Experimental and numerical investigation on hysteresis characteristics and formation mechanism for a variable geometry dual-mode combustor

Accepted Manuscript Experimental and numerical investigation on hysteresis characteristics and formation mechanism for a variable geometry dual-mode combustor

Shuo Feng, Juntao Chang, Chenlin Zhang, Youyin Wang, Jicheng Ma, Wen Bao

PII: DOI: Reference:

S1270-9638(17)30058-5 http://dx.doi.org/10.1016/j.ast.2017.03.040 AESCTE 3979

To appear in:

Aerospace Science and Technology

Received date: Revised date: Accepted date:

10 January 2017 28 February 2017 31 March 2017

Please cite this article in press as: S. Feng et al., Experimental and numerical investigation on hysteresis characteristics and formation mechanism for a variable geometry dual-mode combustor, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.03.040

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Experimental and numerical investigation on hysteresis characteristics and formation mechanism for a variable geometry dual-mode combustor Shuo Feng1, Juntao Chang2, Chenlin Zhang3, Youyin Wang4 Jicheng Ma5, Wen Bao

6

Harbin Institute of Technology, 150001 Heilongjiang, People’s Republic of China Abstract: As part of our efforts to study the hysteresis characteristic and formation mechanism for a variable geometry dual mode combustor, a series of geometry path continuous adjustment experiments and numerical simulation were conducted in the variable geometry dual mode combustor with a Mach number of 3, a divergent ratio ranging from 1.6 to 2.54 and a fuel equivalence ratio ranging from 0.6 to 1.0. Experimental results indicated that the wall static pressure of feature points had an obvious hysteresis phenomenon with the geometry path continuous variation. For given feature points, the hysteresis becomes much smaller with the increasing of the divergent ratio. The interaction between the oblique shock train motion and combustion heat release distribution was adequately considered to explicate the mechanism of the hysteresis formation. It was found that the hysteresis phenomenon was produced from the unstable positive feedback effect of the oblique shock train motion in the isolator. Moreover, the effect of the hysteresis on the total pressure loss, irreversible entropy loss and combustion performance were investigated numerically and experimentally. It was therefore strongly believed that the study of the hysteresis characteristic and formation mechanism on the combustion performance could be very significant to improve combustion performance for the variable geometry dual mode combustor, especially for a wide range of flight Mach numbers.

1 PhD Candidate, School of Energy Science and Engineering, [email protected]. 2 Professor, Academy of Fundamental and Interdisciplinary Sciences, [email protected]. ˄Corresponding author˅ 3 PhD Candidate, School of Energy Science and Engineering, [email protected]. 4 PhD Candidate, School of Energy Science and Engineering, [email protected]. 5 PhD Candidate, School of Energy Science and Engineering, [email protected]. 6 Professor, School of Energy Science and Engineering; [email protected].

Keyword: variable geometry dual mode combustor; hysteresis characteristic; formation mechanism

Nomenclature A

=

exit cross section, m2

F

=

thrust, N

Fd

=

thrust coefficient, 2F/ȡV2A

h1

=

vertical distance from wedge slider to upper wall of combustor, m

h2

=

height of isolator, m

Ma

=

Mach number

P

=

static pressure, Pa

Pt

=

total pressure, Pa

s

=

static entropy, J/K

Tt

=

total temperature, K

V

=

velocity of entrance airflow, m/s

x

=

axial distance from isolator entrance , m

y

=

longitudinal distance of combustor, m

Y

=

specify species in mole fractions

ȟ

=

divergent ratio

φ

=

fuel equivalence ratio

ρ

=

density of entrance airflow, kg/m3

Subscripts -1

= first of same divergent ratio

-2

= second of same divergent ratio

↔

= back and forth movement

1. Introduction Many different launch vehicle concepts have been proposed in the search for a space shuttle replacement [1-4]. Each of these vehicles promised to reduce costs and increase reliability over current launch vehicles. They can be operated over a wide range of flight Mach numbers if a multiple integrated propulsion system [5-6] was used. Dual-mode combustor as one of the most important component for the engineering implementation of the combined cycle engine of space mission vehicles [7], has attracted a great deal of attention. For example, Huang et al [8-9] studied the combustion characteristics of a rocked-based combined-cycle engine combustor numerically. Optimum combustion performance is always influenced by the interactions between the combustor geometry and heat release for a wide range of flight Mach numbers. Much work has been done on this particular aspect in recent years. The supersonic combustion with cavity-strut injection [10] was investigated to improve the combustion performance in a model scramjet engine by studying the ignition schemes [11-12] and the fuel transport and mixing progress [13]. Several approaches have been proposed in the literature for fulfilling this required adaptation between combustor geometry and combustion heat release. For example, the French and Russian proposed a variable geometry combustor by simply moving the engine cowl in the flight test program [14-17] since 2003. A large part of technology development effort had been led on ground and then was dedicated to combustor to ensure the high performance. The combustor was validated using the basic experimental results taken from literatures. Avrashkov et al [18] in detailed studied about the gas-dynamic structure of the variable geometry dual mode combustor. The results showed that a serious tests of the variable geometry dual mode combustor of a wide-range ramjet engine had a stable operation and the required performance over the entire model

