粘性散逸
Theory and Physics
What is Viscous Dissipation?
Professor, I have a vague image that viscous dissipation is "heat generation due to viscosity," but what exactly is this phenomenon?
It's a phenomenon where the kinetic energy of a fluid is irreversibly converted into internal energy (heat) by the action of viscous forces. At the molecular level, it's the dissipation of kinetic energy due to molecular friction accompanying shear between fluid layers.
To give some everyday examples:
- Tire rubber heating up during driving (deformation dissipation of a viscoelastic body)
- High temperatures during Space Shuttle atmospheric re-entry (viscous dissipation inside the shock wave)
- Localized heating at the gate in polymer injection molding
- Slight temperature rise of water in a dam spillway
The water temperature rises even from a dam's falling water!
Theoretically, yes. If all the potential energy of water falling from a height $h = 100\,\text{m}$ were converted to heat:
This almost matches the value Joule measured experimentally in the 19th century.
Derivation of the Viscous Dissipation Function
Writing the viscous dissipation term in the energy equation accurately:
For incompressible Newtonian fluids, this becomes:
In tensor notation:
The important point here is that $\Phi \geq 0$ always holds. This is a requirement of the second law of thermodynamics; viscous dissipation is a one-way process that always converts kinetic energy into heat.
So it's an "irreversible process."
Brinkman Number — An Indicator of Viscous Dissipation Importance
The dimensionless number used to judge whether viscous dissipation can be neglected is the Brinkman number.
$\mu$ is viscosity, $U$ is characteristic velocity, $k$ is thermal conductivity, $\Delta T$ is characteristic temperature difference.
| Br Value | Interpretation | Example |
|---|---|---|
| $\text{Br} \ll 1$ | Viscous dissipation negligible | Normal water flow in a pipe |
| $\text{Br} \sim O(1)$ | Viscous dissipation should be considered | Polymer processing, lubricant films |
| $\text{Br} \gg 1$ | Viscous dissipation dominant | High-speed flow, atmospheric re-entry vehicles |
So we only need to include the dissipation term in the energy equation when the Br number is large, right?
Exactly. Unnecessarily including the dissipation term increases computational cost and worsens convergence, so prior judgment using the Br number is important.
Discovery History of Viscous Dissipation — From Joule Heating to Fluid Friction Heat (Stokes' Contribution)
The person who formulated viscous dissipation in fluids (the phenomenon where flow energy is converted to heat) hydrodynamically was George Gabriel Stokes (G.G. Stokes). In his 1845 paper "On the Theories of the Internal Friction of Fluids in Motion," he mathematically described the relationship between the viscous stress tensor and energy dissipation, clarifying the physical meaning of the viscous term in the N-S equations. Before this, Joule (1843) had demonstrated electrical-to-heat conversion (Joule heating) in solid resistors, and fluid friction heat can be positioned as its fluid dynamics counterpart. Aerodynamic heating in high-speed projectiles (ballistic missiles, spacecraft re-entry) originates from this viscous dissipation, and Stokes' 150-year-old theory is directly connected to the design of the Apollo capsule's heat shield.
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? Looking only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost drops significantly, solving first in steady-state 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 becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misunderstanding: "Convection and conduction are similar things" → They're 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, right? 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. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flow (slow, viscous), diffusion is dominant. Conversely, in high Re number flow, convection overwhelms and diffusion becomes a supporting role.
- Pressure term $-\nabla p$: When you push a syringe plunger, 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 becomes the force pushing the fluid. Dam water release works on the same principle. On a weather map, where isobars are densely packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, it might be due to mixing up absolute/gauge pressure.
- Source term $S_\phi$: Warmed air rises upward—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed up 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," expressed by the source term. 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 room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian fluid assumption: Linear relationship between shear stress and strain rate (non-Newtonian fluids require viscosity models)
- Incompressibility assumption (for Ma < 0.3): Treat density as constant. For Mach number 0.3 and above, consider compressibility effects
- Boussinesq approximation (Natural Convection): Consider density changes only in the buoyancy term, using constant density in other terms
- Non-applicable cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (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 coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition |
| CFL number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Incorporating Viscous Dissipation into the Energy Equation
When calculating viscous dissipation in CFD, which equations do you modify and how?
