Mixing-layer Confluent Flow

Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 99 (2015) 320 – 326

“APISAT2014”, 2014 Asia-Pacific International Symposium on Aerospace Technology, APISAT2014

Effect of Combustion Heat Release on the Stability of Confined Boundary/mixing-layer Confluent flow Liu Zhiyonga,* , Shang Qinga, Liu Xiaoyongb, Fei Lisenb, Liu Fengjunb a

China Academy of Aerospace and Aerodynamics, Beijing, 100074, China b Beijing Power Machinery Institute, Beijing, 100074, China

Abstract The issue of mixing enhancement of the fuel and oxidizer in the combustor of a ramjet has received more and more attention. Various injection ways were designed and justified to improve the mixing process by producing more multiplescale vortex structures. Under the restriction of compact configuration of the combustor in an integral rocket dual combustion ramjet, a better way to enhance mixing is turning the laminar flow to turbulent flow. The present study focuses on the stability analysis of the boundary/mixing-layer confluent flow affected by the combustion heat release in the combustion chamber. Two types of basic flows are formed for linear stability analysis both from a theoretical model and numerical computation. Eigenvalue spectra and eigenfunctions are obtained and compared. The results show that the heatrelease effect stabilizes the confluent mixing flow. © Published by Elsevier Ltd. This © 2015 2014The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA). Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA)

Keywords: Confluent flow; mixing layer; linear stability; heat release

1. Introduction Mixing flows have a number of industrial applications. The mixing of fuel and oxidizer is of great significance to combustion and researchers pay a lot attention to mixing enhancement in power devices, such as engines, fuel cells etc. In the combustor of an integral rocket dual combustion ramjet, the mixing area lies adjacent to the wall thus the

* Corresponding author. Tel.: +86- 13717770747;. E-mail address: [email protected]

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA)

doi:10.1016/j.proeng.2014.12.541

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Liu Zhiyong et al. / Procedia Engineering 99 (2015) 320 – 326

mixing process is badly affected by the confinement of walls. The present study was incited by this specified application and instability analysis was investigated to explore the mechanism of mixing augmentation. Many progresses about seeking the mechanism of mixing enhancement in mixing layers have been made in recent decades. Kumar et al. [1] reported an oscillating shock can increase the mixing-layer turbulence levels and thus enhance mixing. Computational investigations [2] have shown that streamwise vorticity induced by baroclinic torques in the mixing layers enables mixing enhancement. While relatively less attention has been paid to the confined mixing layer which is a better model of actual devices, especially combustors. Linear stability theory (LST) and direct numerical simulation (DNS) were adopted by Greenough et al. [3] to study the wall effect on the instability of compressible mixing layers. They found two types unstable modes, i.e. K-H mode and supersonic wall mode. Hu [4] studied the development process of disturbance in confined mixing layers and found that even if the disturbances of two modes propagate linearly and separately, the disturbance energy grow oscillatorily, and periodic structures were discovered. Hudson et al. [5] compared the computational and experimental results of confined compressible mixing layers and reported that the growth rate corresponds well with the experiment in the inlet area, while discrepancy appears downstream. They considered the difference resulted from the wall effect and fluid viscosity. Liu et al. [6] used a boundary layer to model the wall effect and studied the instability features of half unbounded wake/boundary layer confluent flow. Different distances between the wake and the boundary layer were investigated and the best spacing for mixing enhancement was found. Based on the primary research on the instability of compressible boundary/mixing layers confluent flow in a twodimensional planar channel [7], the present study focuses on the effect of heat release on the linear stability of the confluent flow. Two mean flows generated both by a theoretical model and numerical computation are considered. The hydrodynamic computation software Fluent® is adopted to conduct numerical simulations and the mixing of ethylene and air is simulated. Detailed stability characteristics are compared and analyzed. 2. Technical approaches Compressible linear stability equations are adopted under the assumption that the basic flow is locally parallel, and the derivation can be readily found in the literature [7]. The spatial scales in Cartesian coordinate system are nondimensionalized by G Xe* x* / ue* , pressure by Ue*ue 2 and other quantities by corresponding free stream values of outer flow. We assume that the viscosity and thermal conductivity satisfy Sutherland's law. The ideal gas assumption is also adopted. Instantaneous flow variables are decomposed into a base and a fluctuation quantity and the disturbances can be written as

 u, v, w, p, T

ªuˆ  y , vˆ  y , w ˆ  y , pˆ  y , Tˆ  y º eiD x E z Zt ¬ ¼

(1)

(1) in which α and β are streamwise and spanwise wavenumbers, ω is circular frequency. The boundary conditions can be written as

­ ° y  H , uˆ ® ° ¯ y H , uˆ

vˆ vˆ

wˆ Tˆ wˆ Tˆ

0

(2)

0

where H is half-width of the channel. Spatial instability is investigated in the present study and thus ω is real while α and β are complex. A fourth-orderaccurate difference method [8] is applied to discretize the stability equations and then a system of homogeneous equations can be got, i.e.

