Nonpremixed Combustion in an Accelerating Transonic Flow Undergoing Transition

Transcription

1 43rd AIAA Aerospace Sciences Meeting and Exhibit, Jan 1-13, 25/Reno, NV Nonpremixed Combustion in an Accelerating Transonic Flow Undergoing Transition Felix Cheng,Feng Liu, and William A. Sirignano Department of Mechanical and Aerospace Engineering University of California, Irvine, CA The flow through a turbine passage is modelled by a mixing layer of fuel and oxidizer streams going through a channel with imposed streamwise pressure gradients. Due to the strong favorable pressure gradients, the flow accelerates from low subsonic to low supersonic speed and at the same time undergoes transition from laminar flow to turbulence. In this study, we focus on the transitional stage of this unsteady, accelerating, reacting, and compressible mixing layer. The full Navier-Stokes equations coupled with multiple species equations and chemical reactions are solved numerically. No turbulence model is employed since the transitional flow is fully deterministic. Inlet perturbations determined from linear stability analysis are introduced at the inlet to excite the mixing layer. The stabilizing effect of streamwise acceleration is shown. The production of both positive and negative vorticity in the reacting case is identified. The interactions between regions of unlike vorticity are characterized. Flame tearing is also characterized. I. Introduction Designers of jet engines are attempting to increase the thrust-to-weight ratio and to widen the range of engine operation. Since the flow in a turbine passage is accelerating and power is extracted from the flow, it is possible to add heat without raising the flow temperature beyond the turbine-blade material limit. Sirignano and Liu 1,2 showed by thermodynamic analysis that the thrust of aircraft turbojet and turbofan engines can be increased significantly with little increase in fuel consumption by intentionally burning fuel in the turbine stages. For ground-based gas turbines, benefits have been shown to occur in power-to-weight ratio and efficiencies. 1 An exothermic chemical reaction in the accelerating flow through a turbine passage offers, therefore, an opportunity for a major technological improvement. Besides improving thrust or efficiency, this technology also shows a promise in reduced pollutants formation due to the reduction of peak temperature in the combustion process of the accelerating flow. The geometry of the turbine passage is rather complex. In order to simplify the problem while retaining the major physics in the present study, we model the flow through the turbine passage by a mixing layer of fuel and oxidizer streams going through a channel with imposed streamwise pressure gradients. The problem therefore reduces to the study of an accelerating, compressible, and reacting mixing layer. There has been little previous research on steady-state multidimensional flows with mixing and chemical reactions in the presence of strong pressure gradients. Research has been done on high-speed, nonaccelerating, reacting flows. A comprehensive literature review was done by Sirignano and Kim. 3 In that paper, they also obtained similarity solutions for laminar, two-dimensional, mixing, reacting, and nonreacting layers with a favorable pressure gradient in the primary flow direction. Fang, Liu and Sirignano 4 extended that study to mixing layers with arbitrary pressure gradients by using a finite-difference method for the boundary layer equations. The influence of pressure gradients, initial temperature, initial pressure, Graduate Student Researcher, Member AIAA. Professor, Associate Fellow AIAA. Professor, Fellow AIAA. Copyright c 25 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. 1 of 17

