WAVES IN REACTING FLOWS

CNRS/Inst. de Mecanique des Fluides de Toulouse Toulouse, Prance 

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The main reason why numerical waves have not been discussed much in the CFD community is that most RANS codes use excessive artificial viscosity and large turbulent viscosity levels (due to turbulence models) which kills all numerical waves. They also kill all acoustic waves and all hydro-dynamic modes and cannot be used for the present needs of combustion research. Methods which can compute accurately waves in reacting flows must use centered schemes and LES (or DNS) formulations in order to avoid damping all waves (physical and numerical). A convenient way to illustrate this point is to compare the various viscosities pla3ung a role in a CFD code ... 

Example 4.8 Chemical reactions and reacting flows The extension of the theory of linear nonequilibrium thermodynamics to nonlinear systems can describe systems far from equilibrium, such as open chemical reactions. Some chemical reactions may include multiple stationary states, periodic and nonperiodic oscillations, chemical waves, and spatial patterns. The determination of entropy of stationary states in a continuously stirred tank reactor may provide insight into the thermodynamics of open nonlinear systems and the optimum operating conditions of multiphase combustion. These conditions may be achieved by minimizing entropy production and the lost available work, which may lead to the maximum net energy output per unit mass of the flow at the reactor exit. 

Like in classical acoustics in cold flows [304], wave equations can also be derived for reacting flows but they are much more complex [270 340]. Af-... 

Numerical Waves in High-Fidelity Simulations of Reacting Flows... 

Figure 6, which is for the same simulation as Fig. 4, shows sectional averages of the pressure, particle velocity, and density as a function of position with the dashed lines denoting the initial values in the undisturbed crystal. These quantities peak at the front and then relax during the reaction and expansion behind the shock. The shockfront shape is in accord with ZND continuum theory for unsupported planar detonations, which predicts a von Neumann peak near the front followed by a reacting flow and a (Taylor) rarefaction wave. The peak pressure around 1.0 eV/A (Fig. 6, top), which corresponds to an effective pressure of approximately 400 kbar, and maximum particle velocity of 4.8 km/s (Fig. 6, middle) are consistent... 

The equations of non-equilibrium reacting flows derived in the state-to-state, multi-temperature and one-temperature approaches were applied for calculations of distributions and macroscopic parameters in particular flows of air components behind shock waves, in nozzles, in non-equilibrium boundary layer (see Nagnibeda Kustova (2009) and references in this book). On the basis of obtained distributions, global reaction rates (92) were calculated in relaxation zone behind the shock wave Kustova Nagnibeda (2000) and in nozzle expansion Kustova et al. (2003) in different approaches. The results obtained for the relaxation zone behind the shock wave at the following free stream conditions Tq = 293 K, Pq = 100 Pa, Mq = 15 are presented in Fig. 3. 

In this Chapter, the theoretical models for non-equilibrium chemical kinetics in multi-component reacting gas flows are proposed on the basis of three approaches of the kinetic theory. In the frame of the one-temperature approximation the chemical kinetics in thermal equilibrium flows or deviating weakly from thermal equilibrium is studied. The coupling of chemical kinetics and fluid dynamics equations is considered in the Euler and Navier-Stokes approximations. Chemical kinetics in vibrationaUy non-equilibrium flows is considered on the basis of the state-to-state and multi-temperature approaches. Different models for vibrational-chemical coupling in the flows of multi-component mixtures are derived. The influence of non-equilibrium distributions on reaction rates in the flows behind shock waves and in nozzle expansion is demonstrated. 

The behavior of a reactive wave depen ds on the flow of its reacting and product-gases. The conservation laws lead to systems of partial differential equations of the first order which are quasilinear, ie, equations in which partial derivatives appear linearly. In practical cases special symmetry of boundary and initial conditions is often invoked to reduce the number of independent variables. 

In the normal mechanism the reaction runs simultaneously over the entire cross-section of the tube the curves presented in 11.5 illustrate the change in pressure, temperature and composition. We axe fully justified in using an approach in which we consider all quantities characterizing the state to be dependent only on the distance of the point from the shock wave front. In the case of the SM, in the mechanism which we have proposed here for rough tubes, in each intermediate cross-section part of the substance has not reacted at all (the core of the flow) and part of the substance has completely reacted (the peripheral layers) the states of the two parts— composition, temperature, specific volume—are sharply different. The only common element is the pressure, which is practically identical in a given cross-section in the two parts of the flow (in the compressed, but unreacted mixture and in the combustion products), but which changes as combustion progresses from one cross-section to another. 

The wave field produced in the steady, two-dimensional flow of a reacting gas past a wavy wall has been treated in [63] and [64]. Lick [65] has obtained solutions to the nonlinear, steady, two-dimensional conservation equations governing the flow of a reacting gas mixture about a blunt body. Reviews of these and other studies may be found in [1], [2], and [66]-[71]. 

B. T. Chu, Wave Propagation in a Reacting Mixture, 1958 Heat Transfer and Fluid Mechanics Institute, Stanford Stanford University Press, 1958,80-90 Wave Propagation and the Method of Characteristics in a Reacting Gas Mixture with Application to Hypersonic Flow, Tech. Note No. 57-213, Wright Air Development Center (1957). 

Optional for paraffin sections if cross-reacting antibody follows Rinse slides in TBS and apply an intermediate micro-wave heating at 300 W power in 150 mL of citrate buffer (pH 6) for 10 min. Cool in flowing tap water. 

The first thing that we found was that defects had to be a minimum size to have any effect whatsoever. A vacancy, or even a divacancy, can be passed over by a shock wave without so much as a hiccup. However, beyond a certain size, a void (or vacancy cluster) can produce not only a warp in the shock front, but serious heating upon its collapse. Such an overheat can subsequently lead to initiation of chemical reaction originating at the void, even for a shock wave that would not have been strong enough to produce initiation in a perfect crystal (see Fig. 12). For materials that don t react chemically, such hotspots can still be nucleation sites for other shock-induced phenomena, such as plastic flow or polymorphic phase transformations. 

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