Multiobjective design optimization of the performance for the cavity flameholder in supersonic flows

Accepted Manuscript Multiobjective design optimization of the performance for the cavity flameholder in supersonic flows Wei Huang, Jun Liu, Li Yan, Liang Jin

PII: DOI: Reference:

S1270-9638(13)00149-1 10.1016/j.ast.2013.08.009 AESCTE 2947

To appear in:

Aerospace Science and Technology

Received date: 15 March 2013 Revised date: 27 July 2013 Accepted date: 16 August 2013

Please cite this article in press as: W. Huang et al., Multiobjective design optimization of the performance for the cavity flameholder in supersonic flows, Aerospace Science and Technology (2013), http://dx.doi.org/10.1016/j.ast.2013.08.009

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Multiobjective design optimization of the performance for the cavity flameholder in supersonic flows Wei Huang*, Jun Liu, Li Yan, Liang Jin Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha, Hunan 410073, People’s Republic of China

Abstract: The wall-mounted cavity makes a great difference to the flameholding mechanism in the supersonic flow, and it has been widely employed in the design process of the scramjet flowpath. However, the cavity would bring additional drag force to the engine, and the temperature within the cavity is very high. Therefore, both the drag force and the temperature should be minimized in the design process of the cavity flameholder. In the current study, the non-dominated sorting genetic algorithm (NSGA II) coupled with the Kriging surrogate model has been employed to optimize the nonreacting flow field around the cavity flameholder in the supersonic freestream with a Mach number of 3.0, and the test cases have been selected by the orthogonal table. At the same time, the numerical results have been compared with the experimental data obtained by Gruber et al., and four grid scales have been utilized to perform the grid independency analysis as well. The obtained results show that the wall static pressure profiles predicted by the numerical approaches show very reasonable agreement with the experimental data, and the static pressure along the floor face of the cavity is underpredicted. This may be induced by the inaccuracy of the turbulence model. The Pareto front for the multiobjective design optimization results is obtained, and there must be a compromise between the drag force and the area-weighted average temperature. The optimized drag force increases with the increase of the length-to-depth ratio and with the decrease of the swept angle, and the optimized area-weighted average temperature increases with the decrease of the length-to-depth ratio and with the increase of the swept angle. The drag force of the acoustic cavity is the smallest in the range

*

Lecturer, Corresponding author, E-mail: [email protected], Phone: +86 731 84576452, Fax: +86 731 84576449 -1-

considered in the current study. However, it would generate the strongest pressure oscillation. Keywords: Aerospace propulsion system; cavity flameholder; multiobjective design optimization; supersonic flow; drag force; area-weighted average temperature

1. Introduction The wall-mounted cavity has been widely employed in the scramjet engine to prolong the residence time of the mixture in the supersonic flow, and it has a very important impact on the combustion process between the fuel and the supersonic freestream [1][2]. At the same time, it would bring additional drag force to the scramjet engine [3]. Further, due to the vortices formed within the cavity, the temperature in the cavity is very high, and the vortices are generated because of the low velocity. This may enhance the requirement of the thermal protection for the cavity. Therefore, the drag force and the temperature within the cavity should be both minimized. Luo et al.[4] have estimated and compared the drag forces of three different geometric configurations of the cavity flameholder, namely the classical rectangular, triangular and semi-circular, and the cavities are with the fixed depth and length-to-depth ratio. They have found that the triangular cavity imposes the most additional drag force on the scramjet engine, and the drag force of the classical rectangular one is the least. At the same time, the particle image velocimetry (PIV) technique has been applied to study the flow structure within the rectangular, triangular and semi-circular cavity with a fixed length-to-depth ratio of 2.0 [5]. Based on the research work mentioned above, the variance analysis method has been employed by Huang et al. [6] to investigate the influences of the geometric parameters on the drag force of the heated flow field around the cavity flameholder, and they have found that the ratios of the length-to-upstream depth and the downstream-to-upstream depth must be foremost in the design process of the cavity flameholder. A cavity flameholder with a large ratio of the length-to-upstream depth brings large additional drag force in the reacting flow field. -2-