2 initial velocity, and transport properties were studied. Mehring, Liu, and Sirignano 5 performed a numerical study on a reacting, turbulent, and accelerating mixing layer based on the laminar boundary layer calculations by Fang, Liu, and Sirignano. 4 Cai et al. 6 developed a finite-volume method for solving the two-dimensional, compressible, Favre-averaged Navier-Stokes equations with chemical reactions using the Baldwin-Lomax turbulence model. Although the steady-state calculations provide important insight into the fundamental physics of multidimensional mixing layers with chemical reactions and strong pressure gradients, they are unable to describe the mechanism by which the flows evolve from laminar to turbulence. Motivated by this, our primary objective of this paper is to investigate the instability of two-dimensional, reacting, accelerating shear layers from the linear stage to the early transitional stage, and its effects on the combustion process. As a result, the understanding of the instability of non-reacting and non-accelerating shear layers serves as a foundation of our research. The instability of non-reacting, non-accelerating mixing layers has been studied extensively by experiments, stability theory, and numerical simulations. A comprehensive overview of those works could be found in Ho and Huerre. 7 Studies of the development of vortical structures in unstable mixing layers are of particular interest to our research. The roll-up of the vortex sheet, the pairing, and merging of the vortices have tremendous influence on the mixing of two fluids, the growth of the boundary layers, and hence the combustion process. An early theoretical work was done by Michalke, 8 in an attempt to study the formation of vortices in free boundary layers by means of stability theory using a hyperbolic-tangent velocity profile. By examining the vorticity distribution of the perturbed flow, they found a tendency of roll-up of the vortex layer. Roll-up of a laminar boundary layer and pairing of vortices were also observed by Freymuth 9 in experiments. The interaction of two-dimensional vortices in a mixing layer was studied by Winant and Browand 1 experimentally. Pairings of discrete vortices were observed, and they occurred continuously throughout the channel. It was also found that the momentum thickness had a linear growth with downstream distance in the region where pairing of vortices occurred repeatedly. The merging of vortices in unforced mixing layers appears in an unpredictable fashion, because the flows are excited by random noises. However, one can manipulate the merging process by applying external forcing. Ho and Huang, 11 by perturbing the mixing layer with subharmonic frequencies, managed to control the number of vortices that merge simultaneously. Moreover, the locations of merging became localized. Davis and Moore 12 performed a numerical study on vortex merging in mixing layers, and found consistent results with Ho and Huang. 11 They were able to predict and control the merging patterns of the discrete vortices. In reacting mixing layers, the stability characteristics could be different from that in non-reacting mixing layers due to the modifications of the flow profiles caused by the chemical reactions. The effects of heat release on the instability of reacting mixing layers were studied by Shin and Ferziger 13 by linear stability theory. An inviscid, low Mach number stability equation was derived and solved numerically. With a sufficient amount of heat release, multiple unstable modes (referred to as the outer modes) in addition to the central mode (the unstable mode associated with the central inflection point of the mean-velocity profile) were found. The outer modes arose due to the modifications of the density and velocity profiles caused by the chemical reactions. Both the outer and central modes were suppressed by heat release, but the outer modes were suppressed less than the central mode and could be the dominant unstable modes. Shin and Ferziger 14 extended their previous work to include compressibility effects. Multiple supersonic unstable modes were found in both the non-reacting and reacting flows when the phase velocity of the disturbance was supersonic relative to the freestream. The supersonic modes became less unstable with increasing Mach number but more unstable with increasing heat release. These modes seemed not to enhance mixing of the two streams. Direct numerical simulations (DNS) of reacting mixing layers show general agreements with linear stability analysis. McMurty et al. 15 performed direct numerical simulations of a low Mach number, two-dimensional mixing layer undergoing a single-step exothermic chemical reaction. The results showed that the rate of chemical products formed, the thickness of the mixing layer, and the entrainment of mass into the layer all decreased with increasing rate of heat release. In another three-dimensional DNS study by McMurty et al., 16 reduction in the entrainment of the reactants and the global reaction rate were again observed. They found that the baroclinic torque and thermal expansion in the mixing layer produced changes in the flame vortex structures that caused more diffuse vortices than in the constant-density non-reacting case. The rotation rates of the large-scale structures were lowered, resulting in reduced growth rate and entrainment of the unmixed fluids. 2 of 17

3 Hermanson and Dimotakis 17 also found consistent results with linear stability theory and DNS by conducting experiments to examine the effects of heat release in a planar reacting mixing layer formed between two streams. The adiabatic temperature in the experiments corresponded to moderate heat release. Their results showed that the growth rate of the layer decreased slightly with increasing heat release, and the overall entrainment of mass into the layer was substantially reduced. They also found that the mean structure spacing decreased with increasing temperature, and suggested that the vortex amalgamation was inhibited. A. Flow Description II. Governing Equations and Solution Methods The flow going through a turbine passage can be simulated by a mixing layer passing through a curved channel. The curved walls of the channel impose transverse and streamwise pressure gradients on the flow, therefore allowing us to study the effects of transverse and streamwise accelerations through the turbine passage. As a first approach to this problem, the geometry is further reduced to a planar mixing layer passing through a converging-diverging channel in which only a streamwise pressure gradient is imposed. The flow properties only vary in the x and y directions, and are uniform in the z direction. The side view of the channel is shown in Figure 1. x represents the streamwise direction and y the transverse direction. Slip conditions are applied to the upper and lower impermeable walls so that wall boundary layers need not be resolved. The top view of the channel is shown in Figure 2 where θ is the channel width. It varies in x only and serves to provide streamwise pressure gradients to the flow. A steady, quasi one-dimensional isentropic flow calculation is used to determine θ. First, a desired linear variation of pressure along the channel is supplied, then, the corresponding local area to throat area ratio is obtained by the area-mach number relationship, and finally, θ is readily obtained for a specified inlet or throat area. In the cases of zero pressure gradient, θ is set to be constant, and the channel is referred to as a straight channel in this paper. U 1, r 1, T 1 y x U 2, r 2, T 2 Figure 1. Side view of the converging-diverging channel q (x) B. Governing Two-Dimensional Equations The flow within a turbine-blade row is at high speed and usually transonic. Figure 2. Top view of the There are large gradients of pressure, density and velocity in the flow field. converging-diverging channel In order to study the mutual interaction between the combustion process and the unsteady fluid motion, we adopt the two-dimensional, compressible, multi-component Navier-Stokes equations with chemical source terms as our governing equations. The equations written in conservation form are as follows: (θw) t + (θf) x + (θg) y (θf µ) x (θg µ) y = θs, (1) where w is the vector of the conservative variables of mass, momentum, and energy, the vectors f and g are the inviscid fluxes, f µ and g µ the viscous fluxes; and s is the source term. These terms are given as w = ρ ρu ρv ρe ρ i, s = p θ θ x p θ θ y Q ω i, (2) z x 3 of 17

