w. cederbond and f.k. lu- mach reflection induced detonation in a reactive flow

8/3/2019 W. Cederbond and F.K. Lu- Mach Reflection Induced Detonation in a Reactive Flow

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Mach Reflection Induced Detonation in a Reactive

Flow

W. Cederbond and F.K. Lu

Aerodynamics Research Center, Mechanical and Aerospace Engineering Department, Box

19018, University of Texas at Arlington, Arlington, TX 76019, USA

Abstract. A comparison of a chemically reactive flow versus a non-reactive flow is made inthis work to show the possibility of the presence of a detonation wave associated with a Machstem also known as a Mach reflection wave. A reactive, inviscid and unsteady flow over atwo-dimensional wedge is observed. Then, it is compared to a non-reactive flow over the samegeometry and under the same conditions. A range of deflection angles and incoming flow Machnumbers is used in this study. The Euler equations are discretized using a finite-volume approachand a two-step explicit Runge-Kutta integration scheme is implemented together with a point-implicit treatment of the source terms to obtain a time-accurate solution. In addition, Roes flux-difference splitting scheme extended to non-equilibrium flow is used for the cell face fluxes, andthe MUSCL approach is used for higher-order spatial accuracy. For the purpose of constructingan efficient numerical tool, while maintaining a reasonable accuracy, a two-step global model fora hydrogen-air mixture was used.

1 Introduction

Consider that M 1 in Fig. 1 is only slightly above the minimum Mach number necessary fora straight, attached shock wave at the given deflection angle θ. For this case, the obliqueshock is simply a straight, attached incident shock. However, the Mach number decreasesacross a shock (i.e., M 2 < M 1). This decrease may be enough such that M 2 is not above

the minimum Mach number for the required deflection angle θ through the reflectedshock. In such a case, a solution for a straight reflection shock wave is not possible but aMach reflection occurs, as depicted in Fig. 1. Here, the originally straight incident shockbecomes curved as it nears the upper boundary and becomes a normal shock wave there.This allows the streamline at the wall to continue parallel to the boundary behind theshock intersection. In addition, a curved reflected shock branches from the normal shockand propagates downstream [1].

The shock at the head of the wave is a jump discontinuity. In order to capture thisdiscontinuity, and to study the flow with high accuracy, a numerical algorithm had to beimplemented and an accurate scheme had to be chosen. Most programs use the upwind orflux-split algorithms that are known to yield accurate solutions of shock-wave dominatedflows due to their superior shock capturing properties.

In this study, a two-dimensional time-accurate numerical simulation model is usedfor oblique shock waves [2] The code formulates the physical phenomena as preciselyas possible including chemical and thermal non-equilibrium, and numerically solves theresulting mathematical formulation as accurately as possible as well. The code uses acombination of point-implicit scheme [3] that treats the chemical source terms implic-itly and all other terms explicitly, and a local ignition averaging is applied to the globaltwo-step reaction model for efficient time-accurate solution of a propagating detonationwave. The partition of internal energy is based on the two-temperature model, and the



2 Cederbond and Lu

Fig. 1. Mach wave reflection

vibrational energy of each species is obtained by subtracting out fully-excited transla-tional and rotational energy from total internal energy. For an accurate capture of theshock wave both in time and space, Roes flux-difference split scheme is combined with theRange-Kutta integration scheme. The chemical reaction for a stoichiometric N2–O2–H2

flow is described by a simple two-step reaction involving five species, as follows:

H2 + O2 → 2OH

2OH + H2 → 2H2O

Several different configurations are investigated in this work, with the goal of deter-mining the occurrence of detonation behind the Mach stem of a reactive gas flow. Oncedetonation is detected, a comparison is made with an inert gas flow to show that thedetonation present in the reactive gas flow is solely due to chemical reaction in the flow.The results are then validated against theoretical Chapman–Jouguet values. Extensivecalculations and simulations were performed with different mesh sizes to select the propermesh size that provided adequate and reasonable CPU time without compromising the

resolution of the physical process.For simplicity, a pair of wedges of opposite family and with equal angles of deflectionθ is chosen and a two-dimensional, inviscid, non-conducting unsteady flow is assumed.In addition to the range of deflection angles, a range of incoming flow Mach numbers M 1form a matrix of simulations to cover the most susceptible cases where detonation is evenpossible. Only the combination of θ, from 5 through 20◦, and M 1, from 1.16 through 3,that are most likely to induce detonation associated with a Mach stem are considered inthis study.

