the transmission fluid is a liquid, the mesh should be defined based on BS. Figure 1 illustrates a cut of the temperature field solution for liquid metal and water. In the first case, the size of the temperature scales are larger than the velocity scales. In addition, Figure 1 also illustrates the sizes of the spots for the liquid metal. These spots are the horizontal cuts of the tri-dimensional plums. It is also possible to notice that the spots for water are smaller, because the bachelor scales decrease by increasing Pr. Hence, for the liquid metals the mesh should reproduce KS, while for the water, it is better to discretize a mesh on BS.

References

• This article is based on the lecture notes written by the author during the Industrial Aerodynamics course attended at Dallara Academy.