4 ρu ρv ρuu + p ρuv f = ρvu, g = ρvv + p, ρhu ρhv (3) ρ i u ρ i v τ xx f µ = τ yx uτ xx + vτ yx q x N i=1 ρ iu di h i, (4) ρ i u di τ xy g µ = τ yy uτ xy + vτ yy q y N i=1 ρ iv di h i. (5) ρ i v di In the above equations t is time; ρ i is density for species i; p is pressure; µ is molecular viscosity; u and v are the flow velocity components in the x and y directions, respectively. The streamtube thickness function θ(x) is equivalently the channel height in the third dimension in order to vary the cross-sectional area of a channel with constant width so that a streamwise pressure gradient may be imposed on the flow. Other quantities are defined in the following equations: [ τ ij = 2µ S ij 1 ] u k δ ij, (6) 3 x k h = µ T q j = C p, (7) Pr x j n T Y i h i, and h i = C pi dt, (8) T i=1 H = h (u2 + v 2 ), (9) E = H p ρ, (1) Q = ρ i V di = ρd i N ω i h i, (11) i=1 ( ρi ρ ) = ρd i Y i, (12) where τ ij is the shear stress tensor, h i is the heat of formation of species i at the reference temperature T, C pi is the specific heat at constant pressure of species i, Y i is the mass fraction of species i, V di is the diffusion velocity of species i, and D i is the diffusion coefficient for species i. For the calculations in this paper, the Schmidt number is assumed to be 1. Methane (CH 4 ) is used for the current computations although the method is not restricted to only one type of fuel. The combustion process is described by a one-step overall chemical reaction as follows: CH 4 + 2O N 2 CO 2 + 2H 2 O N 2. The chemical kinetics rate for the fuel is ω F = W F Ae Ea/RT [CH 4 ] a [O 2 ] b, (13) where the brackets [ ] represent molar concentration in mol/cm 3. According to Westbrook and Dryer 18, for methane, A = sec 1, E a = 48.4 kcal/mol, a =.3, b = of 17

5 In the solution procedure, the average gas constant R, molecular weight W, and viscosity coefficient µ can be obtained by the following equations: R = N R i Y i, (14) i=1 N 1 W = 1 Y i, (15) W i µ = i=1 N µ i (T)Y i. (16) The molecular viscosity coefficient of each species µ i is obtained by the Sutherland-law 19 : i=1 ( ) 3 µ i T 2 T + 11 = µ i T T + 11, (17) where T o is the reference temperature and µ i is the corresponding viscosity coefficient of species i. C. Numerical Method The governing equations shown in the previous section are a system of multidimensional, nonlinear, partial differential equations for which no analytical solution exists. In order to obtain accurate solutions with as little simplification as possible, we seek numerical solutions of Eq. (1) via a finite-difference method. Before discretization, Eq. (1) in the physical domain is transformed to a uniform computational domain by the following transformation relations: ξ = ξ (x,y) x = x(ξ,η), (18) η = η(x,y) y = y (ξ,η). (19) The transformed governing equation in the computational domain is written as follows: (θŵ) t + (θˆf) ξ + (θĝ) η (θˆf µ ) ξ (θĝ µ) η = θs J, (2) where ŵ = w/j, ˆf = (ξ x f + ξ y g)/j, ĝ = (η x f + η y g)/j, ˆf µ = (ξ x f µ + ξ y g µ )/J, ĝ µ = (η x f µ + η y g µ )/J, and J is the Jacobian of the grid transformation. The transformation was employed for the ease of treating curved walls and/or non-uniform meshes. Here, only non-uniform Cartesian meshes are used. A flux-splitting algorithm similar to Steger and Warming 2 is used for spatial discretization of the inviscid flux. The inviscid flux is split into positive and negative parts according to the signs of the local eigenvalues of the flux Jacobian; the derivatives of the positive and negative parts are evaluated by backward difference and forward difference, respectively. Central difference is used for the viscous flux and a Runge-Kutta multistage scheme is implemented for time marching. D. Linear Stability Equations Linear stability theory allows us to determine the unstable modes of a mixing layer. The unstable modes are then imposed at the inlet and serve as excitations to the mixing layer. The linear stability equations are derived from the compressible Euler equations. The chemical and viscous terms have been neglected and uniform properties such c p,c v, and γ are assumed. The flow variables are decomposed into a mean and a small perturbation. Substituting the mean and perturbation quantities into the compressible Euler equations, neglecting the products of small perturbations, and making the assumption of parallel mean flow ( v =, ū/ x = ), one arrives with the linearized equations as follows: ρ t + ρ u x + ū ρ x + ρ v y ρ + v =, (21) y 5 of 17

