Two-Dimensional Supersonic Flow with Perpendicular Injection of the Gas

The flow around jets has been comprehensively studied by many investigators. Mathematical models have been successfully realized for flow of perfect gases [1-4], but the practical design of supersonic ramjet (scramjet) engines requires an understanding of jet interaction with a crossflow for multispecies gases. The flow field in such devices is complicated by geometrical factors, turbulent fuel-air mixing, chemical reaction, shock waves, and separation regions ahead of and behind the jet. Despite successful numerical models of such flows, the detail of the flow physics of the combustion in the real devices through the use of computational tools is a difficult problem. Therefore the investigators studied some physical phenomena or proposed the numerical method which are important for solution of the problem of the combustion. For example, this problem has been modeled by Drummond and Weidner [1-2] during that process Reynolds-averaged multispecies Navier-Stokes equations which were coupled with Baldwin-Lomax turbulence model have been solved by using the explicit predictor-corrector scheme of MacCormack. Drummond and Weidner modeled injection of fuel from two slots and studied influence of the slots width on the process of mixing. Their comparison of the pressure on the wall to the experiment data showed a satisfactory fit. Grasso and Magi have also solved the Favre-averaged Navier-Stokes equations for a transverse gas injection in supersonic air streams [4]. The governing equations are solved by a finite volume approach with the k ε − model equations coupled. A high-order TVD method has been developed to multispecies turbulent gas mixtures and the effect of the free stream Much number on the flow structure have been studied. The numerical investigation of supersonic mixing of hydrogen with air has been performed in the work [5-6]. For the process the main flow air entering through a finite width of inlet and gaseous hydrogen is injected perpendicularly from the side wall. The explicit TVD scheme has been used to solve the system of two-dimensional Navier-Stokes equations. In this study the enhancement of mixing and good flame-holding capability of a supersonic combustor have been investigated by varying the distance of injector position from left boundary. Upstream recirculation can evolve which might be activated as a good flame holder. One of the ways to solve Navier-Stokes equation for multispecies flow was proposed by Shuen and Yoon [ 7], during which they had developed numerical method using high-order


Introduction
The flow around jets has been comprehensively studied by many investigators. Mathematical models have been successfully realized for flow of perfect gases [1][2][3][4], but the practical design of supersonic ramjet (scramjet) engines requires an understanding of jet interaction with a crossflow for multispecies gases. The flow field in such devices is complicated by geometrical factors, turbulent fuel-air mixing, chemical reaction, shock waves, and separation regions ahead of and behind the jet. Despite successful numerical models of such flows, the detail of the flow physics of the combustion in the real devices through the use of computational tools is a difficult problem. Therefore the investigators studied some physical phenomena or proposed the numerical method which are important for solution of the problem of the combustion. For example, this problem has been modeled by Drummond and Weidner [1][2] during that process Reynolds-averaged multispecies Navier-Stokes equations which were coupled with Baldwin-Lomax turbulence model have been solved by using the explicit predictor-corrector scheme of MacCormack. Drummond and Weidner modeled injection of fuel from two slots and studied influence of the slots width on the process of mixing. Their comparison of the pressure on the wall to the experiment data showed a satisfactory fit. Grasso and Magi have also solved the Favre-averaged Navier-Stokes equations for a transverse gas injection in supersonic air streams [4]. The governing equations are solved by a finite volume approach with the k ε − model equations coupled. A high-order TVD method has been developed to multispecies turbulent gas mixtures and the effect of the free stream Much number on the flow structure have been studied. The numerical investigation of supersonic mixing of hydrogen with air has been performed in the work [5][6]. For the process the main flow air entering through a finite width of inlet and gaseous hydrogen is injected perpendicularly from the side wall. The explicit TVD scheme has been used to solve the system of two-dimensional Navier-Stokes equations. In this study the enhancement of mixing and good flame-holding capability of a supersonic combustor have been investigated by varying the distance of injector position from left boundary. Upstream recirculation can evolve which might be activated as a good flame holder. One of the ways to solve Navier-Stokes equation for multispecies flow was proposed by Shuen and Yoon [ 7], during which they had developed numerical method using high-order www.intechopen.com lower-upper symmetric successive overrelaxation scheme. The higher approximation scheme for solution the Favre-averaged Navier-Stokes equations based on the ENO scheme had been constructed in [8][9]. The interaction of a planar supersonic air flow with perpendicularly injected hydrogen jet across the slot from the duct walls have been studied. The influence of the jet Mach number and the ratio of the jet and flow pressure on shock wave structure of the flow and the jet penetration depth was shown.
The k ε − turbulence model developed by Jones and Launder had been successfully used to numerically calculate supersonic flows [10][11], where the effect of compressibility on the structure of turbulence must be considered. The compressibility correction to the dissipation-rate term in the k transport equation has been used by Rizzeta [10][11] to simulate a supersonic turbulent flowfield. In this present work, a plane supersonic flow in a channel with perpendicular injection of jets from slots located symmetrically on the lower and upper walls is modeled numerically. For convenience of computations, symmetry is assumed so that only the lower half of the channel is modeled, with jet injection from the lower wall only. The flow pattern is shown in Figure 1. The influence of the compressibility effect on the shock wave structure and on the vortex system ahead and behind of the jet are studied in the context of the k ε − and algebraic Baldwin-Lomax models of turbulence. The detailed physical analysis of the supersonic jet interaction with a crossflow for multicomponent gases depending on parameters of the flowfield as the Mach numbers of the air stream, parameters of turbulence, pressure ratio are performed. Such an analysis can improve understanding of the relevant flow structures responsible for the generation of the pressure field and for the mixing of the injectant with the cross flow. Ultimately this analysis will contribute to understanding of scramjet fuel injection systems. The choice of the range of parameters under consideration is determined by the available experimental data of the processes of hydrogen combustion in the range of regime parameters under consideration, which will allow comparisons with experimental data of other authors in the future.