The above approach has been used to evaluate the influences of the geometric parameters of the cavity flameholder on the aero-propulsive performance of the integrated hypersonic vehicle as well [7], and the obtained results have shown that the cavity flameholder with its swept angle 45º is the best configuration in the range considered. Further, the transient phenomenon of a buoyancy-induced periodic flow and mixed-convection heat transfer in the lid-driven arc-shape cavities with different geometric configuration has been performed numerically [8], and the periodic flow pattern can be observed only if the inertial and buoyant forces are of approximately equal strength. However, in the opinion of the authors, the optimization of the flow field of the cavity flameholder in supersonic flows has been rarely investigated [3], although the influences of the geometric parameters of the cavity flameholder on the performance of the engine have been performed by many researchers, as well as the different flow field structures. Meanwhile, the relationships between the design variables and the objective functions are not clearly, and they must be explored deeply to point out the way for the design of the cavity flameholder. In the current study, the nonreacting flow field around the cavity flameholder in the supersonic flow has been optimized by the non-dominated sorting genetic algorithm (NSGA II) coupled with the Kriging surrogate model, and the numerical results have been compared with the experimental data obtained by Gruber et al. [9]. The grid sensitivity analysis has been performed as well. Further, in the optimization process, the length-to-depth ratio, the height and the swept angle have been taken as the design variables, and the drag force and the area-weighted average temperature have been considered as the objective functions. The combustion properties of the cavity flameholder are not taken into consideration at this moment, as well as the unsteady flow field properties in the cavity flameholder.

-3-

2. Physical model and numerical method 2.1 Physical model The experimental configuration of the cavity flameholder, as studied by Gruber et al. [9], is employed to provide data for the validation of the flow field structure within the cavity, as shown in Fig.1. Fig.1 shows the schematic view for the cavity flameholder configuration employed in the current study, and there are four geometric parameters, namely the depth (D), the length (L), the height (h) and the swept angle (ș). However, the depth of the cavity flameholder has been set to be constant in the current study, namely D = 8.9mm, and only three design variables have been considered in the following discussion, i.e. the length-to-depth ratio (L/D), the height (h) and the swept angle (ș). In the experimental model employed by Gruber et al.[9], the length-to-depth ratio is set to be 3.0 or 5.0, the height is 0.0 and the swept angle is 90° or 30°. In the optimization process, each design variable has three levels, i.e. L/D ‫{ א‬3, 5, 7}, h ‫{ א‬1mm, 3mm, 5mm}, ș ‫{ א‬30°, 45°, 60°}, and the specific experimental arrangement is given is Table.1. Table.1 shows the design of experiments for the configuration of the cavity flameholder employed in the current study, and the test cases are arranged by the orthogonal table L9(34) [10]. The supersonic freestream flows from left to right with a Mach number of 3.0, a total pressure of 690kPa and a total temperature of 300K, see Table.2, and the intersectional point between the upstream wall and the leading edge of the cavity flameholder has been set to be the origin of the coordinate system, see Fig.1. Table.2 depicts the boundary conditions for the supersonic freestream.

2.2 Numerical method In the current study, the two-dimensional Reynolds-averaged Navier-Stokes (RANS) equations are solved with the density based (coupled) double precision solver of Fluent [11], and the RNG k-İ turbulence model is employed to simulate the flow fields of the cavity flameholders in the supersonic flow. The RANS equations are selected -4-

