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Theory and Physics
Overview
Professor, "real gas effects" essentially mean cases where the ideal gas assumption doesn't hold, right? In what situations does this become a problem?
Broadly speaking, there are two main situations. One is at high temperatures where gas molecules undergo vibrational excitation, dissociation, or ionization (e.g., hypersonic re-entry, plasma, etc.). The other is at high pressure and low temperature where intermolecular forces and molecular volume can no longer be ignored (e.g., supercritical CO₂ cycles, LNG processes, etc.). In both cases, the deviation from the ideal gas equation of state $pv = RT$ is fundamental.
We measure the deviation using the compressibility factor $Z$, right?
Exactly. The compressibility factor is defined as
For an ideal gas, $Z = 1$. For high-pressure natural gas, $Z \approx 0.8$, and for supercritical CO₂, $Z$ can drop as low as 0.2-0.5.
Equation of State (EOS)
What kind of equations of state are used instead of the ideal gas law?
Let's look at some representative EOS (Equation of State).
van der Waals equation:
Here, $a$ represents intermolecular attraction, and $b$ represents the excluded volume. This is historically important but has limited accuracy. In practice, Peng-Robinson (PR) or Soave-Redlich-Kwong (SRK) are widely used.
Peng-Robinson EOS:
Here, $\kappa = 0.37464 + 1.54226\omega - 0.26992\omega^2$, and $\omega$ is the acentric factor.
So if you have the critical temperature $T_c$, critical pressure $p_c$, and acentric factor $\omega$, you can use it for any substance, right?
Yes. For CO₂, $T_c = 304.1$ K, $p_c = 7.38$ MPa, $\omega = 0.225$. However, the PR-EOS tends to underestimate liquid density by 5-15%. Sometimes the Peneloux volume correction term is added to compensate for this.
High-Temperature Air Thermochemical Model
How are real gas effects on the high-temperature side modeled?
For air, the following phenomena occur sequentially depending on temperature.
| Temperature Range | Phenomenon | Effect |
|---|---|---|
| < 800 K | Ideal gas-like | $\gamma \approx 1.4$ |
| 800-2500 K | Vibrational excitation of O₂ | $\gamma$ decreases |
| 2500-4000 K | Dissociation of O₂ | Generation of O atoms |
| 4000-9000 K | Dissociation of N₂ | Generation of N atoms |
| > 9000 K | Ionization | Generation of e⁻, N⁺, O⁺ |
Assuming chemical equilibrium, the composition of each chemical species is determined by minimizing the Gibbs free energy. For finite-rate reaction models, Arrhenius parameters for each reaction are specified.
The specific heat ratio $\gamma$ changing with temperature seems like it would have a big impact.
Exactly. When the post-shock temperature exceeds 3000 K, $\gamma$ drops from 1.4 to 1.1-1.2. This causes the shock wave angle and density ratio to differ significantly from ideal gas predictions.
What van der Waals realized in 1873—molecules have "size"
The theoretical foundation for real gases was laid by the Dutch physicist Johannes van der Waals. In his 1873 doctoral thesis, he proposed the revolutionary idea for its time that "gas molecules have a finite size and there are attractive forces between them." The van der Waals equation of state derived from this was the first practical model capable of describing gas behavior in high-pressure, low-temperature regions. Modern engineering uses Peng-Robinson and Redlich-Kwong equations, which are refinements of van der Waals' concept. So, a doctoral thesis from 150 years ago directly influences the choice of equations of state in today's compressible CFD.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, 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's looking only at "after sufficient time has passed and the flow has settled down"—in other words, setting this term to zero. Since computational cost drops significantly, solving first with a steady-state assumption 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 due to air, the "carrier," transporting 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 misconception: "Convection and conduction are similar" → 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 they naturally mix. 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 makes the diffusion term stronger, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelms, 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 plunger side is high pressure, and the needle tip is low pressure—this pressure difference is 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. "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. When you switch to compressible analysis and suddenly get strange results, it might be due to mixing up absolute/gauge pressure.
