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Detonation in supersonic radial outflow
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A. Kasimov and S. Korneev
King Abdullah University of Science and Technology
Introduction
Steady-state profiles
We analyzed standing converging two-dimensional detonation waves in a radially expanding supersonic flow of the combustible mixture. Such detonation may appears in the divergent flow of a compressible reactive gas between two parallel walls. Assuming no friction losses the flow between walls is described by the system of Euler equations, which in the laboratory frame are:
We calculated the steady-state profiles for various mixture models. At first we consider one stage Arrhenius kinetics with ideal gas equation of state. The reaction rate and equation of state are:
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where determines the progress of the Geometry of the system chemical reaction. Two-stages model We impose radially symmetric incoming assumes additional equation for the supersonic flow boundary condition at the progress of the induction parameter: source of radius . The gas from the source expands adiabatically and at the distance there shock. The shock heats mixture up and initiate formation where and is reaction and induction of induction zone. When the induction rates. The total energy is given by: process is finished, reaction starts to release heat and reaction zone is formed. where the form of internal energy is specified by the model. Consider steady-state radially symmetric solution of the system. The system of equations can be reduced to two conservation laws and three ODEs. It completely defines the solution together with Rankine-Huguenot conditions at the shock and boundary conditions at the source. The system also assumes regularization conditions at the sonic point which can be used to determine the radius of the shock. The figure below shows general structure of the steady-state solution.
General scheme of the radially symmetric steady-state solution
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Numerical simulations
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Result of 2D simulations Figure on the top shows the results of 2D simulations of system with Arrhenius kinetics. The simulation was started with steady-state solution which has very thin induction zone. The solution expands and to keep the detonation in the bounded area we surrounded by obstacles. Figure below shows the result of 2D simulation of twostages model for hydrogen oxygen mixture. The solution collapses.
The steady-state profiles for Arrhenius kinetics is shown on the figure below. Arrhenius kinetics with local density dependence and ideal gas equation of state was also considered. Calculated steady-state profiles for this case has similar structure and scales as for regular Arrhenius law. More realistic two-stages reaction model for hydrogen-oxygen mixture was investigated. The model was developed by Nikolaev Steady-state radially symmetric pressure profiles at el. by reducing all steps of the Two types of the solutions which are common for regular Arrhenius hydrogen-oxygen reaction to a kinetics and modified. The pressure profiles are given for the regular simple one step expression. The Arrhenius kinetics. Parameters for the left hand-side figure are: reaction rate of the model describes rate of change of mean molecular and for the right hand-side figure difference in parameters are: weight of the mixture
where is mean molecular weight, and is dissociation energy, is mean molecular weight in completely dissociated state and is mean molecular weight in completely recombined state. The equation of state does not remain ideal gas anymore. Adiabatic index now depends on the temperature and on the molecular weight.
Conclusions
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This equation of state applies only to the reaction zone, the induction zone is considered as ideal adiabatic flow. Figure below shows the steady-state profiles for the two-stages model.
Steady-state profiles for the model with two stages for hydrogen-oxygen stoichiometric mixture Flow velocity, pressure and temperature profiles of the steady-state solution. The figures are in dimensional units. The pressure at the source is density is flow velocity is
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The detonation exists for a reasonable range of flow parameters.
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Depending on the mixture, the detonation may collapse or may expand.
Contacts
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Prof. Aslan Kasimov
[email protected]