because they can be solved on coarser meshes and permit the simplification of steady flow when compared with other numerical approaches, namely the detached eddy simulation, large eddy simulation and direct numerical simulation [12], and the RNG k-İ turbulence model is preferred due to its low requirement on grid density near the walls [13][14][15]. The second order spatially accurate upwind scheme (SOU) with the advection upstream splitting method (AUSM) flux vector splitting is utilized, and the Courant-Friedrichs-Levy (CFL) number is kept at 0.5 with proper under-relaxation factors to ensure stability. The standard wall functions are introduced to model the near-wall region flow, and the no-slip conditions are assumed for the walls of the channel. At the outflow, all the physical variables are extrapolated from the internal cells due to the flow being supersonic. The air is assumed to be a thermally and calorically perfect gas, and the Sutherland law with three coefficient method of viscosity is employed. The solutions can be considered as converged when all the residuals reach their minimum values after falling for at least three orders of magnitude, and the difference between the computed inflow and the outflow mass flux is required to drop below 0.001kg/s [16]. At the same time, the computational domain is structured by the commercial software Gambit, and the computational domain is multi-blocked in order to cluster the grid around the leading and trailing edges of the cavity flameholder. Four grid scales are employed to perform the grid sensitivity analysis in the flow field of the cavity flameholder, namely Grids 1-4 in Table.3. Table.3 shows the grid information for the grid sensitivity analysis. The grid spacing at the wall is set to be 0.001mm, which results in a value of y+ smaller than 1.2 for all of the flow fields, see Table.3. In Table.3, Zone II represents the region within the cavity flameholder, and Zone I represents the other regions in the supersonic channel.

3. Code validation and grid independency analysis Fig.2 displays the static pressure distributions along the walls of the cavity flameholder with L/D = 3.0 without -5-

swept angle, L/D =3.0 with the swept angle being 30° and L/D = 5.0 without swept angle, and Fig.3 shows the static pressure distribution along the wall of the combustor with L/D being 3.0 and no swept angle. In Fig.2, the effect distance includes the cavity upstream leading edge from the separation corner, the cavity floor face, and the cavity downstream trailing edge [2][6]. It has been normalized by the depth of the cavity. In Figs.2-3, the wall pressure is normalized by the static pressure of the freestream. The numerical predictions obtained by the different grid scales have been compared with the experimental data of Gruber et al.[9], and this is one aspect for the framework to built confidence in computational simulation predictions [17]. It is clearly shown that the predicted results all show reasonable agreement with the experimental data, see Fig.2, and the static pressure distributions along the floor face and the rear face of the cavity flameholder have been slightly underpredicted. This implies that the numerical method is suitable to simulate the nonreacting flow field around the cavity flameholder, and this small discrepancy may be induced by the inaccuracy of the turbulence model. However, the static pressure distribution along the front fact of the cavity flameholder shows very good agreement with the experimental data. At the same time, it is observed that the grid scale makes only a slight difference to the static pressure distributions along the walls of the cavity flameholder and the combustor, see Figs.2-3. Thus, in order to reduce the computational cost, the Grid 2 has been employed to simulate the test cases arranged by the orthogonal table. Fig.4 shows the streamline and static pressure contour of the cavity flameholder with L/D being 3.0 and no swept angle. It is clearly observed that a large eddy occupies nearly the whole region within the cavity flameholder, and two small corner eddies are formed as well, see Fig.4(a). This implies that the experimental model employed by Gruber et al. [9] is a open cavity, and it is consistent with the rules defined by Kim et al. [18]. A strong shock wave is formed in the trailing edge of the cavity, with a weak one generating in the leading edge [4], see Fig.4(b).