6 ρ u t + ρū u ū + ρv x y + p =, (22) x ρ v t + ρū v x + p =, (23) y ( ) ( ) ) R T ρ γ 1 t + ū T T ( u + v + p x y x + v =, (24) y p = R ρt + Rρ T, (25) where () represents a mean quantity and () represents a small perturbation. The small perturbations are now modelled by the general form of two-dimensional travelling waves. They are given by ( (u,v,ρ,p,t) (x,y,t) = û, ˆv, ˆρ, ˆp, ˆT ) (y)e i(αx ωt), (26) where α is the wave number, and ω is the angular frequency. In temporal analysis, α is real and ω is complex. In spatial analysis, α is complex and ω is real. In the present study, only spatial analysis is considered. After substituting the travelling wave models into the above linearized equations, the system of equations can be reduced to a single first-order ordinary differential equation by the method adopted by Day. 21 This method introduces a composite variable χ, defined as χ iαˆp ˆv. (27) Now, the final form of the linearized stability equation expressed in terms of χ is written as ( ) dχ gχ + dū dy + χ dy α 2 ρ (ū c) =, (28) ū c where the complex phase velocity c and the new variable g are defined as c ω α, (29) g (ū c)2. (3) 1 ρ γ p The boundary conditions are determined from the asymptotic behaviors of the linearized equations, they are ρ χ(y = ± ) = α (ū c). (31) g In the spatial analysis, a known value of angular frequency ω is input to the stability equation (28) and the corresponding eigenvalue α and the eigenfunction χ are being sought. The mean velocity and density profiles are modelled by the hyperbolic tangent functions as follows: u (y) = Ū 2 ρ(y) = ρ 2 [ 1 + λ tanh ( y )], (32) 2δ θ [ s ( )] y 1 + s tanh, (33) 2δ θ where λ (U 1 U 2 )/(U 1 + U 2 ), Ū (U 1 + U 2 )/2, ρ (ρ 1 + ρ 2 )/2, s ρ 2 /ρ 1, and δ θ is a reference thickness. Equation (28) is integrated numerically by a fourth-order Runge-Kutta scheme. With an initial guess of α, the integration starts from y = and y = with the appropriate boundary condition (31) and ends at y =. A matching condition χ( + ) = χ( ) is enforced at y =, and a Newton-Rasphson iterative method is used to update the value of α. Convergence is obtained when χ( + ) χ( ) 1 1, and the values of α and χ from the last integration are used to evaluate the eigenfunctions. 6 of 17

7 III. Computational Results Mixing layers undergoing transition exhibit early development of large coherent structures. In this paper, our main objective is to study the transitional stage of accelerating and chemically reacting mixing layers. For the flows without forced disturbances, the onset of instability relied on background noise, and no large coherent structures were observed within the computational domain in some of our computations. In order to excite the development of the unstable structures, the use of forced disturbances was motivated. The forced disturbances have been proven to be efficient in exciting the mixing layers, and provide a means of manipulating the development of the unstable structures. The solutions to the inviscid parallel-flow stability equation shown in Section IID give the frequency and the eigenfunctions for the disturbances given by Eq. (26), with the real parts taken as the inlet disturbances. The growth rate of an instability mode varies with its frequency and, in practice, the frequency is chosen such that the maximum growth rate is attained. As an illustration, the plot of growth rate versus frequency for the non-accelerating cases is shown in Figure 3. The frequency for the peak growth rate is.22, which is referred to as the fundamental frequency. The corresponding eigenfunctions for u and p are shown in Figures 4 and 5, respectively. The computational results for the non-accelerating and accelerating mixing layers with and without chemical reaction are presented in this section. The inflow velocity and density in all cases are given by the tanh profiles in Eqs. (32) and (33), respectively, and the thickness of the profiles is controlled by a reference thickness δ θ. For a constant-density mixing layer, δ θ is indeed the momentum thickness corresponding to the velocity profile given in Eq. (32). However, this is only an approximation for mixing layers with varying density profiles. For the cases shown below, the value of δ θ is specified and is the same for the velocity and density profiles. For the non-accelerating cases in Sections IIIA and IIIB, δ θ =.125 m, the physical domain has a length of.2 m (16 δ θ ), and a height of 1. m (8 δ θ ). The domain is discretized by 21 grid points in the x-direction and 121 grid points in the y-direction. The grid points are non-uniformly distributed such that a higher density of grid points are placed near the inlet and across the dividing streamline. The largest grid size in the x-direction has been carefully adjusted; one wavelength of the fundamental frequency is resolved in well over 2 grid points. For the computations of the accelerating mixing layers, practical limitation on the grid size was encountered. Due to the streamwise acceleration, we found that the development of instability was suppressed and hence a channel with much longer length was required. The ideal way would have extended the channel length while keeping the grid size unchanged; however, the number of grid points would increase tremendously for which the computational time would become impractical. In order to maintain the computational efficiency, coarser grids with respect to the reference thickness were used instead. For the accelerating cases in Sections IIIC and IIID, δ θ was reduced to.125 m, while the length and height are.3 m (24 δ θ ) and.1 m (8 δ θ ), respectively. The number of grid points in the x-direction is 241, and in the y-direction is 121. Due to the reduction in resolution, one wavelength of the fundamental frequency for the accelerating mixing layers is resolved in approximately 12 grid points. However, this grid setting would have allowed only 2 grid points to resolve one wavelength for the non-accelerating cases had those cases been calculated for this longer domain. As a consequence, a direct quantitative comparison between the non-accelerating and accelerating cases is unavailable. The non-accelerating cases mainly serve as the base cases for qualitative comparison. The major flow conditions for the 4 cases presented below are summarized in Table 1. acceleration reaction u 1 (m/s) u 2 (m/s) T 1 (K) T 2 (K) p 1 = p 2 (atm) δ θ (m) l/δ θ h/δ θ Case A no no Case B no yes Case C yes no Case D yes yes Table 1. Summary of the flow conditions 7 of 17