Mathematical model
The two-dimensional Favre-averaged Navier-Stokes equations for multi-species flow with chemical reactions are: where , k ε are the turbulent kinetic energy and rate of dissipation of turbulent kinetic energy respectively. k P is turbulence production term, t M is the turbulence Mach number, D , E ε are the low-Reynolds number terms, and k Y is the mass fraction of k th species, 1... kN = , where N is the number a component of a mix of gases. The thermal equation for multi-species gas is: where k W is the molecular weight of the species. The total energy equation is The enthalpy of the gas mixture is calculated according to where h is the specific enthalpy of k th species. Specific heat at constant pressure for each component pk c is: molar specific heat pk C is given in terms of the fourth degree polynomial with respect to temperature in the JANAF Thermochemical Tables [12]. The system of equations (1) is written in the conservative, dimensionless form. The governing parameters are the entrance parameters at x =−∞, which are the pressure and total energy normalized to 2 u ρ ∞∞ , the enthalpy normalized to 0 / RT W ∞∞ , the molar specific heat normalized to 0 R , the turbulence kinetic energy k normalized to 2 u ∞ , the dissipation of turbulence kinetic energy ε normalized to 3 / uh ∞ and the considered lengths normalized to the slot width h .

Transport properties
The coefficient of dynamic viscosity is represented in the form of the sum of l µ -molecular viscosity and t µ -turbulent viscosity: The Wilke formula is used to determine the mixture viscosity coefficient by µ is the molecular viscosity of species i, which is defined by following formula [13, p.16 In (5)

Initial and boundary conditions
At the entrance: ,ε k at the entrance: the , k ε profile at the entrance is defined by using Baldwin-Lomax's algebraic turbulence model for known averaged physical parameters of the flow, where the coefficient of turbulent viscosity contains local velocity gradients in the transverse direction. Making an assumption /1 The boundary layer on the wall is given in the input section, and the velocity profile approximated by power law. On the slot: ( 0 / np p ∞ = is the jet pressure ratio, 0 p is the jet pressure, and p ∞ is the flow pressure). On the lower wall the no-slip condition is imposed, for the temperature the adiabatic condition are imposed; on the upper boundary the condition of symmetry is assumed; on the outflow the nonreflecting boundary condition [9] is used.

