Energy Equation of Fluids
Theory and Physics
Overview
Professor, when is the energy equation for fluids necessary?
It is essential when you want to determine the temperature field. Designing heat exchangers, cooling electronic devices, combustion, natural convection—all require solving the energy equation. For isothermal flow, the NS equations and continuity equation alone form a closed system, but when temperature is involved, the energy equation is added.
Energy Equation (Temperature Form)
The energy equation for the temperature field in incompressible flow is as follows.
Here, $c_p$ is the specific heat at constant pressure, $k$ is the thermal conductivity, $\Phi$ is the viscous dissipation function, and $\dot{q}$ is the internal heat generation rate.
What is viscous dissipation?
It's the term where kinetic energy is converted to heat by viscosity. For incompressible flow,
In typical engineering flows, viscous dissipation is negligibly small, but it cannot be ignored for high-viscosity fluids (like polymer melts) or ultra-high-speed flows.
Eckert Number
The importance of viscous dissipation is evaluated by the Eckert number.
If $Ec \ll 1$, viscous dissipation is negligible. For example, in an air flow ($U = 50$ m/s, $\Delta T = 20$ K), $Ec \approx 0.12$, which is small. On the other hand, in polymer extrusion ($U = 0.1$ m/s, $\mu = 1000$ Pa·s), dissipation can be the main cause of temperature rise.
Energy Equation (Enthalpy Form)
How does it change for compressible flow?
The energy equation using total enthalpy $h_0 = h + \frac{1}{2}|\mathbf{u}|^2$ is used.
For compressible flow, the equation of state $p = \rho R T$ (ideal gas) is added, closing the system of equations. The four fields—density, velocity, pressure, and temperature—are all coupled.
Dimensionless Parameters
| Parameter | Definition | Physical Meaning |
|---|---|---|
| Prandtl Number Pr | $\nu/\alpha = \mu c_p/k$ | Ratio of momentum diffusivity to thermal diffusivity |
| Nusselt Number Nu | $hL/k$ | Ratio of convective to conductive heat transfer |
| Peclet Number Pe | $Re \cdot Pr$ | Ratio of advection to diffusion |
| Eckert Number Ec | $U^2/(c_p\Delta T)$ | Kinetic energy / Thermal energy |
So, while the Pr number is determined by the fluid properties, the Nu number is obtained as a calculation result, right?
Exactly. Air has $Pr \approx 0.71$, water has $Pr \approx 7$, and engine oil has $Pr \approx 100$–1000. A larger Pr number means a thinner thermal boundary layer, requiring finer meshes to accurately resolve the temperature gradient near the wall.
History of the Energy Equation—From Joule's Heat-Work Equivalence Experiment (1843) to Fluid Mechanics
The concept of the mechanical equivalent of heat included in the fluid energy equation was established by the British scientist James Joule in 1843 through his water stirring experiment. The Joule constant, 1 cal = 4.186 J, remains a fundamental constant in thermodynamics. Its integration into fluid mechanics came from combining Fourier's heat conduction equation (1822) with the Navier-Stokes equations, and the standard form of the energy equation was established in the 1850s. In particular, the "Viscous Dissipation" term—the effect where velocity gradients are converted to heat—cannot be ignored in supersonic or high-viscosity flows and is a term that must be enabled in modern CFD for high-accuracy analysis. While it is often omitted in low-speed, low-viscosity engineering CFD, it's important to recognize its applicability limit (Br=ηU²/(kΔT)>0.1).
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, the water comes out spluttering and unstable, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. This significantly reduces computational cost, so solving first with a steady-state approach is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order-of-magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while it naturally mixes. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelmingly dominates, and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: The "pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed upward by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside" and are expressed by source terms. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated room in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach numbers above 0.3, compressibility effects must be considered
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms
- Non-applicable Cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flows (shock capturing required), free surface flows (VOF/Level Set, etc., required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³@20°C, Water: approx. 998 kg/m³@20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Discretization of the Energy Equation
How is the energy equation solved numerically?
In the finite volume method, the advection and diffusion terms are discretized as cell face fluxes.
A characteristic point is that the energy equation is a scalar transport equation and can be solved as a linear problem if the velocity field is known. An approach is possible where the energy equation is solved as a post-processing step after solving the NS equations.
Advection Scheme Selection
Is the advection scheme for the temperature field the same as for the velocity field?
When the Pe number (Peclet Number) is large, advection dominates, and central differencing causes numerical oscillations. Generally, as follows.
| Pe Number Range | Recommended Scheme | Remarks |
|---|---|---|
| Pe < 2 | Central Differencing (CD) | Stable for diffusion-dominated flows |
| Pe > 2 | Upwind Differencing (Upwind) | Has numerical diffusion |
| High Accuracy | QUICK, TVD (MUSCL, etc.) | Balances accuracy and stability |
Thermal Boundary Conditions
There are mainly three types of thermal boundary conditions at walls.
| Boundary Condition | Mathematical Expression | Applications |
|---|---|---|
| Fixed Temperature (1st Kind) | $T_{wall} = T_0$ | Cooling water wall, constant temperature bath |
| Fixed Heat Flux (2nd Kind) | $-k\frac{\partial T}{\partial n} = q_w$ | Heater, heat-generating surface |
| Convective Heat Transfer (3rd Kind) | $-k\frac{\partial T}{\partial n} = h(T - T_\infty)$ | Heat exchange with external environment |
Temperature Field in Turbulence
For turbulent flow, is a model also needed for the temperature field?
In RANS, modeling of the turbulent heat flux $\overline{u_i'T'}$ is required. The most common is the eddy diffusivity model.
$Pr_t$ (Turbulent Prandtl Number) is typically set to 0.85–0.9. The temperature profile near the wall (e.g., Jayatilleke's wall function) is also important.
How much does the turbulent Pr number affect the results?
It affects wall heat transfer rate by about 10–20%. Especially for liquid metals ($Pr \ll 1$, $Pr_t \approx 1$–4), the standard value of 0.85 is inaccurate. For liquid metals, dedicated models are necessary.
Discretization of the Energy Equation—"Implicit or Explicit?" is Determined by the Rate of Temperature Change
When solving the fluid energy equation numerically, the choice between explicit and implicit methods greatly affects computational efficiency. Explicit methods are simple to implement but are bound by the stability condition for heat diffusion Δt≤ρcΔx²/(2k). In models mixing insulation and metal, where thermal conductivity differs by a factor of 100, the stable Δt is constrained by the thinnest metal cell, causing the overall computation time to explode. Switching to an implicit method at this point allows Δt to be set much larger. "The equation form is the same," but changing the discretization strategy can change computation time by tens of times—this is the reality of numerical methods.
Upwind Differencing (Upwind)
1st-order Upwind: Large numerical diffusion but stable. 2nd-order Upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
2nd-order accuracy, but numerical oscillations occur for Pe > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Scheme (MUSCL, QUICK, etc.)
Suppresses numerical oscillations via limiter functions while maintaining high accuracy. Effective for capturing shocks or steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is the stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES
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