Convection

Natural Convection

In a gravitational field, natural convection occurs due to temperature differences which affect the density, and thus the relative buoyancy, of the fluid. Denser components fall, while lighter (less dense) components rise, leading to bulk fluid movement.

Natural convection is more likely and more rapid when any of the following conditions apply:

• a greater variation in density between two mixing fluids
• a larger acceleration due to gravity that drives the convection
• a larger distance through the convecting medium.

Natural convection is less likely and/or less rapid when any of the following conditions apply:

• a more rapid diffusion (which acts to diffuse away the thermal gradient that causes the convection)
• a more viscous (sticky) fluid.

To quantify when natural convection is possible, the dimensionless Rayleigh number is defined and can be viewed as the ratio of buoyant and viscous forces times the ratio of momentum and thermal diffusivities. Natural convection becomes possible when the Rayleigh number exceeds ${10}^5$ .

Forced Convection

In forced convection, fluid movement results from external sources (for example, a fan, pump, or the action of a propeller), and is typically used to increase the rate of heat transfer at a surface in cooling or heating applications.

In cases of mixed convection (that is, natural convection and forced convection occurring together) you can determine how much is due to forced convection and how much is due to natural convection. The relative magnitudes of the Rayleigh (Ra), Prandtl (Pr), and Reynolds (Re) numbers indicate which form of convection dominates:

If $\frac{Ra}{Pr{\mathrm{Re}}^2}»1$ , natural convection dominates, forced convection is neglected.

If $\frac{Ra}{Pr{\mathrm{Re}}^2}«1$ , forced convection dominates, natural convection is neglected.

If $\frac{Ra}{Pr{\mathrm{Re}}^2}\simeq 1$ , mixed convection occurs, neither forced or free convection dominate, so both are taken into account.

Convective Heat Transfer Coefficients

In Eqn. (1662), ${\dot{q}}_s″$ , $T_s$ , and $T_{ref}$ are fundamental in nature, while the heat transfer coefficient is a constant of proportionality that relates the three fundamental parameters. While ${\dot{q}}_s″$ and $T_s$ are unambiguous, there is some latitude in the choice of the fluid temperature $T_{ref}$ . Depending upon the choice of $T_{ref}$ , the heat transfer coefficient is different in order to satisfy Eqn. (1662). Therefore, heat transfer coefficients cannot be defined without also defining a reference temperature: there is an infinite number of heat transfer coefficient and reference temperature combinations that give the same surface heat flux upon convergence.

In Simcenter STAR-CCM+, the conceptual centerpiece for modeling convective heat transfer at the wall for turbulent flows stems from the wall functions (WF). Consider the expression for the local surface heat flux:

You can then equate the terms from Eqn. (1662) and Eqn. (1663) and define an $h$ in terms of the local flow conditions.

In this equation:

• ${\rho }_f$ is the fluid density
• $C_{p,f}$ is the fluid-specific heat capacity
• $u_{\tau }$ is a velocity scale that is based on the wall shear stress
• $T^{\text{+}}$ is the dimensionless temperature
• $y^{\text{+}}=\left(u_{\tau }{y}_c\right)/{\upsilon }_f$ is a Reynolds number
• $y_c$ and $T_c$ are the normal distance and temperature of the near-wall cell, respectively

The Wall Functions (WFs) are a set of semi-empirical functions that are used to satisfy the flow physics in the near-wall region (the boundary layer). They give relationships for $T^{\text{+}}$ and $u_{\tau }$ in terms of the laminar and turbulent Prandtl numbers, the dimensionless near-wall flow velocity, and the turbulent kinetic energy.

Four heat transfer coefficients are defined in Simcenter STAR-CCM+:

• Local Heat Transfer Coefficient
• Specified y+ Heat Transfer Coefficient
• Heat Transfer Coefficient
• Virtual Local Heat Transfer Coefficient

These heat transfer coefficients are all grounded to varying degrees in the wall function treatment. The local heat transfer coefficient is the one that is used internally within the Simcenter STAR-CCM+ solver. The other three heat transfer coefficients are computed during post-processing. In the case, of multi-phase flow, volume-fraction-weighted versions of these heat transfer coefficients and reference temperatures are defined for use with co-simulation or coupling to other CAE codes.