F D ^M` 0

(3)

Müller method and QZ algorithm are employed to calculate the eigenvalue α, and further the corresponding

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eigenfunction φ. The basic flow consists of two flows with different velocities. Boundary layers lie adjacent to the walls between the outer flows and the walls and the mixing layers lie near the centerline between the outer flows and inner flow. For the boundary layer flow the velocity and temperature profiles are obtained from similarity equations with adiabatic wall. For mixing layers the velocity is of hyperbolic tangent form, i.e. T1 1  1  ET  E hr

1  u J 1 2  M1 1  u  u  Eu 1  Eu 2

(4)

in which R=(1-U2)/(1+U2) is dimensionless speed ratio and σ is a positive parameter determining the gradient of velocity. The temperature profile is obtained from a theoretical model based on large activation energy asymtotics[9], i.e. for the outer flow: u  Eu J  1 2 T2 ET  1  ET  E hr  M1 1  u  u  Eu (5) 1  Eu 2 and for the inner flow:

1  R ˜ tanh V y (6) 1 R where γ is the ratio of heat specific, βT and βu are the ratios of temperature and velocity in the outer main flow to those in the inner main flow. βhr is the coefficient of heat release. Larger value implies stronger effect of heat release. The pressure is assumed uniform in the wall-normal direction as p 1/ J M12 . In the present study we set σ=0.4, and γ=1.4. u  y

3. Results and discussion Spatial linear stability of the confluent flow is studied and the geometry of the channel is two-dimensional with half width H=106. The width of inner flow is 135.5 thus the mixing layers lie at ym=f67.75. In the free stream of outer flow the velocity U*1=300m/s and temperature T*1=700K. While in the inner flow U*2=350m/s and T*2=2200K. The convective Mach number Mc=0.041 in this case. During the numerical computation, actual size of the plate dividing the two flows is considered and the width D=7.5 and length L=106. The velocity and temperature of basic flow are shown in Fig. 1 (Note that the length is scaled by 1000¥ in the computation domain).

(a)

(b)

Fig. 1. Numerical results of basic flow.(a) velocity; (b) temperature

3.1. Stability features without heat-release effect The mixing flow without heat release is investigated in this section. For the theoretical model, the coefficient of heat release is set to be 0. Correspondingly, combustion is not taken into consideration in the numerical

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computation. The profiles of velocity and temperature are displayed in Fig. 2. A major difference between the two velocity profiles lies in the wake region caused by the dividing plate. This results in the appearance of additional inflectional points, which indicates more complicated stability characteristics.

3

3

2.5 U T

2

U T

2

1.5

1

1

0.5

0

-100

-50

0

50

0

100

-100

-50

Y

0

50

100

Y

(a)

(b)

Fig. 2. Basic flow without heat release. (a) theoretical model ;(b) numerical computation

The eigenvalue spectra (Fig. 3) confirm the previous presumption. For the basic flow from theoretical model, only one unstable mode exists on the spectra map. Whereas, the computational flow exhibits more instabilities; three unstable modes appear on this map. The most unstable mode, or the main unstable mode, is of most interest. Comparably, the growth rate of the main unstable mode is almost triple as large as that of the theoretical model. Even the second most unstable mode has larger growth rate than the latter. Thus the computational basic flow is more unstable.

0.015 computational theoretical 0.01

-Di 0.005

0

0.09

0.1

0.11

Dr

0.12

0.13

Fig. 3. Eigenvalue spectra in absence of heat release, ω=0.1, β=0, Re=1.5E4

The corresponding eigenfunctions of two basic flows also differ from each other obviously, as shown in Fig. 4. There is a major difference between the two flows on the symmetry characteristics. Both the real and imaginary parts of the streamwise disturbing velocity of the computational flow are symmetric, while the other is antisymmetric. Detailed comparison reveals that the disturbance fluctuates more remarkably near the mixing layers for the main unstable mode of computational basic flow.

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0.04

0.2

u_real u_imag

u_real u_imag

0.02

0.1

0 0

-0.02 -0.1

-0.04 -0.2

-100

-50

0

50

-100

100

-50

0

50

100

Y

Y

(a)

(b)

Fig. 4. Eigenfuctions of u, Re=1.5E4, ω=0.1, α=0.0984-0.0144i, β=0 (a)theoretical model; (b) main unstable mode of computational basic flow

3.2. Stability features with heat-release effect The mixing combustion flow of ethylene and air is numerically simulated. The velocity and temperature profiles at x=0.17 are adopted as basic flow to conduct linear stability analysis (Fig. 5(b)). The theoretical model gives the basic flow with heat-release coefficient £hr=2.0 (Fig. 5(a)). Under the assumption of large activation energy, the mixing layer is thinner than that of basic flow without heat release. While the basic flow from computation has comparably thick mixing layer. For the numerical results, the heat release resulting from combustion make the temperature mildly higher, and two peaks appear at the inner side of confluent flow beside the mixing layers. The influence of wake region is also lessened by heat release according to the velocity profile. The effect of heat release slightly affects the basic flow as the selected profiles locate at a somewhat upstream position where combustion is not strong enough.