4. Result and discussion -6-

The drag forces and the area-weighted average temperatures of the test cases are shown in Table.1, and the drag force of Case 7 is the smallest in the range considered in the current study. Its value is 22.90N. At the same time, it is observed that the swept angle and the length-to-depth ratio both have an important impact on the drag force performance of the cavity flameholder. The drag force of the cavity flameholder increases with the increase of the length-to-depth ratio and with the decrease of the swept angle, and this may be induced by the large static pressure imposing on the trailing edge of the cavity flameholder, see Fig.5. Fig.5 depicts the wall static pressure profile comparisons for different cavity flameholder configurations. However, the area-weighted average temperature varies a litter in the range considered in the current study, see Table.1, and its range is from 284.5K to 287.0K. At the same time, the area-weighted average temperature increases slightly with the increase of the swept angle, and this may be induced by the largest right corner eddy formed within the cavity flameholder, see Fig.6. Fig.6 depicts the static pressure contours and streamlines for different cases employed in the current study, and it is clearly shown that the height increases with the increase of the swept angle. Meanwhile, the area of the right corner eddy increases with the increase of the height. Further, the drag force of the cavity configuration employed by Gruber et al.[9] is only 20.58N, and it is smaller than those employed in the current study. This may imply that the drag force of the acoustic cavity is the smallest. However, the acoustic cavity must induce the strongest pressure oscillation in supersonic flows [19], and this is not beneficial for the combustion in the supersonic flow. Thus, it should be combined with the angled cavity configuration in the scramjet engine to achieve the flameholding mechanism. In order to clarify the relationship between the design variables and the objective functions, the nonreacting flow field around the cavity flameholder is optimized by the non-dominated sorting genetic algorithm (NSGA II) [20] coupled with the Kriging surrogate model [21], and the same algorithm has been applied in the flow fields of the single expansion ramp nozzle [22] and the transverse injection in the scramjet combustor [23] successfully. Fig.7 shows the flowchart of the multiobjective design optimization process. The Kriging surrogate model is an -7-

interpolation technique based on the statistical theory, and it takes full account of the relevant characteristics of the variable space, containing the regression and nonparametric parts [24].

( )

( )

( )

y x ( ) = F T x( ) β + z x( ) , ( i = 1,", n ) i

i

i

(1)

Herein, n is the number of sample points, ȕ is the regression coefficient, x( ) is the sample points and i

( )

F T x( ) i

is the deterministic function that is a global approximation of the design space represented by the

( )

polynomial of x. z x( ) i

is the error of the random distribution, and it could provide the approximation of the

local deviation. To improve the prediction accuracy and the generalization ability of the model, the mean of the prediction error must be zero, and the mean square error of the prediction error must be the minimum. The Pareto front for the multiobjective design optimization results is obtained, see Fig.8. Fig.8 depicts the pareto front for the multiobjective design optimization results in the nonreacting flow field of the cavity flameholder. It is observed that the drag force increases with the decrease of the area-weighted average temperature, and this implies that there must be a compromise between these two objective functions in the design process of the cavity flameholder. Table.4 illustrates the optimized results obtained by the single-objective design optimization approach, and it is obvious that the optimized drag force for the single-objective design optimization approach is much smaller than those obtained by the test cases considered in the current study. The optimized area-weighted average temperature obtained by the single-objective design optimization approach is a bit smaller than that obtained in the Case 1. At the same time, it is observed that the optimized area-weighted average temperature increases with the decrease of the length-to-depth ratio, and it increases with the increase of the swept angle, see Figs.9(a) and (b). Fig.9 shows the relationships between the design variables and the optimized area-weighted average temperature. However, the value of the height has only a slight impact on the optimized area-weighted average temperature, see Fig.9(c), and the lower value can be beneficial for the obtainment of the optimized area-weighted average temperature. -8-

Fig.10 shows the relationships between the design variables and the optimized drag force of the cavity flameholder, and it is obvious that the optimized drag force increases with the increase of the length-to-depth ratio and with the decrease of the swept angle, see Figs.10(a) and (b). This is just opposite to the variable trend of the optimized area-weighted average temperature mentioned above. The value of the height makes only a slight difference to the optimized drag force as well, see Fig.10(c).