range of flight conditions. Moreover, a great deal of nonlinear and hysteresis characteristics work have conducted by some researches for the scramjet. Yu et al [19] studied a class of nonlinear phenomena by the cusp topological model in the hypersonic flow and supersonic combustion progress of scramjet. The catastrophe strength and hysteresis width of mode transition were studied for the physical mechanism of the catastrophe and hysteresis of mode transition in scramjet engine. In addition, the stability boundaries of combustion mode transition was predicted and analyzed by the model. Bao et al [20] studied the nonlinear catastrophic and hysteresis characteristic of combustion mode transition by changing the fuel mass flow rate along two adverse paths in a liquid-fueled strut-based scramjet combustor. The nonlinear catastrophe and hysteresis behaviors of mode transition was observed in the experiment. Zhu et al [21] studied the flame stabilization and propagation by a series of fuel adjustment experiments in a dual mode scramjet with two-staged-strut injectors. The sudden changes in the wall pressure profiles and an obvious hysteresis phenomenon were found in the fuel adjustment procedure under the lower inflow enthalpy condition. There was a much smaller hysteresis phenomenon under the higher inflow enthalpy condition. Chen et al [22] studied the supersonic isolator flow behavior by adjusting the cowl convergence angles and blockage ratios. Pressure profiles indicated that isolator with a 8°cowl convergence angle had the largest pressure endurance. And the flow decelerated to sonic at the exit due to the increasing of the blockage ratio to nearly chock the whole isolator. Cui et al [23-24] investigated the nonlinear and hysteresis phenomenon by introducing the topological rule for the scramjet. However, there have been few studied on the hysteresis characteristics and hysteresis formation mechanism for a dual mode combustor in the open literature. Especially, the effect of the hysteresis on combustion performance by adjusting the geometry path variation was not investigated

for the variable geometry dual mode combustor. Therefore, the French-Russian variable geometry dual mode combustor [18] was introduced to study the hysteresis characteristic and hysteresis formation mechanism numerically and experimentally by adjusting geometry path variation with a Mach number of 3, a divergent ratio ranging from 1.6 to 2.54 and a fuel equivalence ratio ranging from 0.6 to 1.0. The divergent ratio of the combustor was changed along two adverse paths: one was increasing the divergent ratio and one was decreasing the divergent ratio. Particular attention was focused on the hysteresis phenomena. The wall static pressure of feature points was observed by adjusting the geometry path variation with different fuel equivalence ratios. The time response and path response of the feature point pressure were obtained due to the hysteresis effect. The formation mechanism of the hysteresis was explicated by the interaction between the oblique shock train motion and combustion heat release distribution. The effect of the hysteresis on the total pressure loss, irreversible entropy loss and thrust performance were also obtained numerically and experimentally.

2. Combustor Setup and Computational Fluid Dynamic Method 2.1 Combustor Setup The current set of experimental results were obtained using the supersonic combustion research facility of Harbin Institute of Technology in China. A schematic of the variable geometry dual-mode combustor could be found in Fig.1.The inflow air could be heated to a total temperature of 1505K through the combustion of alcohol-air. Additional oxygen was injected to maintain a 0.21% mole fraction in the heated products. The air entered the isolator segment of the combustor at a Mach number of 3.0. A programmable logic controller was used to trigger various events related to fuel injection timing during the experiments. The divergent ratio of the combustor was defined by h2/h1.

The divergent ratio of the combustor was selected and adjusted with ξ = 1.6 ↔ 1.85 and ξ = 2.0 ↔ 2.54

by the movement of the wedge slide. The fuel equivalence ratio was varied from 0.6 to 1.0. The pressure tap ports of the supersonic combustion research facility were located along the length of the combustor, instrumented with transducers providing a full range of 0~1000kPa, an uncertainly of 0.25% full scale, and a maximum sampling frequency of 1000 Hz. The combustor had four sections (labeled as I, II, III and IV in Fig.1). Section I was a constant cross section isolator and the struts were located on the central part of Segment I. In order to minimize the aerodynamic disturbance in the flow field and struts drag, a thin strut was employed with a blockage ratio of 3%. The combustor sections included a symmetric divergent section II, constant section area III and an inner nozzle section IV. When the combustor was operated under the ramjet mode in a low flight Mach number to avoid the thermal blockage, the combustor was required to maintain a great divergent ratio to obtain high combustion performance. When the combustor was operated under the scramjet mode at a high flight Mach number, the divergent ratio was required to be small for obtaining high combustion performance.