3

3

2.5

2

U T

2

U T

1.5

1

1

0.5

0

-100

-50

0

50

100

0

-100

-50

0

Y

Y

(a)

(b)

50

100

Fig. 5 Basic flow with heat release. (a) theoretical model, βhr=2.0; (b) numerical computation

The stability eigenvalue spectra are computed for the two basic flows (Fig. 6). Parameters are the same with Fig. 3. Three unstable modes still exist for the computational basic flow. The main unstable mode has growth rate twice

Liu Zhiyong et al. / Procedia Engineering 99 (2015) 320 – 326

as large as that of the theoretical model. Discrepancies of the growth rate come forth for each of the unstable modes when the heat-release effect is taken into consideration. Typically for the computational basic flow, the growth rate of main unstable mode decreases as much as 40%. For the theoretical model, the growth rate of the unstable mode slightly decreases about 8%.

0.01

0.008

computational theoretical

0.006

-Di 0.004

0.002

0

0.08

0.1

0.12

Dr

Fig. 6. Eigenvalue spectra with heat-release effect, ω=0.1, β=0, Re=1.5E4

This result indicates that the heat-release effect stabilize the basic flows both from the theoretical model and the numerical computation. The suppression on the instability is especially effective for the ethylene/air combustion flow. The shape functions of velocity disturbances are depicted for the basic flows from theoretical model with/without heat release (Fig. 7). Evidently, both disturbing velocity components of the no-heat-release case are larger than those of the heat-release case. This also indicates that the heat-release effect suppresses the disturbing activities, especially near the mixing layers.

0.05 u_with heat release v_with heat release u_without heat release v_without heat release

0.04

0.03

0.02

0.01

0

-100

-50

0

50

100

Y

Fig. 7 Eigenfunctions of disturbing velocity (theoretical model)

The effect of heat-release coefficient on the stability of basic flow from theoretical model is also investigated, as shown in Fig. 8. The maximum of growth-rate curve decreases, and correspondingly the unstable frequency range narrows with the increase of heat-release coefficient. Both signs imply that the heat-release effect stabilize the

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confluent flow. Mc=0.041 heat release coefficient Ehr=0.0

0.006

Ehr=1.0 Ehr=2.0 Ehr=3.0 Ehr=4.0 Ehr=5.0

0.005

0.004

-Di 0.003

0.002

0.001

0

0

0.05

0.1

Z

0.15

0.2

Fig. 8. Variation of growth rate with heat-release coefficient (theoretical model)

4. Conclusions The linear stability of two-dimensional compressible confluent boundary/mixing layers flow has been studied. Heat release is taken into consideration to preliminarily measure its effect on the stability of basic flows both from a theoretical model and numerical simulation. The ethylene and air are adopted to simulate the combustion mixing flow by numerical computation. Linear stability theory gives the eigenvalue spectra which reveal that three unstable modes exist for the computational basic flow while only one for the theoretical model. The main unstable mode of the computational flow has growth rate three times as large as that of flow from theoretical model. Therefore, the computational flow is more unstable than the theoretical one. The shape functions of disturbances are also compared for both flows. The heat-release effect is then taken into account. Linear stability analysis shows that the growth rate of main unstable mode decreases up to 40% than no-heat-release case for the computational flow. Meanwhile, for the basic flow from theoretical model, the disturbing growth rate decreases with the increase of heat release coefficient. And the frequency range of linear instability becomes narrow correspondingly. Conclusively, heat release effect stabilizes the confluent mixing flow. Linear stability analysis lays foundation for the mechanism research of mixing enhancement for the specified confined confluent flow. Further investigations are expected to simulate the evolution of disturbances with Parabolized Stability Equation method in the near future. References [1] A. Kumar, D. M. Bushnell, M. Y. Hussaini, A mixing augmentation technique for hypervelocity scramjets, Journal of Propulsion and Power, 5 (1987) 514-522. [2] J. P. Drummond, M. H. Carpenter, D. W. Riggins, M. S. Adams, Mixing enhancement in a supersonic combustor, AIAA Paper 89-2794, 1987. [3] J. A. Greenough, J. J. Riley, M. Soetrisno, D. S. Eberhardt, The effects of walls on a compressible mixing layer, AIAA Paper 89-0372, 1989. [4] F. Q. Hu, A numerical study of wave propagation in a confined mixing layer by eigenfunction expansions, ICASE Report No. 93-8, 1993. [5] D. A. Hudson, L. N. Long, P. J. Morris, Computation of a confined compressible mixing layer, AIAA-95-2173, 1995. [6] W. W. Liou, F. J. Liu, Compressible linear stability of confluent wake/boundary layers, AIAA Journal 41(2003) 2349-2356. [7] Z. Y. Liu, X. J. Yuan, X. Y. Liu, L. S. Fei, F. J. Liu, Stability analysis of supersonic boundary/mixing layers confluent flow, Chinese Journal of Theoretical and Applied Mechanics 46(2014) 28-36. [8] M. R. Malik, S. Chuang, M. Y. Hussaini, Accurate numerical solution of compressible linear stability equations, ZAMP 33(1982) 189-301. [9] T. L. Jackson, M. Y. Hussaini, An asymptotic analysis of supersonic reacting mixing layers, ICASE Report No. 87-17, 1987.