2 Mathematical Formulation

2.1 Governing Equations

In a two-dimensional, Cartesian coordinate system, the Euler equations take the followingform:

∂ U

∂t+

∂ F

∂x+

∂ G

∂y= S (1)



Mach Reflection Induced Detonation 3

where U is the vector of conserved variables, F and G are the convective flux vectors,and S is the vector of source terms. The vectors are written as

U =

⎡⎢⎢⎢⎢⎣

ρs

ρu

ρv

ρevρE

⎤⎥⎥⎥⎥⎦

, F =

⎡⎢⎢⎢⎢⎣

ρsuρu2 + p

ρuv

ρuev(ρE + p)u

⎤⎥⎥⎥⎥⎦

, G =

⎡⎢⎢⎢⎢⎣

ρsvρuv

ρv2 + p

ρvev(ρE + p)v

⎤⎥⎥⎥⎥⎦

, S =

⎡⎢⎢⎢⎢⎣

ws

00

wv

0

⎤⎥⎥⎥⎥⎦

(2)

In this equation, the subscript s ranges from 1 to N s, where N s is the number of species.The first row represents species continuities, followed by the two momentum conservationequations for the mixture. The next row describes the rate of change in the vibrationalenergy, and the final row is the total energy conservation equation. In addition, u and v

are the velocities in the x and y directions respectively, ρ is the mixture density, p is thepressure, ev is the vibrational energy, and E is the total energy per unit mass of mixture.In addition, ρs is the sth species density, ws is the mass production rate of species s per

unit volume, and wv is the vibrational energy source term [2].

2.2 Thermodynamic Properties

A general representation of species internal energy includes a portion of the internalenergy in thermodynamic equilibrium and the remaining portion in a non-equilibriumstate. The equilibrium portion of the internal energy is the contribution due to the trans-lational and internal modes that can be assumed to be in equilibrium at the translationaltemperature T . The remaining non-equilibrium portion is the contribution due to inter-nal modes that are not in equilibrium at the translational temperature , but may beassumed to satisfy a Boltzmann distribution at a different temperature [2]. The speciesinternal energy based on the two-temperature model can be written as follows:

es = eeq,s(T ) + ev,s(T v) (3)

where eeq,s the equilibrium is portion of the internal energy and ev,s is the vibrationalenergy which is not in thermodynamic equilibrium.

2.3 Chemical Kinetics

High temperature flows typically involve some chemical reactions, and the time scale inwhich the chemical reactions take place is important in the estimations of the flow fieldproperties, especially if the flow speed is sufficiently large that the flow timescale is com-parable to the chemical reaction timescale. For accurate modeling of a detonation wave,especially in the detonation front where rapid chemical reactions take place in the shockcompressed region, species continuity equations based on the chemical kinetics should

be solved together with fluid dynamic equations to account for the possible chemicalnon-equilibrium [3].

2.4 Vibrational Energy Relaxation

The energy exchange between vibrational and translational modes due to intermolecularcollisions can be described by the Landau–Teller formulation where it is assumed that



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the vibrational level of a molecule can change by only one quantum level at a time. Theresulting energy exchange rate is given by

Qv,s = ρse∗

v,s(T )− ev,s

< τ s >(4)

where e∗v,s

(T ) is the vibrational energy per unit mass of species s evaluated at the localtranslational-rotational temperature, and < τ s > is the averaged Landau–Teller relax-ation time of species s [3].

2.5 Numerical Formulation

A discretized set of equations is derived in this section from the governing partial differ-ential equations using the finite-volume method. The advantage of this method is its useof the integral form of the equations, which ensures conservation, and allows the correcttreatment of discontinuities. Non-equilibrium flows involving finite-rate chemistry and

thermal energy relaxation often can be very difficult to solve numerically because of thestiffness. The stiffness in terms of time scale can be defined as the ratio of the largestto the smallest time scale. The point implicit treatment is known to reduce the stiffnessof the system by effectively rescaling all the characteristic times in the flow fields intothe same order of magnitude. Temporal accuracy can be added by using Runge–Kuttaintegration schemes instead of first-order accurate Euler integration. The flux-differencesplit algorithm is used to solve a local Riemann problem at the cell interface in order todetermine the cell-face flux.