Local Heat Transfer Coefficient

In Simcenter STAR-CCM+, the local surface heat transfer coefficient is always calculated (and used internally) in the code using the Wall Functions and Eqn. (1665). However, there can be times when you want to view or export heat transfer coefficients that are defined in other ways. For example, to compare the results to previous work or to use with other applications. In general, you can accomplish this by post-processing the simulation results and defining the one-to-one linear mapping (based on equating the surface heat fluxes):

$$h_{user}=h\left(\frac{T_s-T_c}{T_s-T_{ref}}\right)$$

(1664)

In this way, you can define your own $h$ and $T_{ref}$ pair, where $h$ , $T_c$ , and $T_s$ are known from the simulation, and you pick $T_{ref}$ to suit your requirements.

If you equate the Local Heat Transfer Coefficient term from Eqn. (1662) and Eqn. (1663):

$$h_1=\frac{{\rho }_f\left(y_c\right)C_{p,f}\left(y_c\right)u_{\tau }}{T^{\text{+}}\left(y^{\text{+}}\left(y_c\right)\right)}$$

(1665)

and using $h_1$ and total heat flux ${q"}_{\text{total}}$ in Eqn. (1662), solve for reference temperature:

$$T_{\text{ref},1}=T_s-\frac{{q"}_{\text{total}}}{h_1}$$

you have the pair ( $h_1$ and $T_{\text{ref},1}$ ) which can be viewed or exported.

Specified y+ Heat Transfer Coefficient

The first built-in post-processing option in Simcenter STAR-CCM+ calculates the heat transfer coefficient as in Eqn. (1665), but at a user-specified $y^{\text{+}}$ value instead of the value that is associated with the near-wall cell:

$$h_2=\frac{{\rho }_f\left(y_c\right)C_{p,f}\left(y_c\right)u_{\tau }}{T^{\text{+}}\left({y^{\text{+}}}_{\text{use}r}\right)}$$

(1666)

Then, using $h_2$ and total heat flux ${q"}_{\text{total}}$ in Eqn. (1662), solve for reference temperature:

$$T_{\text{ref},2}=T_s-\frac{q_{\text{total}}″}{h_2}$$

(1667)

The result is the pair ( $h_2$ and $T_{ref,2}$ ) which can be viewed or exported.

This definition of the reference temperature does not guarantee that $T_{ref,2}$ lies within the minimum and maximum temperature range of the local fluid region. If $T_{ref,2}$ lies outside this temperature range, then either the minimum or maximum temperature is used in its place, and the heat transfer coefficient is modified accordingly.

Heat Transfer Coefficient

The second built-in post-processing option in Simcenter STAR-CCM+ calculates the surface heat flux using Eqn. (1663), and then recasts Eqn. (1662) as:

$$h_3=\frac{q_s″}{T_s-T_{ref,3}}$$

(1668)

The heat transfer coefficient is calculated using a user-defined reference temperature ($T_{ref,3}$).

The result is the pair ($h_3$ and $T_{ref,3}$) which can be viewed or exported.

Virtual Local Heat Transfer Coefficient

The third built-in post-processing option in Simcenter STAR-CCM+ does not need to solve the energy equation in the fluid.

Instead, the virtual local heat transfer coefficient is calculated using only flow field information as:

$$h_4=\frac{{\rho }_f\left(y_c\right)C_{p,user}{u}_{*}}{T^{+}\left(y^{+}\left(y_c\right)\right)}$$

(1669)

where $T^{+}$ is calculated using the Standard Wall Functions for Temperature. For theses wall functions, you specify the molecular Prandtl number ${\mathrm{Pr}}_{user}$ and the turbulent Prandtl number ${\mathrm{Pr}}_{t,user}$ as reference values.

Since thermal material properties are not available, you specify the specific heat $C_{p,user}$ as reference value. The fluid density ${\rho }_f$ is obtained locally from the simulation.

The result is $h_4$ which can be viewed or exported.