8 A. Non-accelerating, non-reacting mixing layer A mixing layer formed by merging initially parallel air and fuel streams is studied. No pressure gradient is imposed. The air stream is composed of nitrogen and oxygen, and the fuel stream is purely methane. The mean inlet velocity and density profiles are given by Eqs. (32) and (33), respectively. Two different frequencies for the forced disturbances are used for this case; they are the fundamental, which causes roll-ups of the mixing layer, and the first subharmonic, which induces pairings of like-rotating vortices. The fundamental, which corresponds to the frequency at which the maximum amplification rate is attained, has a nondimensional frequency ω of.22 (15 hz), and the first subharmonic has ω1 of.11 (525 hz). The instantaneous vorticity and temperature contours are shown in Figures 6 and 7 respectively to illustrate the effects of the instability. The figures have been enlarged for better visualization, and the thicknesses of the layers are actually much thinner compared to the channel height. Roll-ups of the vorticity layer occur; however, the vortices do not rotate around each other and engage in pairing within the length constraint. From the temperature contours, the mixing layer shows wavy structures induced by the forced disturbances, and the structures seem to grow larger as they propagate, but again, merging of the structures is not observed. In a similar case that we have studied but not presented here, a larger velocity difference (δu U 1 U 2 ) between the freestreams caused the vorticity layer to break into discrete vortices. Also, rolling and pairings of the vortices were clearly seen. The smaller δu in this case therefore is not large enough to fully excite the development of the instability in the physical length considered. However, it is necessary to use this relatively smaller δu in order to reduce the amount of curving of the dividing streamline in other accelerating cases where streamwise pressure gradients are imposed. A displacement thickness δ has been defined to help examine the growth of the mixing layer. It is defined as: δ h/2 y ( 1 ρu ) dy ρ 1 u 1 + y h/2 ( 1 ρu ) dy ρ 2 u 2, (34) where y (x) is the y-location of the time-averaged dividing streamline which splits the mass flow rate into the same proportion as the inlet dividing streamline does, and () denotes a time-averaged property. The growth of the displacement thickness for this case is shown in Figure 11. The growth rates agree quite well initially with the exponential curve-fit, but deviate slightly further downstream. This trend indicates that the instability modes are undergoing early development and the transitional stage has not been reached yet. Fourier analysis has been used to investigate the frequency response of the mixing layer to the forced disturbances. To study the evolution of an individual mode as it propagates downstream, it is useful to define an integrated modal energy as E m = 1 2 h/2 h/2 ( ρ û m 2 + ˆv m 2) dy, (35) where û m and ˆv m are the Fourier transform coefficients at the frequency ω m of the u and v velocity components, respectively. Figure 8 shows that the integrated energy for the fundamental mode dominates over the 1 st subharmonic initially, but decays gradually further downstream, and eventually is dominated by the continuously growing 1 st subharmonic. Since the energy for the 1 st subharmonic has not reached its saturation point, this indicates that pairing of vortices has not occurred. In the larger δu cases mentioned previously, a very similar trend was found, but the 1 st subharmonic continued to grow and reached a saturation point at which pairing of vortices occurred periodically. B. Non-accelerating, reacting mixing layer The non-accelerating mixing layer in Section IIIA including chemical reactions is studied in this section. The flow conditions, the imposed disturbances, and the geometry remain the same as the non-reacting case. The instantaneous vorticity contours are shown in Figure 9. The contours on the upper half plane near the inlet represent a region of counter-clockwise (i.e. positive) vorticity, where the contours on the lower half plane correspond to the more dominant clockwise (negative) vorticity. The counter-clockwise vorticity is created by the sudden ignition of the flame very close to the inlet where local peaks in streamwise velocity exist. However, those peaks dissipate tremendously further downstream. Neither the counter-clockwise nor clockwise vortices engage in pairing of like vortices. They appear to be more diffuse than the non-reacting case; the strengths of the vortices reduce quickly as they propagate downstream. 8 of 17

