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Fluid mechanics Definition

Zeldovich–von Neumann–Döring detonation model

Builds on DetonationShock wave in gasesRankine–Hugoniot conditionsEuler equations

Definition

Consider a gas mixture with one irreversible reaction, described by a progress variable λ∈[0,1]\lambda \in [0, 1], with λ=0\lambda = 0 unburnt and λ=1\lambda = 1 burnt, and heat of reaction q>0q > 0. The reaction rate is a given function ω(p,v,λ)≥0\omega(p, v, \lambda) \ge 0 of the pressure, the specific volume and λ\lambda. The ZND model of a detonation with speed DD is a steady planar flow, seen in a frame moving with the front. It consists of the following parts.

  1. Ahead of the front, the unburnt gas is uniform, with state (p1,v1)(p_1, v_1) and λ=0\lambda = 0.
  2. At the front there is a non-reacting shock wave. Its jump conditions take the gas to the von Neumann state on the Rayleigh line p−p1=−m2(v−v1)p - p_1 = -m^2 (v - v_1), where m=D/v1m = D / v_1.
  3. Behind the shock there is a reaction zone, where the one-dimensional reactive Euler equations hold. In this zone the gas moves along the same Rayleigh line, and λ\lambda grows from 00 to 11 by dλ/dt=ωd\lambda / dt = \omega along particle paths.

Remarks

In the reaction zone the state at progress λ\lambda lies on the intersection of the Rayleigh line with the partial Hugoniot curve with heat release λq\lambda q. At λ=1\lambda = 1 the gas reaches the full reactive Hugoniot curve. A steady solution reaching λ=1\lambda = 1 exists for D≥DCJD \ge D_{CJ} under the usual convexity assumptions, and for D=DCJD = D_{CJ} the final state is the Chapman–Jouguet state, where the flow is sonic.

Example: take a mixture whose rate ω\omega is positive for λ<1\lambda < 1, and the Chapman–Jouguet speed. In the ZND wave the pressure jumps at the shock to the von Neumann spike. Then it decreases monotonically to the Chapman–Jouguet pressure as λ\lambda increases to 11.

Non-example: the Chapman–Jouguet model, which treats the detonation as a single discontinuity from the unburnt to the burnt state, has no reaction zone and fails clause 3.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Conservation law, concept, no evidence levelConservation lawconceptDetonation, concept, no evidence levelDetonationconceptEuler equations, concept, no evidence levelEuler equationsconceptShock wave in gases, concept, no evidence levelShock wave in gasesconceptWeak solution, concept, no evidence levelWeak solutionconceptRankine–Hugoniot conditions, statement, no evidence levelRankine–Hugoniot cond…statementZeldovich–von Neumann–Döring detonation model, definition, no evidence levelZeldovich–von Neumann…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Conservation law concept
  2. Detonation concept
  3. Euler equations concept
  4. Shock wave in gases concept
  5. Weak solution concept
  6. Rankine–Hugoniot conditions statement
  7. Zeldovich–von Neumann–Döring detonation model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/ZND_detonation_model
About this page
Tag12EPT5G5QV
URIm2://12EPT5G5QV
Kinddefinition
WritingW0 stub
MSC76L05, 80A25, 76V05
DefinesZeldovich–von Neumann–Döring detonation model
UsesDetonation, Shock wave in gases, Rankine–Hugoniot conditions, Euler equations
Changed3 November 2026

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