Mixed Convection (Combined Convection) — Flow Regime Determination by Richardson Number

That's correct. "Mixed convection" or "combined convection" is a state where forced flow (from fans or pumps) and buoyancy-driven flow (due to density differences from temperature variations) are simultaneously effective.

The key point is that there exists a zone where assuming only forced convection or only natural convection leads to significant errors. For example, in fan cooling of electronics, buoyancy effects cannot be ignored in low-speed regions where fan velocity is weak. Conversely, in data center hot aisles, server exhaust heat creates rising air currents that interfere with the forced airflow from the HVAC system.

Yes, there is. It's the Richardson number (Ri). It is defined as $\mathrm{Ri} = \mathrm{Gr}/\mathrm{Re}^2$. It's a dimensionless number representing the ratio of buoyancy effects to inertial effects; this single value determines the character of the flow.

The definition of the Richardson number and the classification of flow regimes are as follows:

For example, consider the area near a server rack exhaust vent. Let's assume exhaust temperature $T_s = 60\,^\circ\mathrm{C}$, ambient temperature $T_\infty = 25\,^\circ\mathrm{C}$, exhaust vent height $L = 0.5\,\mathrm{m}$, and HVAC supply air velocity $U_\infty = 0.8\,\mathrm{m/s}$. Using the volumetric expansion coefficient for air $\beta \approx 1/T_\mathrm{avg} \approx 1/316\,\mathrm{K}^{-1}$:

$\mathrm{Ri} = \dfrac{9.81 \times (1/316) \times 35 \times 0.5}{0.8^2} \approx \dfrac{0.543}{0.64} \approx 0.85$

Since $\mathrm{Ri} \approx 0.85$, it's precisely in the mixed convection regime. Calculating with only forced convection equations would underestimate heat transfer, and using only natural convection would miss the influence of flow velocity.

It's not simple addition; you use the Churchill-Usagi correlation. It takes the form of a power sum:

Physically speaking, the boundary layers for forced and natural convection do not develop independently; they interfere with each other. Simple addition ($n=1$) overestimates this interference effect. $n=3$ is the "empirically optimal value" that best matches experimental data, systematized by Churchill in 1977.

Writing it concretely:

Yes, it does. If $\mathrm{Nu}_{\mathrm{natural}} \ll \mathrm{Nu}_{\mathrm{forced}}$, then even when cubed it's almost zero, so $\mathrm{Nu}_{\mathrm{mixed}} \approx \mathrm{Nu}_{\mathrm{forced}}$. Conversely, if $\mathrm{Ri} \gg 1$, then $\mathrm{Nu}_{\mathrm{mixed}} \approx \mathrm{Nu}_{\mathrm{natural}}$. In other words, this correlation has the correct asymptotic behavior at both limits. That's the excellent point of Churchill's correlation.

It's easy to understand by considering a heated vertical wall. Near the wall, a buoyant flow of warmed fluid rising is generated.

• Aiding flow: When the forced flow blows from bottom to top, it's in the same direction as the buoyant flow → the boundary layer thins and Nu increases.
• Opposing flow: When the forced flow blows from top to bottom, it's opposite to the buoyant flow → the boundary layer thickens and Nu decreases; in the worst case, separation or reverse flow occurs.
• Transverse flow: When the forced flow is horizontal, it's orthogonal to buoyancy → three-dimensional secondary flows are generated.

It's very bad. For example, if you design a vertical electronic board with airflow from top to bottom, a recirculation zone can form in the lower part of the board (where temperature is higher) due to the competition between buoyancy and forced flow, leading to localized hot spots. In an actual analysis of an automotive ECU I handled in the past, there was a case where an IC in the center of the board experienced thermal runaway when configured for opposing flow. Simply changing the design to an aiding configuration improved $\Delta T$ by over 15K.

The basics are the Navier-Stokes equations, but the key point is adding a buoyancy term to the momentum equation. The most widely used is the Boussinesq approximation, where density variation appears only in the buoyancy term, and density is considered constant elsewhere. It's valid for small temperature differences ($\beta \Delta T \ll 1$).

Exactly. Let me mention three important points to note:

• Applicability limit of the Boussinesq approximation: A guideline is $\beta \Delta T < 0.1$. For air, if $\Delta T > 30\,\mathrm{K}$, you should consider a non-Boussinesq model (variable density model).
• Gravity direction setting errors: If you set the gravity vector direction incorrectly in a CFD solver, the buoyancy direction reverses, yielding completely meaningless results. It's an incredibly common beginner mistake.
• Choice of reference temperature $T_0$: It's common to use inlet or ambient temperature. An inappropriate reference temperature distorts the magnitude of the buoyancy term.

In mixed convection, the momentum and energy equations are strongly coupled through the buoyancy term. Therefore, the points for discretization are more severe than in ordinary CFD:

• Spatial Discretization: Use at least second-order or higher schemes (2nd-order upwind, QUICK, MUSCL, etc.) for the convection term. First-order upwind difference has excessive numerical diffusion, which smears out the delicate flow patterns caused by buoyancy.
• Pressure-Velocity Coupling: SIMPLE family (SIMPLE, SIMPLEC) is standard. When buoyancy is strong, a Coupled Solver is often more stable.
• Temporal Discretization: For unsteady analysis (often oscillatory in opposing flow), second-order implicit methods are recommended. Keep the CFL number below 1.

The convection term in CFD is completely different in nature from those in structures. The numerical diffusion of first-order upwind is proportional to mesh coarseness, so buoyant plumes (rising currents) become blurred and disappear. If you get a result like "buoyancy effects appear to be zero even though there's a fan," first suspect the discretization scheme.

To conclude, standard k-ε is not recommended. In mixed convection, the generation and suppression of turbulence by buoyancy are important, but the treatment of the buoyancy term in standard k-ε is insufficient. The recommended order is as follows:

Yes, it's off by default. When this setting is turned on, a buoyancy-induced turbulence production term $G_b = -\beta g_i \dfrac{\mu_t}{\mathrm{Pr}_t} \dfrac{\partial T}{\partial x_i}$ is added to the turbulent kinetic energy equation (k-equation). This is a setting that must be enabled for mixed convection. If you run the analysis with it off, the enhancement/suppression of turbulence by buoyancy is completely ignored, leading to significant errors, especially in opposing flow.

Mixed convection is indeed difficult to converge. Because buoyancy and inertia are competing, the solution tends to oscillate. Here are some effective techniques in practice:

1. Initial Condition Strategy: First, converge a solution for forced convection only without buoyancy ($g=0$), then use that as the initial condition and gradually turn on gravity (increase $g$ stepwise using under-relaxation).
2. Under-relaxation Factors: Lower momentum from 0.5→0.3, pressure from 0.3→0.2. Keep energy relatively high at 0.8–0.9.
3. Steady → Unsteady Switch: In opposing flow, there are cases where only an unsteady solution exists inherently. If it doesn't converge in steady state, don't hesitate to switch to unsteady analysis.
4. Residual Monitoring Points: Don't just monitor overall residuals; also monitor "wall Nu" and "velocity profiles at specific cross-sections" to confirm physical convergence.