9 Due to the chemical reactions, the temperature of the flow increases tremendously, and the maximum temperature in the flowfield is found to be 41 K. Temperature profiles of the mixing layer exhibit a peak near the dividing streamline, resulting in a local minimum in the density profiles accordingly. The instantaneous flame structure is illustrated in Figure 1, where the darkened area corresponds to the hot region in which combustion takes place. Similar to the non-reacting cases, the forced disturbances cause waviness in the flame, but do not grow strongly enough to cause breaking of the layer. The growth of the mixing layer is shown in Figure 11. The displacement thickness is approximately proportional to x 1/2, which indicates that the growth is governed by laminar diffusion rather than inviscid instability. This is because the high temperature in the flame region causes an increase in viscosity and hence reduces the Reynolds number. The displacement thickness is found to be larger than that in the non-reacting case, which the opposite of the results obtained by other researchers. In those studies, the growth of the boundary layers was dominated by the interactions between vortices, and the decrease in boundary-layer thickness caused by the heat release was mainly attributed to the reduced rotation rates of the large-scale structures. In our cases, however, pairing of vortices has not occurred in the given length, and the growth of the non-reacting and reacting mixing layers is governed by different mechanisms mentioned previously. C. Accelerating, non-reacting mixing layer An accelerating mixing layer formed by merging initially parallel air and fuel streams is studied. A streamwise pressure gradient is imposed by varying the the width (θ) of the channel as shown in Figure 2. The resultant pressure gradient is 22.2 atm/m. Forced disturbances are used to excite the mixing layer. However, due to the changes in the mean flow profiles along the channel caused by the acceleration, the optimal frequency determined by the inlet flow conditions without pressure gradients does not necessarily cause roll-ups of the layer. Therefore, we have determined the frequency and mode shape of the disturbances by the local conditions at various x-locations and use the one that produces the most unstable structures. However, it must be clarified that the chosen disturbances are by no means the optimal ones for the physical situation; they are simply the most favorable ones of those considered and serve the purpose of inducing large-scale structures. Two frequencies for the forced disturbances are used. They are determined at an x-location where Ū = 338 m/s, and δu = 191 m/s. A nondimensional frequency ω of.6 (2586 hz) is used as the fundamental frequency, and ω 1 of.3 (1293 hz) is used as the first subharmonic. Note that the fundamental frequency here does not correspond to the peak amplification rate. The instantaneous vorticty contours for this case are shown in Figure 12. Small waviness in the vorticity layer develops initially, followed by the formation of the highly concentrated vorticity indicated by the darkened areas. Roll-ups of the layer occur at about x = 12 δ θ, and the structures grow rapidly thereafter but pairing of vortices does not occur within the chamber length. The development of the large-scale structures can also be seen by looking at the instantaneous temperature contours near the dividing streamline shown in Figure 13. The mixing characteristics of the layer has been modified significantly by the instability. Figure 14 shows the instantaneous mass-fraction distributions for O 2 and CH 4 at a certain x-location. Local maximum and minimum exist across the layer for both the oxidizer and fuel streams, contrasting the monotonic distributions of mass fractions in laminar diffusion layers. The growth of the mixing layer is shown in Figure 19. The displacement thickness grows slowly in a region between the inlet to x 13 δ θ. This region corresponds to where the small waviness of the vorticity layer is observed before the roll-ups occur. At x 15 δ θ where the roll-ups have started to occur, the displacement thickness increases rapidly, and the growth rate is close to a linear relation with the downstream distance. This linear relation indicates that the mixing layer is undergoing its transitional stage. The evolution of the instability modes shown in Figure 15 is quite different from that in the nonaccelerating case. Note that the 1 st subharmonic at the inlet has a lower amplitude than the fundamental, but is not resolved in the figure. Neither the fundamental nor the 1 st subharmonic is amplified from the inlet to x 1 δ θ, showing that the growth of the mixing layer in the region where small waviness is observed cannot be dominated by the instability. Both modes start to grow a little further downstream, and the 1 st subharmonic is amplified sharply and dominates over the fundamental. This indicates that the acceleration favors the 1 st subharmonic over the fundamental. It is also observed that the acceleration introduces a threshold beyond which the instability modes start growing rapidly. 9 of 17