2.6 Geometric Configuration and Grid Study

The geometric configuration used in this study is shown in Fig. 2 below. The two-dimensional wedge is placed in the supersonic, reactive, inviscid, unsteady flow. Thedeflection angle is varied between 5 and 25◦. The height of the domain used for thecomputational simulation is varied between 0.05 and 0.1 m. The length of the wedge isvaried between 0.1 and 0.2 m, depending on the deflection angle θ. The domain is variedbetween 101× 51 and 201× 101 grid points, based on a mesh size of 1 mm.

Fig. 2. Geometric configuration.

The incoming supersonic flow comprises a premixed stoichiometric hydrogen–air mix-ture. The initial pressure and temperature of the flow are fixed at p1 = 2 atm andT 1 = 700 K respectively. The Mach number M 1 is varied between 1.16 and 6.0.



Mach Reflection Induced Detonation 5

3 Results

A flow past a 15◦ wedge is shown in Fig. 3 via isobars. The incoming Mach number is 1.9,

while the flow pressure and temperature are 2 atm and 700 K respectively. The heightand length of the domain remain the same as the previous cases, 0.075 m and 0.15 mrespectively.

Fig. 3. Example: M 1 = 1.9, θ = 15◦, p1 = 2 atm, T 1 = 700 K, h = 0.075 m, l = 0.15 m.

This is one of the few cases examined that is interesting in achieving the objectiveof this. At t = 0.270 ms, two regular reflection waves are seen first. At t = 0.275 ms, adetonation kernel appears at the upper right corner of the domain. The detonation kernel

is in the form of a third reflection wave. At t = 0.280 ms, the detonation becomes a fullydeveloped wave propagating upstream forming a Mach stem with the second reflectionwave. Simultaneously, a normal wave starting at the upper wall, connecting the incidentwave with the first reflection wave is appearing, creating a Mach reflection (MR) whichis clearly shown in the upper enlarged picture next to the frame. At t = 0.295 ms,the phenomenon sought is making its first appearance; a detonation wave behind theMach stem, developing instantly to a full detonation wave as seen at t = 0.300ms. Atthis moment, two detonation waves are present. The first detonation wave propagates



6 Cederbond and Lu

upstream while the second detonation wave at the Mach stem is getting longer. Finallyboth waves meet at t = 0.305 ms. The next frame at t = 0.310 ms reveals that the firstdetonation wave is overtaking the Mach stem induced detonation wave as it almost seems

to be stationary. Also shown even clearer at t = 0.375 ms where the first detonation waveis clearing the domain to the left, the MR-induced detonation wave is remaining as it isinteracting with the detonation shock formed past the first detonation flame.The mixtureis ignited at the center of the tube (0.15 m).

4 Conclusions

A Mach reflection induced detonation has been examined numerically. Inviscid, non-heat-conducting flow equations are fully coupled with the chemical kinetics of the reactionsfor a general description of the chemical non-equilibrium. Vibrational energy conserva-tion based on the two-temperature model is used to account for the possible thermalexcitation and the relaxation of the vibrational energy mode. The governing equations

are discretized using the finite-volume formulation, and a time-accurate solution is ob-tained from the Runge-Kutta integration scheme with a point-implicit treatment of thesource terms. Roe’s flux-difference splitting scheme extended to non-equilibrium flow isimplemented for the cell face fluxes, and the MUSCL approach is used for higher-orderspatial accuracy.

A Mach reflection (MR) induced detonation was captured and analyzed. The two-dimensional simulation showed complex wave interactions. The wave strength of thedownstream moving detonation is stronger than that of the upstream moving detonation,which is different from that encountered in the one-dimensional case. This differencebetween the one- and two-dimensional cases is due to the geometry and the subsequentflow produced. It was found that the interaction between a steady oblique shock and apropagating normal detonation wave produced a vortex that moved toward the wedgesurface.

References

1. J.D. Anderson, Jr.: Fundamentals of Aerodynamics, 3rd ed., McGraw-Hill, New York (2001)2. H.W. Kim, F.K. Lu, D.A. Anderson and D.R. Wilson: Numerical Simulation of Detonation

Process in a Tube. CFD J., Vol. 12, No. 2, pp. 227–241 (2003)3. T.R.A. Bussing and E.M. Murman: “Finite-Volume Method for the Calculation of Com-

pressible Chemically Reacting Flows,” AIAA J., Vol. 26, No. 9, pp. 1070–1078 (1988)

w. cederbond and f.k. lu- mach reflection induced detonation in a reactive flow

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