Detonation
Theory and Physics
Overview
Professor, what's the difference between a detonation and an explosion?
There are two forms of combustion propagation. Deflagration is subsonic flame propagation, like a typical burner flame. On the other hand, detonation is a phenomenon where a shock wave and a combustion wave combine and propagate at supersonic speeds. The propagation speed reaches about 1800 m/s for methane/air and about 2000 m/s for hydrogen/air.
So, the shock wave and combustion run together?
Yes. The leading shock wave adiabatically compresses the unburned mixture, raising its temperature, and the high temperature causes rapid chemical reactions. The reaction energy sustains the shock wave. This self-sustaining mechanism is the essence of detonation.
Chapman-Jouguet Theory
What is the Chapman-Jouguet (CJ) theory?
CJ theory is a thermodynamic theory for determining the propagation speed of a detonation wave. It adds the heat release $q$ from the chemical reaction to the Rankine-Hugoniot relations across the shock wave and applies the CJ condition (the flow velocity behind the detonation wave equals the local speed of sound).
In a simplified form, the CJ detonation velocity can be roughly written as follows.
Here, $\gamma$ is the specific heat ratio, and $q$ is the heat release per unit mass [J/kg].
There's also the concept of CJ Mach number, right?
The CJ detonation Mach number is given by the following equation.
For hydrogen/air (equivalence ratio 1.0), $M_{CJ} \approx 5.0$, and for methane/air, $M_{CJ} \approx 5.2$.
ZND Structure
What is the internal structure of a detonation wave?
In the ZND (Zel'dovich-von Neumann-Doering) model, the detonation wave has a three-layer structure.
1. Shock front (von Neumann spike): Unreacted gas is shock-compressed. Pressure reaches about twice the CJ value.
2. Induction zone: The delay interval before chemical reactions proceed. Corresponds to the ignition delay time.
3. Reaction zone: Rapid chemical reactions occur, reaching the CJ state.
Does a shorter induction zone length mean a more stable detonation?
Yes. The induction zone length $\Delta_i$ is directly linked to the cell size $\lambda$, and there is an empirical rule: $\lambda \approx (10-30)\Delta_i$. When this cell size is sufficiently small compared to the detonation tube diameter (tube diameter > several $\lambda$), stable detonation propagation is maintained.
So, detonation theory is a fusion of shock wave mechanics and chemical kinetics.
Exactly. To handle it with CFD, both numerical methods that accurately capture shock waves and chemical reaction rates at high temperatures and pressures are required.
Detonation—Why the "Detonation Wave" Propagates at "5–10 Times the Speed of Sound"
There are two types of combustion: "deflagration (subsonic flame propagation)" and "detonation (supersonic propagation where shock waves and combustion are integrated)." Detonation waves are described by Chapman-Jouguet (CJ) theory (1899–1905), and the speed at the "CJ surface," where the combustion gas flow becomes sonic, is the characteristic value. For hydrogen-air mixtures, the detonation velocity is about 2 km/s; for natural gas-air, about 1.8 km/s. In CAE, the propagation, cell structure, and transition (DDT: Deflagration-to-Detonation Transition) of detonation waves are analyzed using high-resolution numerical schemes that incorporate chemical reaction models into the Euler equations.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, the flow becomes steady, right? This "period of change" is described by the temporal term. The pulsation of blood flow due to the heartbeat, or the fluctuation of flow 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 this term is set to zero. Since computational cost is significantly reduced, 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 misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is 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. When viscosity is high, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion dominates. 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. "Where there is a pressure difference, flow is generated"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A common point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results become strange immediately after switching to compressible analysis, it might be due to confusion between absolute/gauge pressure.
- Source term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it is pushed upward 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 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 source terms. What happens if you forget the source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result, like warm air not rising in a room with the heater on 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): Density is treated 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 wave 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 | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition. |
| CFL number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability. |
Numerical Methods and Implementation
Details of Numerical Methods
What numerical methods are needed to solve detonation with CFD?
Numerical analysis of detonation differs significantly from typical RANS combustion analysis. It requires both high-resolution schemes that accurately capture shock waves and detailed chemical reactions at high temperatures and pressures.
Spatial Discretization
What schemes are suitable for resolving shock waves?
For shock wave capturing, high-order accuracy TVD (Total Variation Diminishing) schemes or WENO (Weighted Essentially Non-Oscillatory) schemes are used.
| Scheme | Accuracy | Features | Application |
|---|---|---|---|
| Roe + Minmod | 2nd order | Stable but high numerical diffusion | Initial studies |
| HLLC | 2nd order | Resolves contact discontinuities | General purpose |
| WENO-5 | 5th order | High accuracy but high computational cost | DNS/High-precision calculations |
| MUSCL-Hancock | 2nd order | Good cost-accuracy balance | Practical detonation calculations |
So, WENO is ideal, but MUSCL is sufficient for practical work?
Yes. The combination of MUSCL + HLLC Riemann solver is a practical compromise. However, the cell size should aim for 1/20 or less of the detonation cell width $\lambda$. For hydrogen/air (equivalence ratio 1.0, 1 atm), $\lambda \approx 10$ mm, so a mesh width of 0.5 mm or less is required.
Time Integration
What about time integration?
Since detonation wave propagation is a phenomenon on the order of microseconds, explicit time integration is common. However, the chemical reaction part is stiff, so operator splitting (Strang splitting) is used to separate fluid transport and chemical reactions.
Let me show typical parameter values.
| Parameter | Recommended Value | Remarks |
|---|---|---|
| CFL number | 0.3-0.5 | Conservative for shock wave capturing |
| Chemical reaction solver | CVODE (BDF) | Essential for stiff systems |
| Minimum time step | $10^{-9}$ s | To resolve the von Neumann spike |
| Mesh width | $\lambda/20$ or less | Minimum condition for resolving cell structure |
AMR (Adaptive Mesh Refinement)
Uniform fine meshes would lead to enormous computational cost, right?
That's where AMR shows its power. It refines the mesh only near the detonation wave front, leaving unreacted and reacted regions with coarse meshes. CONVERGE has automatic AMR as standard, and OpenFOAM can also achieve it with dynamicRefineFvMesh. AMR can reduce memory and computation time by 1-2 orders of magnitude.
What should be the refinement criterion for AMR?
It is common to use temperature gradient $|\nabla T|$ or pressure gradient $|\nabla p|$ as refinement sensors. The gradient of OH mass fraction is also effective for tracking the reaction zone.
So, numerical calculation of detonation is a trinity of shock-capturing schemes + stiff chemical reactions + AMR.
Exactly. If any one is missing, practical detonation calculation is not possible.
Why "Supersonic Schemes" are Needed for Detonation Calculations—The CFL 0.3 Wall
The first stumbling block in numerical detonation calculations is the discretization scheme. Since detonation waves propagate at supersonic speeds (equivalent to Mach 5–10), the second-order central difference schemes commonly used in regular combustion CFD diverge immediately. To handle this, shock-capturing schemes like WENO (Weighted Essentially Non-Oscillatory) or Roe's method are essential, and the CFL number must also be kept below 0.3. This means a time step one order of magnitude smaller than typical combustion calculations, increasing computation time by more than 10 times. This is why "detonation is considered one of the most difficult types of CAE calculations to handle."
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 Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy using 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 multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant
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