10 D. Accelerating, reacting mixing layer An accelerating, reacting mixing layer is studied in this section. The flow conditions, the forced disturbances, and the geometry are identical to the non-reacting case in Section IIIC. The chemical reactions and the streamwise acceleration cause interesting developments of the vortex structures not seen in the previous cases. Due to the chemical reactions, clockwise (negative) and counterclockwise (positive) vortices are generated. The instantaneous vorticity contours on the top half in Figure 16 show the counter-clockwise vortices, and on the bottom half show the clockwise vortices. The clockwise vortices have larger magnitudes than the counter-clockwise ones, and the strengths for both of them increase with downstream distance because of the streamwise acceleration. Pairing of like vortices does not occur within the length limitations but some interesting interactions with a pair of unlike vortices are observed. In order to visualize the vortices better, Figure 16 is blown up in Figure 17 where the counter-clockwise vortices are labelled by alphabets and the clockwise vortices are labelled by numbers. The clockwise vortices have positive transverse velocities; the vortex structure labelled 1 will move upward and appears to be the one at 2 at a later time. The counter-clockwise vortices, which have smaller magnitudes than the clockwise ones, undergo both positive and negative transverse velocities; the structure labelled a will be pulled downward by the clockwise vortices and appears as b and c at later times while the structure d will move upward slightly and become e at a later time. Also, the clockwise vortices are continuously strained as they move. The vortex between d-a (later e-b) will elongate tremendously, and will be torn apart eventually, producing vortex c. Due to the acceleration, the flow expands along the channel and the temperature, pressure, and density drop accordingly. The maximum temperature in this case is 3592 K, which is more than 5 K lower than that in the non-accelerating case. The instantaneous flame structure is shown in Figure 18, where the darkened areas correspond to the hot regions. Very small waviness occurs near the inlet and the waves do not grow until passing the = 1 mark. The instability waves then grow sharply, and the flame breaks into large fragments. The locations of the flame fragments are heavily biased toward the air stream; the flames grow larger and at the same time move upward as they propagate downstream. The growth of the mixing layer is shown in Figure 19 and compared with the non-reacting case in Section IIIC. The displacement thickness has a larger magnitude than that in the non-reacting case but the trends are similar for both cases. The thickness increases slowly near the inlet but grows dramatically after the roll-ups have occurred. One of the major differences between the two cases is that the thickness for the reacting case has a more noticeable growth rate than the non-reacting case in a region between the inlet to x 13 δ θ. Beyond that point, the growth rates for those cases are very similar. In the region between the inlet and x 13 δ θ, the contours of vorticity and temperature show that the flow is quite laminar so that the instability waves have not been amplified significantly. This leads us to believe that the initial increase in thickness in that region can be attributed to the increased viscosity due to the chemical reaction. Finally, the evolutions of the instabiliy modes are shown in Figure 2. The behaviors are qualitatively the same as in the non-reacting case. IV. Conclusion Non-accelerating and accelerating mixing layers with and without chemical reactions have been studied. Forced disturbances have been applied on all cases in order to enhance the onset of instability for the mixing layers. The streamwise acceleration has a stabilizing effect for both the non-reacting and reacting mixing layers; roll-ups of the layers are delayed, and a longer channel length is required in order to capture the transitional stage of the layers. Pairing of like-rotating vortices does not occur within the chamber length for the accelerating cases. For the accelerating, reacting mixing layer, periodic regions of positive vorticity and periodic regions of negative vorticity are formed. The interactions between these vorticity regions have been characterized. The development of instability waves causes the flame to tear into large fragments and propagate toward the air stream. The displacement thickness has increased compared to its non-reacting counterpart. 1 of 17

11 Acknowledgments The authors are grateful for partial support of this research by AFOSR through an SBIR subcontract with Innovative Scientific Solutions, Inc. References 1 Sirignano, W. A. and Liu, F., Performance Increases for Gas-Turbine Engines Through Combustion Inside the Turbine, Journal of Propulsion and Power, Vol. 15, No. 1, 1999, pp Liu, F. and Sirignano, W. A., Turbojet and Turbofan Engine Performance Increases through Turbine Burners, Journal of Propulsion and Power, Vol. 17, No. 3, 21, pp Sirignano, W. A. and Kim, I., Diffusion Flame in a Two-dimensional, Accelerating Mixing Layer, Physics of Fluids, Vol. 9, No. 9, 1997, pp Fang, X., Liu, F., and Sirignano, W. A., Ignition and Flame Studies for an Accelerating Transonic Mixing Layer, Journal of Propulsion and Power, Vol. 17, No. 5, 21, pp Mehring, C., Liu, F., and Sirignano, W. A., Ignition and Flame Studies for an Accelerating Transonic Turbulent Mixing Layer, AIAA-Paper 1-19, Cai, J., Icoz, O., Liu, F., and Sirignano, W. A., Ignition and Flame Studies for Turbulent Transonic Mixing Layer in a Curved Duct Flow, AIAA-Paper 1-18, Ho, C. M. and Huerre, P., Perturbed Free Shear Layers, Ann. Rev. Fluid Mech., Vol. 16, 1984, pp Michalke, A., Vortex Formation in a Free Boundary Layer According to Stability Theory, J. Fluid Mech., Vol. 22, No. 2, 1965, pp Freymuth, P., On Transition in a Separated Laminar Boundary Layer, J. Fluid Mech., Vol. 25, No. 4, 1966, pp Winant, C. D. and Browand, F. K., Vortex Pairing: The Mechanism of Turbulent Mixing Layer Growth at Moderate Reynolds Number, J. Fluid Mech., Vol. 63, No. 2, 1974, pp Ho, C. M. and Huang, L., Subharmonics and Vortex Merging in Mixing Layers, J. Fluid Mech., Vol. 119, 1982, pp Davis, R. W. and Moore, E. F., A Numerical Study of Vortex Merging in Mixing Layers, Phys. Fluids, Vol. 28, No. 6, 1985, pp Shin, D. S. and Ferziger, J. H., Linear Stability of the Reacting Mixing Layer, AIAA Journal, Vol. 29, No. 1, 1991, pp Shin, D. S. and Ferziger, J. H., Linear Stability of the Compressible Reacting Mixing Layer, AIAA Journal, Vol. 31, No. 4, 1993, pp McMurty, P. A., Jou, W.-H., Riley, J. J., and Metcalfe, R. W., Direct Numerical Simulations of a Reating Mixing Layer with Chemical Heat Release, AIAA Journal, Vol. 24, No. 6, 1986, pp McMurty, P. A., Riley, J. J., and Metcalfe, R. W., Effects of Heat Release on the Large-scale Structure in Turbulent Mixing Layers, J. Fluids Mech., Vol. 199, 1989, pp Hermanson, J. C. and Dimotakis, P. E., Effects of Heat Release in a Turbulent, Reacting Shear Layer, J. Fluids Mech., Vol. 199, 1989, pp Westbrook, C. K. and Dryer, F. L., Chemical Kinetic Modeling of Hydrocarbon Combustion, Prog. Combust. Sci.,, No. 1, White, F. M., Viscous Fluid Flow, McGraw-Hill, Second ed., Steger, J. L. and Warming, R. F., Flux Vector Splitting of the Inviscid Gasdynamic Equations with Application to Finite-Difference Methods, Journal of Computational Physics, Vol. 4, 1981, pp Day, M. J., Structure and Stability of Compressible Reacting Mixing Layers, Ph.D. thesis, Standford University, Standard, CA 9435, of 17