5. Conclusions In the current study, the non-dominated sorting genetic algorithm (NSGA II) coupled with the Kriging surrogate model has been employed to optimize the cavity flameholder in the supersonic flow with a freestream Mach number of 3.0, and the numerical results have been compared with the available experimental data obtained by Gruber et al.[9]. In the optimization process, the length-to-depth ratio, the swept angle and the height have been taken as the design variables, and the drag force and the area-weighted average temperature have been set as the objective functions. We have come to the following conclusions:

z

The numerical results all show reasonable agreement with the experimental data obtained by Gruber et al, and the static pressure along the floor face of the cavity is slight underpredicted. This discrepancy may be induced by the inaccuracy by the turbulence model.

z

The static pressure distribution along the trailing edge of the cavity flameholder has an important impact on the drag force of the cavity, and the area-weighted average temperature has a strong relationship with the size of the right corner eddy formed in the cavity.

z

The pareto front for the multiobjective design optimization results has been obtained, and the optimized drag force increases with the decrease of the optimized area-weighted average temperature. Thus, there must be a compromise between these two objective functions during the design of the cavity flameholder.

z

The optimized drag force increases with the increase of the length-to-depth ratio and with the decrease of

-9-

the swept angle, however, the optimized area-weighted average temperature increases with the decrease of the length-to-depth ratio and with the increase of the swept angle. The value of the height has only a slight impact both on the optimized drag force and area-weighted average temperature.

Acknowledgements The authors would like to express their thanks for the support from the Science Foundation of National University of Defense Technology (No.JC11-01-02) and the Hunan Provincial Natural Science Foundation of China (No.12jj4047).

References [1] Mishra D P, Sridhar K V. Numerical study of effect of fuel injection angle on the performance of a 2D supersonic cavity combustor. Journal of Aerospace Engineering, 2012, 25: 161-167 [2] Huang W, Pourkashanian M, Ma L, Ingham D B, Luo S B, Wang Z G. Investigation on the flameholding mechanisms in supersonic flows: backward-facing step and cavity flameholder. Journal of Visualization, 2011, 14: 63-74 [3] Ben-Yakar A, Hanson R K. Cavity flame-holders for ignition and flame stabilization in scramjets: An overview. Journal of Propulsion and Power, 2001, 17(4): 869-877 [4] Luo S B, Huang W, Liu J, Wang Z G. Drag force investigation of cavities with different geometric configurations in supersonic flow. Science China Technological Sciences, 2011, 54(5): 1345-1350 [5] Ozalp C, Pinarhasi A, Sahin B. Experimental measurement of flow past cavities of different shapes. Experimental Thermal and Fluid Science, 2010, 34: 505-515 [6] Huang W, Pourkashanian M, Ma L, Ingham D B, Luo S B, Wang Z G. Effect of geometric parameters on the drag of the cavity flameholder based on the variance analysis method. Aerospace Science and Technology, - 10 -

2012, 21: 24-30 [7] Huang W, Luo S B, Liu J, Wang Z G. Effect of cavity flame holder configuration on combustion flow field performance of integrated hypersonic vehicle. Science China Technological Sciences, 2010, 53(10): 2725-2733 [8] Cheng C H, Chen C L. Buoyancy-induced periodic flow and heat transfer in lid-driven cavities with different cross-sectional shapes. International Communications in Heat and Mass Transfer, 2005, 32: 483-490 [9] Gruber M R, Baurle R A, Mathur T, Hsu K Y. Fundamental studies of cavity-based flameholder concepts for supersonic combustors. Journal of Propulsion and Power, 2001, 17(1): 146-153 [10] Tucker A A, Hutto G T, Dagli C H. Application of design of experiments to flight test: a case study. Journal of Aircraft, 2010, 47(2): 458-463 [11] Fluent Inc., Fluent 6.3 User’s Guide, Fluent Inc., Lebanon, NH, 2006 [12] Zingg D W, Godin P. A perspective on turbulence models for aerodynamic flows. International Journal of Computational Fluid Dynamics, 2009, 23: 327-335 [13] Li N, Zhang K Y, Xu J L. Simulation and experiment validation of a two dimensional asymmetric ramp nozzle (in Chinese). Journal of Aerospace Power, 2004, 19: 802-805 [14] Roy C J, Blottner F G. Review and assessment of turbulence models for hypersonic flows. Progress in Aerospace Sciences, 2006, 42: 469-530 [15] Spalart P R. Strategies for turbulence modelling and simulations. International Journal of Heat and Fluid Flow, 2000, 21: 252-263 [16] Huang W, Wang Z G, Li S B, Liu W D. Influences of H2O mass fraction and chemical kinetics mechanism on the turbulent diffusion combustion of H2-O2 in supersonic flows. Acta Astronautica, 2012, 76: 51-59 [17] Roy C J. Review of code and solution verification procedures for computational simulation. Journal of Computational Physics, 2005, 205: 131-156 - 11 -

