Viscous Dissipation
Viscous Dissipation: Theoretical Foundations
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 โ 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 โ 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.
Computational Methods for Viscous Dissipation
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 ฮฆ 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 โ Ensuring Energy Conservation Accuracy in High-Speed Flows
In high Mach number flow or high viscosity fluid analysis, accurately solving the "viscous dissipation term ฮฆ=ฯ:โu" (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=ฮผUยฒ/(kฮT)", 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.
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