12 .1 α = αiδ θ λ ω = ωδ θ Ū Figure 3. Spatial growth rate versus frequency for the non-accelerating cases. ûr,ûi y δ θ Figure 4. Real and imaginary parts of the x-component of the velocity eigenfunction for the non-accelerating cases., real part; - - -, imaginary part ˆpr, ˆpi y δ θ Figure 5. Real and imaginary parts of the pressure eigenfunction for the non-accelerating cases., real part; - - -, imaginary part. 12 of 17

13 Figure 6. Instantaneous vorticity contours for the non-reacting, non-accelerating mixing layer with forced fundamental and 1 st subharmonic Figure 7. Instantaneous temperature contours for the non-reacting, non-accelerating mixing layer with forced fundamental and 1 st subharmonic. 4E-6 3.5E-6 3E-6 2E,2E 1 2.5E-6 2E-6 1.5E-6 1E-6 5E Figure 8. Integrated energies for the non-reacting, non-accelerating mixing layer with forced fundamental and 1 st subharmonic., fund. ; - - -, 1 st sub.. 13 of 17

14 Figure 9. Instantaneous vorticity contours for the reacting, non-accelerating mixing layer with forced fundamental and 1 st subharmonic Figure 1. Instantaneous temperature contours for the reacting, non-accelerating mixing layer with forced fundamental and 1 st subharmonic. δ/δθ Figure 11. Displacement thickness for the reacting and non-reacting, non-accelerating mixing layers with forced fundamental and 1 st subharmonic., data for the reacting case;, curve fit of power 1/2 for the reacting case;, data for the non-reacting case; - - -, exponential curve fit for the non-reacting case. 14 of 17

15 x/δθ Figure 12. Instantaneous vorticity contours for the accelerating, non-reacting mixing layer with forced disturbances x/δθ Figure 13. Instantaneous temperature contours for the accelerating, non-reacting mixing layer with forced disturbances Yo2, YCH4 Figure 14. Instantaneous mass-fraction profiles for the accelerating, non-reacting mixing layer with forced disturbances., O2 ; - - -, CH4. 15 of 17

16 E, 2E E x/δθ Figure 15. Integrated energies for the accelerating, non-reacting mixing layer with forced disturbances., fund. ; - - -, 1st sub x/δθ Figure 16. Instantaneous vorticity contours for the accelerating, reacting mixing layer with forced disturbances. 1 d e a b c x/δθ Figure 17. Enlarged instantaneous vorticity contours for the accelerating, reacting mixing layer with forced disturbances. 16 of 17

17 Figure 18. Instantaneous temperature contours for the accelerating, reacting mixing layer with forced disturbances. 5 4 δ/δθ Figure 19. Displacement thickness for the accelerating, reacting and non-reacting mixing layers with forced disturbances., reacting; - - -, non-reacting 2.5E-4 2.E-4 2E,2E 1 1.5E-4 1.E-4 5.E-5.E Figure 2. Integrated energies for the accelerating, reacting mixing layer with forced disturbances., fund. ; - - -, 1 st sub.. 17 of 17