Method of the solution
To take into account the flow in the boundary layer, near the wall, and near the slot, i.e., in regions of high gradients, more accurately, we refine the grid in the longitudinal and transverse directions by the transformations where is the Jacobian of mapping.. System (7) linearized with respect to the vector U in form: Here,

1~+
where the vectors 1 21 are written in the following form: According to a principle of construction ENO scheme the system (7) for integration on time is formally represented as: Here m E  , m F  is called the modified flux vector. It consists from the original flux vector (  E ,  F and additional terms of third-order accuracy , The second term n RHS η is written similarly. In approximation of derivatives in convective and diffusion terms, we use second-order central-difference operators. The numerical solution of the system (7) is calculated in two steps. The first determines the dynamic parameters and second determines the mass species. Then it is necessary to define Jacobian matrix which, in a case of the thermally perfect gas, represents a difficult task. This problem is connected with the explicit representation of pressure through the unknown parameters. Here pressure is determined by introducing an effective adiabatic parameter of the gas mixture. www.intechopen.com The temperature is found by using the Newton-Raphson iteration [8][9].

Numerical results
The parameters of coordinate transformation (7) have the form: β and τ are refinement factors ( β >1 and τ >1), a is the height of the computational domain in the generalized coordinates, and c x is the point with respect to which grid refinement is performed. The use of algorithm (13) with the Baldwin-Lomax turbulence model has been tested by authors and results were stated in reference [8]. Numerical simulation was performed for perpendicular injection of a gas into twodimensional channel with the channel height of -7,62 cm, length -15 cm and width of slot 0,0559 cm. The slot was located at a distance of 10 cm from entrance. The parameters were The Figure 2 shows the influence of turbulence model on the character of interaction of the jets with the supersonic air flow. In this figure, the shaded pictures of pressure calculated by Baldwin-Lomax and k ε − turbulence models with and without compressibility correction are represented. As is well known, the physics flow is the following: due to deceleration of the main flow, the pressure ahead of the jet increases, and a bow shock wave is formed. An oblique shock wave emanates upstream from the bow shock wave. A second shock wave, named the barrel shock emerged behind the oblique shock wave. The bow shock wave, oblique shock wave and barrel shock wave intersect at one point to form a complicated λ -shaped structure.
All of these shock waves are presented in the Figure 2 for different freestream Mach numbers. Although the qualitative picture of the shock waves are similar for both turbulence models, the evident divergence of the λ -shaped structure is observed in the turbulent boundary layer, which is more prominent for smaller M ∞ Figure 2 (a-c). The differences among the models in the boundary layer are also evident in the local Mach number contours. The distribution of the iso-Mach line contours is shown in Figure  3. In all cases the sonic velocity of the jet becomes supersonic at a certain distance because of the acceleration immediately after injection. Then, the boundary of the supersonic region is closed, forming a barrel which is a consequence of interaction of the shock waves system in the jet itself. With increasing freestream Mach number the jet penetration is reduced, due to the reduction of the hydrogen momentum with respect to the incoming air momentum and consequently a significant decrease in the barrel size is observed. The iso-Mach line contours in the subsonic region ( Figure 3) were compared, and quantitative differences among the models can be seen. It is obvious that increasing of M ∞ reduces these differences.      The influence of the pressure ratio on the upstream flow structure was also studied. An analysis of the influence of the jet pressure ratio on the bow shock wave shows that the angle of inclination of the latter increases with increasing pressure ratio behind the shock wave and ahead of it. The effect of the jet pressure ratio on the separation-region length is illustrated in Figure 9 for 3.75 M ∞ = .
Here results are represented which were calculated by the convergence of the dimensionless residual norm for gas density lnε = -5.

Conclusion
In the present work, the numerical model and computer code have been developed for a k ε − turbulent model that includes compressibility correction to simulate turbulent supersonic air flow with transfer injection of the hydrogen from slots of the walls. The numerical method is based on the ENO scheme. Comparison is made between solution by using algebraic Baldwin-Lomax's turbulence model and k ε − model. The influence of the compressibility effect on the shock wave structure is studied. The interaction of the supersonic air flow with transfer injection of jets depending on the freestream Mach number and pressure ratio have been studied. In spite of similarity in the shock wave structure for both turbulence model, the numerical experiments have shown that the sizes of counter-rotating primary and secondary upstream vortices in the region upstream of the jet differ. The developed computer code allows to study of mixing flowfield behavior with various injector geometries and reacting flows which is important problems in the design of supersonic combustion ramjet (scramjet) engines.