Direct Numerical Simulation of a Transitional Jet in Crossflow with Mixing and Chemical Reactions

The paper describes DNS of a jet in crossflow at Re=650 and jet-to-crossflow velocity ratio R=3.3. Laminar boundary conditions are employed which makes the present flow configuration advantageous for benchmarking. Use is made of local grid refinement in the region of transition to turbulence. Comparison with Re=325 is carried out for some appropriate quantities. Nine passive scalars, reacting and non-reacting, are introduced. This allows to systematically study the influence of the Schmidt number and the Damköhler number with first order reactions. The presence of chemical reactions is a particular feature of the present study. Quantitative results for mean flow and turbulent quantities are provided and discussed. Also, the interaction between turbulence and scalar is quantified. Reference data are made available for later benchmarking. INTRODUCTION The configuration of a jet issuing from a pipe into a crossflow (JICF) appears frequently in chemical, pharmaceutical, environmental and combustion engineering, to name but a few application areas. The complex vortical structures of this flow and its good mixing capabilities make it a target of intense investigation for both experimental and numerical groups (Margason 1993). It is hence useful to consider such a situation as a benchmark configuration to assess statistical turbulence models as well as Large Eddy simulations (LES). Earlier work of the present authors was concerned with turbulent inflow conditions of the jet (Fröhlich et al. 2004, Denev et al. 2005a, 2005b). In the present paper, the lowReynolds number regime is considered with laminar pipe flow, where steady boundary conditions can be applied. This is advantageous for two reasons: First, it removes the ambiguity of specifying in a benchmarking simulation the turbulent fluctuations at the inlet. Technical details in this respect may render comparisons of results from different groups difficult, as well as tracing deficiencies in a single computation back to a particular feature of the numerical method. Second, the transition of the jet from laminar to turbulent, now occurring inside the computational domain, provides a challenge for turbulence models (Denev et al. 2006). Kelso et al. (1996) and Lim et al. (2001) performed experiments in the low-Reynolds number regime. They are however mainly concerned with flow visualization and analysis of coherent structures. Recent Direct Numerical Simulations (DNS) are reported by Muppidi and Mahesh (2005a, 2005b, 2006) for a case with velocity ratio R=u∞/wjet=5.7 and Reynolds number Rejet=wjetD/ν=5000, with “∞” indicating crossflow conditions and D the diameter of the pipe. This value of Rejet requires turbulent boundary conditions at some point upstream of the outlet of the jet. In the third of these references, a passive nonreactive scalar with Schmidt number Sc=1.49 was introduced with the jet. The present work aims to supply a reference solution in a similar spirit which differs from the above simulations mainly in the following: (1) The Reynolds number is lower, such that the pipe flow is laminar. (2) Reacting and nonreacting passive scalars are introduced to provide reference data for mixing and chemical reactions. This feature is unique as there is no similar attempt in the literature the authors are aware of. The parameters have been selected such that the transition zone is short which is advantageous when used as benchmark to reduce the computational effort. FLOW CONFIGURATION AND PARAMETERS Fig. 1 presents the flow configuration and the orientation of the axes. The reference length is the pipe diameter, so in non-dimensional units D=1. The other geometric quantities in the figure are Lx=20, Ly=Lz=13.5, lx=3, lz=2. The reference velocity is the crossflow velocity u∞ set to unity in the computations so that the Reynolds number is Re= u∞D/ν. In the dimensionless simulations 1/Re takes the place of the viscosity and is an input parameter. The inflow condition for the crossflow is steady. The shape of the boundary layer at this position is given as a function of the distance from the closest channel wall dn as uin(y,z)=1.0exp(-4.5dn), resulting in a boundary layer with δ99=1.03D. In the pipe, a distance lz=2 upstream of the jet outlet a parabolic velocity profile corresponding to fully developed laminar pipe flow is imposed. This length of the pipe is sufficient according to literature data and own preliminary simulations. No-slip boundary conditions were applied at all solid walls. The present simulation was performed with Re=650. For comparison, a second simulation with Re=325 was also undertaken. The velocity ratio was R=wjet_bulk/u∞=3.3 in both cases, which ensures the jet trajectory to remain remote from the wall. The inflow conditions were not changed when changing the Reynolds number. The transport of several scalars, reacting and nonreacting, was simulated. Since these are all passive, they can be computed all together in the same simulation. An overview is provided in Table 1: Mixing is studied by means of three passive non-reacting scalars, introduced with the jet, having different Schmidt number. Furthermore, three independent model reactions (Ai+Bi=Pi, i=1, 2, 3) are computed, with species Ai introduced in the pipe, Bi in the crossflow, and Pi being the products. The reaction rates in the equations read rri = Dai Ai Bi. Dalton’s law is fulfilled for each scalar/reaction separately, so that no equation for the product needs to be solved. Since the density is constant and normalized to 1, the concentration variables ci simultaneously represent mass and volume concentration. The parameters in Table 1 were selected to study the influence of different Damköhler numbers (Da=0.5 and 1.0) as well as different Schmidt numbers (Sc=1.0 and 2.0) with reaction. These variations are small to avoid degradation of accuracy (steeper fronts) as well as stability (higher rate of diffusion). Table 1: The reactive and non-reactive scalars computed Scalar Eq. No Schmidt number Damköhler number BC ci=1 in Reaction No 1 1.0 jet no 2 0.5 jet no 3 2.0 jet no 4 1.0 1.0 jet i = 1 5 1.0 1.0 crossflow i = 1 6 1.0 0.5 jet i = 2 7 1.0 0.5 crossflow i = 2 8 2.0 1.0 jet i = 3 9 1.0 1.0 crossflow i = 3 NUMERICAL METHOD AND DETAILS The simulation has been performed with the collocated block-structured Finite Volume Code LESOCC2 (Hinterberger, 2004), developed at the Institute for Hydromechanics of the University of Karlsruhe. Second-order central schemes were used for the spatial discretization of all terms, except the convection term of the species equation where the bounded HLPA scheme was used to maintain the physically correct interval [0;1] for the concentrations. The flow is treated as incompressible and a Poisson equation is solved for the pressure-correction equation. The grid employed consists of 22.3 Mio control volumes in 219 numerical blocks. The blocks located close to the pipe exit were refined by a factor of 3:1 in all spatial directions (Fig. 2). The simulations were carried out on 31 processors of an HP XC4000 parallel cluster with AMD Opteron 2.6 GHz processors with a parallel efficiency of 91%. The computation of one dimensionless time unit took about 316 CPU-hours. At lower Re it was experienced that the simulation can stay fully laminar, so that some perturbations had to be added. Both computations reported here were initialized with an already turbulent flow at a different Reynolds number. After an initial transient towards the statistically steady state averaging was performed over 105.7 dimensionless time units for Re=650 and over 50 time units for Re=325.