[18] Kim K M, Baek S W, Han C Y. Numerical study on supersonic combustion with cavity-based fuel injection. International Journal of Heat and Mass Transfer, 2004, 47: 271-286 [19] Zhu Y, Ouyang H, Du Z. Experimental investigation of acoustic oscillations over cavities. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2010, 224: 697-704 [20] Deb K, Pratap A, Agarwal S, Meyarivan T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 2002, 6: 182-197 [21] Jeong S, Murayama M, Yamamoto K. Efficient optimization design using Kriging model. Journal of Aircraft, 2005, 42: 413-420 [22] Huang W, Wang Z G, Ingham D B, Ma L, Pourkashanian M. Design exploration for a single expansion ramp nozzle (SERN) using data mining. Acta Astronautica, 2013, 83-10-17 [23] Huang W, Yang J, Yan L. Multi-objective design optimization of the transverse gaseous jet in supersonic flows. Acta Astronautica, 2014, 93: 13-22 [24] Yao S B, Guo D L, Sun Z X, Yang G W, Chen D W. Multi-objective optimization of the streamlined head of high-speed trains based on the Kriging model. Science China Technological Sciences, 2012, 55(12): 3495-3509

- 12 -

Figure captions Fig.1 Schematic view for the cavity flameholder configuration employed in the current study. Fig.2 Static pressure distribution along the walls of the cavity flameholder, (a) L/D = 3.0 without swept angle, (b)

L/D = 3.0 with the swept angle being 30° and (c) L/D = 5.0 without swept angle. Fig.3 Static pressure distribution along the walls of the combustor with L/D being 3.0 and no swept angle. Fig.4 Streamline and static pressure contour of the cavity flameholder with L/D being 3.0 and no swept angle. Fig.5 Wall pressure profile comparisons for different cavity flameholder configurations. Fig.6 Static pressure contours and streamlines for different cases employed in the current study. Fig.7 Flowchart of multiobjective design optimization process. Fig.8 Pareto front for the multiobjective design optimization results. Fig.9 The relationships between the design variables and the optimized temperature. Fig.10 The relationships between the design variables and the optimized drag force.

- 13 -

Table captions Table.1 Design of experiments for the configuration of the cavity flameholder employed in the current study. Table.2 Boundary conditions for the supersonic freestream. Table.3 Grid information for grid sensitivity analysis. Table.4 Optimized configurations for the cavity flameholder.

- 14 -

supersonic flow

L D

ș h

Fig.1 Schematic view for the cavity flameholder configuration employed in the current study.

- 15 -

2.5

2

Pw/P0

Grid 1 Grid 2 Grid 3 Grid 4 Exp

1.5

1

Front face

0.5

0

Floor face

1

Rear face

2

3

4

Effective distance

5

(a) L/D = 3.0 without swept angle 2.5

Grid 1 Grid 2 Grid 3 Grid 4 Exp

Pw/P°

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

3

Effective distance

3.5

4

4.5

5

(b) L/D = 3.0 with the swept angle being 30° 2.5 Grid 1 Grid 2 Grid 3 Grid 4 Exp

Pw/P°

2

1.5

1

0.5

0

1

2

3

4

Effective distance

5

6

7

(c) L/D = 5.0 without swept angle Fig.2 Static pressure distribution along the walls of the cavity flameholder, (a) L/D = 3.0 without swept angle, (b) L/D = 3.0 with the swept angle 30° and (c) L/D = 5.0 without swept angle.