- Source term $S_\phi$: Heated air rises—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 generated by a gas stove flame, Lorentz force applied to 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 source terms. What happens if you forget a source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—you get a physically impossible result where warm air from a heater in a winter room doesn't rise.
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): 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, be careful with 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
Numerical Implementation of EOS
When incorporating a real gas EOS into CFD, are there any computational issues?
There are several important points. For ideal gases, conversion from conservative variables ($\rho, \rho\mathbf{u}, \rho E$) to temperature or pressure can be done analytically, but for PR-EOS and others, iterative calculations become necessary.
Specifically, when internal energy $e$ and density $\rho$ are given, temperature $T$ is found from
using Newton's method. Here, $e_{departure}$ is the departure function derived from the EOS; for PR-EOS it is
This iterative calculation is required for each cell × time step, so computational cost increases.
How much slower does the calculation become compared to ideal gas?
Generally about 2-5 times slower. Since EOS evaluation counts dominate, using a look-up table method for speedup is common in practice. A 2D table of temperature and pressure is pre-computed, and at runtime, only interpolation is performed.
Numerical Difficulties with Supercritical Fluids
What's numerically difficult about the supercritical state?
Near the critical point, thermodynamic properties change rapidly. Constant-pressure specific heat $c_p$ peaks near the pseudo-critical temperature, and density also changes abruptly. These steep changes cause numerical oscillations or divergence.
For example, under supercritical CO₂ conditions (p = 8 MPa, T ≈ 305 K), $c_p$ can jump to more than 10 times its normal value. Density can change drastically from 700 kg/m³ to 200 kg/m³ with just a few Kelvin change.
The changes are that drastic? Special techniques in the numerical scheme are needed then.
Exactly. The recommended approaches are as follows.
- Table method: Generate property tables from the NIST REFPROP database and access them via bilinear interpolation
- Implicit method: Due to large density changes, implicit pressure-density coupling is essential for stability
- Mesh resolution: Refine mesh in regions where pseudo-critical transition occurs
- Time step: Use adaptive time stepping to handle rapid density changes
Extension of Riemann Solvers
Can Roe scheme or HLLC be used for real gases as well?
They can be used, but modifications are needed. The ideal gas assumption must be removed when calculating the Roe average state. In the real gas Roe scheme, the average speed of sound is calculated using a generalized $\Gamma = v/(c_p)(\partial p / \partial T)_v$ as in
(e.g., Vinokur's formulation). HLLC has less dependence on the EOS and is easier to implement, so it is often preferred for real gases.
Considering implementation effort, HLLC is more practical, right?
Yes. OpenFOAM's rhoCentralFoam is based on the KNP scheme and is independent of the EOS, so it has the advantage that real gas calculations can be done simply by swapping the EOS.
The "Not Enough Grid Points" Problem in Real Gas Look-up Tables
Calculating real gas equations of state (van der Waals, Peng-Robinson, etc.) in real-time slows down CFD by tens of times. Therefore, in practice, look-up tables (LUT) are used where thermodynamic quantities are pre-computed on temperature-pressure grid points. The problem is the trade-off: "Too few table grid points lead to large interpolation errors, too many increase memory and loading time." An "adaptive table" that concentrates dense grids in high-temperature, high-pressure regions where shock waves pass and uses coarse grids in calm regions is efficient, but designing it itself becomes a piece of technical know-how.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe > 2. Suitable for low Reynolds number, diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy via limiter functions. Effective for capturing shock waves and 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 methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is typically judged when residuals for the continuity equation, momentum, and energy each drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Pressure: 0.2-0.3, Velocity: 0.5-0.7 are typical initial values. If diverging, lower the relaxation factor. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5-20 iterations is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and then velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It's similar to two people leveling a shelf.
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