Add the viscous dissipation function $\Phi$ as a source term to the energy equation.
In finite volume method discretization, $\Phi$ is evaluated as an integral over the cell volume:
Calculate the shear rate from the velocity gradient at each cell center, evaluate $\Phi = \mu |\dot{\gamma}|^2$, and add it as a source term to the energy equation.
Don't we need to modify the momentum equations?
Basically, no. However, if the temperature rise due to viscous dissipation affects viscosity (temperature-dependent viscosity), then feedback occurs to the momentum equations as well. In this case, the energy and momentum equations need to be solved coupled.
Viscous Dissipation in Turbulent Fields
In turbulent fields, the treatment of viscous dissipation differs from laminar flow. In the RANS (Reynolds-averaged) framework:
$\overline{\Phi}_{\text{mean}}$ is dissipation from the mean velocity field, $\varepsilon$ is the turbulent kinetic energy dissipation rate (the very $\varepsilon$ from the $k$-$\varepsilon$ model).
In fact, $\varepsilon$ is automatically considered as a source term in the energy equation for most turbulent problems. The dissipation term in the turbulent kinetic energy equation is exactly this:
$P_k$ is the turbulent kinetic energy production term, $\varepsilon$ is the dissipation rate. This $\varepsilon$ ultimately becomes heat.
So the $\varepsilon$ in the $k$-$\varepsilon$ model is directly connected to viscous dissipation!
Exactly right. The final stage of the turbulent kinetic energy cascade is viscous dissipation ($\varepsilon$). In Kolmogorov scaling, the smallest scale where dissipation occurs is:
Notes on Numerical Accuracy
Let's summarize points to note in the numerical calculation of viscous dissipation.
| Item | Points to Note | Countermeasures | ||
|---|---|---|---|---|
| Velocity gradient accuracy | $\Phi$ is proportional to the square of the velocity gradient, so gradient accuracy directly affects it | Use second-order or higher schemes | ||
| Mesh resolution | Regions where wall shear is maximum | Make wall mesh sufficiently fine | ||
| Distinguishing from numerical dissipation | Numerical viscosity from upwind differencing also generates non-physical dissipation | Reduce numerical dissipation with high-order schemes | ||
| Non-Newtonian fluid | $\Phi = \eta(\dot{\gamma}) | \dot{\gamma} | ^2$ is model-dependent | Verify implementation of Φ consistent with viscosity model |
We need to be careful not to confuse numerical dissipation with physical viscous dissipation.
Especially in LES (Large Eddy Simulation), the SGS (Sub-Grid Scale) model introduces additional dissipation. Separating physical dissipation from model dissipation directly affects accuracy.
Numerical Treatment of Viscous Dissipation — Ensuring Energy Conservation Accuracy in High-Speed Flows
In high Mach number flow or high viscosity fluid analysis, accurately solving the "viscous dissipation term Φ=τ:∇u" (where heat is generated from velocity gradients) in the energy equation is important. This term is proportional to the square of velocity, so it increases rapidly in high-velocity regions. The numerical problem is that in regions with coarse mesh, velocity gradients are underestimated, leading to underestimation of dissipation and breaking energy conservation. Practically, evaluate the importance of dissipation beforehand using the "Brinkman number Br=μU²/(kΔT)", and if Br>0.1, always enable dissipation. Also, in calculations including the dissipation term, temperature and velocity residuals influence each other, so convergence criteria (residual below 1e-6) need to be set more strictly.
Upwind Differencing (Upwind)
1st order upwind: Large numerical diffusion but stable. 2nd order upwind: Improved accuracy but risk of oscillation. Essential for high Reynolds number flow.
Central Differencing (Central Differencing)
2nd order accuracy, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flow.
TVD Scheme (MUSCL, QUICK, etc.)
Suppresses numerical oscillations while maintaining high accuracy via limiter functions. Effective for capturing shock waves or steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multi-physics. 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: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per timestep.