- 16 -

2.5

Grid 1 Grid 2 Grid 3 Grid 4

Pw/P0

2

1.5

1

Upstream wall

0.5 -5

Floor face

0

Downstream wall

x/D

5

10

Fig.3 Static pressure distribution along the walls of the combustor with L/D being 3.0 and no swept angle.

- 17 -

y(m)

0

-0.01

0

0.01

x(m)

0.02

0.03

0.04

(a) Streamline

P(Pa):

3000

11683.4 20366.8 29050.3

y(m)

0.01

0

-0.01

0

0.01

x(m)

0.02

0.03

0.04

(b) Static pressure contour Fig.4 Streamline and static pressure contour of the cavity flameholder with L/D being 3.0 and no swept angle.

- 18 -

2.5 2.4 Case 1 Case 4 Case 7

Case 2 Case 5 Case 8

2.2

2

2 1.8

Pw/P0

Pw/P0

1.6

1.5

1.4 1.2

1

1 0.8 0.6 0

0.5

1

1.5

x/D

2

2.5

3

0

1

2

(a) L/D=3.0

x/D

3

4

(b) L/D=5.0

2.5

Case 3 Case 6 Case 9

2

Pw/P0

0.5

1.5

1

0.5

0

1

2

3

x/D

4

5

6

7

(c) L/D=7.0 Fig.5 Wall pressure profile comparisons for different cavity flameholder configurations.

- 19 -

5

Fig.6 Static pressure contours and streamlines for different cases employed in the current study.

- 20 -

Fig.7 Flowchart of multiobjective design optimization process.

- 21 -

286

T(K)

285.5

285

284.5

284 16

18

20

22

D(N)

24

26

28

30

Fig.8 Pareto front for the multiobjective design optimization results.

- 22 -

3 2.8

60

2.6 2.4

50

θ(°)

2

40

1.8 1.6 30

1.4 1.2 1 284.5

285

T(K)

285.5

20

286

284.6

284.8

(a) L/D vs. T

285

285.2

T(K)

285.4

285.6

285.8

(b) ș vs. T

8

6

h(mm)

L/D

2.2

4

2

0

284.6

284.8

285

285.2

T(K)

285.4

285.6

285.8

286

(c) h vs. T Fig.9 The relationships between the design variables and the optimized temperature.

- 23 -

286

3 60 2.5

θ(°)

2

1.5

1

40

30

18

20

22

D(N)

24

26

20

28

18

20

(a) L/D vs. D

22

D(N)

24

26

(b) ș vs. D

8

6

h(mm)

L/D

50

4

2

0

18

20

22

D(N)

24

26

28

(c) h vs. D Fig.10 The relationships between the design variables and the optimized drag force.

- 24 -

28

Table.1 Design of experiments for the configuration of the cavity flameholder employed in the current study. Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9

ș

L/D

h(mm)

D(N)

T(K)

30° 30° 30° 45° 45° 45° 60° 60° 60°

3 5 7 3 5 7 3 5 7

1 3 5 3 5 1 5 1 3

28.06 50.69 72.41 25.00 44.89 65.48 22.90 42.20 61.43

284.78 285.99 284.92 285.52 286.37 286.25 285.96 286.38 286.53

- 25 -

Table.2 Boundary conditions for the supersonic freestream. Supersonic freestream

P’(kPa)

P0(kPa)

T’(K)

T0(K)

Ma

690

18.784

300

107.143

3.0

- 26 -

Table.3 Grid information for grid sensitivity analysis. Zone I Grid 1 Grid 2 Grid 3 Grid 4

y+

Zone II

x

y

x

y

200 400 800 400

80 80 80 160

50 100 200 100

80 80 80 80

- 27 -

<1.172 <1.183 <1.186 <1.183

Table.4 Optimized configurations for the cavity flameholder.

Dmin Tmin

ș(°)

L /D

h(mm)

Dopt(N)

Topt(K)

61.496 23.654

1.374 2.8443

8.7134 0.12681

17.5046 -

284.653

- 28 -