θ θ

θ

Fig.1 Schematic of variable geometry dual-mode combustor 2.2 Computational Fluid Dynamic method 1)

Computational Fluid Dynamics method

In order to provide extra information for the flow field characteristics analysis of the variable geometry dual-mode combustor, numerical simulation was run with computational fluid dynamics. A Navier-Stokes solver was provided to calculate the steady flow field surrounding the variable geometry dual-mode combustor model. The governing two-dimensional compressible Reynolds-averaged Navier-Stokes (RANS) equations were discretized using the finite volume framework. The RANS equations were considered for their ability to permit the simplification of steady flow when compared with the other numerical methods, namely detached eddy simulation, large eddy simulation and direct numerical simulation [25-27].Advection term was discretized using advection upstream splitting method (AUSM) scheme. The governing RANS equations of continuity, momentum and energy were coupled together using the density based solver. The coupled set of governing RANS equations was discretized in time using an implicit Euler scheme for steady-state simulation, thus time-marching proceeded until a steady-state solution was reached. In spatial discretization methods, second order upwind scheme was used for flow term, turbulent kinetic energy term and specific dissipation rate term. The governing equations are as follows [28]: The continuity equation is given by

∂ (ρu ) = 0 i ∂x i

(1)

The momentum conservation equation is given by

∂ ∂p ∂ ∂u + − τ ij ) + Si (ρu u ) = − (μ i j ∂x j ∂xi ∂x j ∂x j

(2)

The energy equation is given by

∂ ∂ K ∂T (ρu T ) = − ( ) + ST j ∂x j ∂xi c ∂x j

(3)

Where ȡ is the gas density, ui and uj are the velocity components in the xi and xj directions, respectively,

ȝ is the dynamic viscosity, IJij is the Reynolds stress tensor, Si is the general source term in the momentum conservation equation, ST is the viscous dissipation function, T is the temperature and K is the thermal conductivity. The SST (shear stress transport) k-Ȧ is a hybrid of the standard k-İ and the k-Ȧ models. Huang et al [29] performed the effect of the turbulence model on the transverse slot injection flow field in supersonic flows. The results showed that the wall pressure profile with low jet-to-crossflow pressure ratios was predicted accurately by the k-İ turbulence model and the SST k-Ȧ turbulence model for the flow field with high jet-to-crossflow pressure ratio [30]. The turbulence model SST k-Ȧ can improve the ability to predict the flow separation. Menter’s k-Ȧ turbulence model SST with two-equation was implemented with default closure coefficients [31]. However, it was also worth considering that RANS simulations model were very sensitive to the values used for turbulent Schmidt and Prandtl numbers [32]. The reaction and Prandtl number on the reacting flow field of a typical cavity-based scramjet combustor were studied by Huang et al[33].Hassan et al[34]showed the resolved field to derive estimates for the turbulent Schmidt, which was not constant with strong various throughout the flow. Present work took the turbulent Schmidt number as 1.3 to obtain the best pressure profile. Moreover, Huang et al [35-36] performed the chemical kinetics mechanism on the influences of turbulence model in the supersonic flow. Two different chemical reaction mechanisms were employed to study the effect of the chemical kinetics mechanism on the combustion. The results indicated that the chemical reaction mechanism had only a slight impact on the overall performance of the turbulent diffusion combustion. Smirnov et al [37] performed chemical kinetics mechanism on the acetylene with 11 reaction and 9 components. A modified k-omega turbulence model was used to simulate flame acceleration. In the calculations, single-step kinetics and finite-rate/eddy-dissipation were used to model the kerosene turbulence chemical interaction kinetics. Rate data for the C12H23-O2 forward reaction mechanism were adapted from Westbrook et al[38].This mechanism already had a good agreement with that of rate data on the mixing and combustion characteristic of kerosene by Kumaran et al[39]. The air was modeled as a thermally perfect air and the fuel was modeled as a single-specie. Viscosity of the mixture was evaluated using the mass-weighted-mixing law. For the individual fluids in the mixture, these properties were evaluated using the Sutherland’s law and fifth-order polynomials in temperature. The ratio of specific heats was calculated by the temperature polynomial. The injection scheme was the group injection.

The molecular formula of the hydrocarbon was assumed to be C12H23, and the supersonic freestream was composed of 21% O2 and 79%N2. Then the chemical reaction equation for the hydrocarbon was C12H23+17.5O2ĺ12CO2+11.5H2O

(4)

The single chemical reaction rate expression [38] was usually expressed

R1 = R exp( − Ea / T )[ fuel ]a (O2 )b

(5)

Where R was the pre-exponential collision frequency factor, Ea was the activation energy, a and b were the reaction rate coefficients. Table1 Reaction rates for the one-step mechanism [38]

2)

Reaction

R

Ea

a

b

R1

2.587ή109

1.256ή108

0.25

1.5

Boundary Condition and Initial Conditions The boundary type of isolator entrance was chosen as the pressure-inlet. The static pressure, total

pressure and total temperature of the isolator entrance were chosen to be 0.06 MPa, 1.68 MPa and 1505K, respectively. The boundary type of the injection was chosen as the mass-flow-inlet. The injector exit static pressure were at 1.1MPa. The kerosene fuel was injected at 600K from the fuel injection port. For the simulation of supersonic combustion, the kerosene was gaseous. In general, the liquid hydrocarbon fuel requires quick vaporization before mixing and the subsequent combustion [40]. Their relatively long ignition delay times typically exceeded the residence time of the gas flow within combustor [41-42]. For the gaseous hydrocarbon fuel, the diffusivity is higher. In addition, in order to improve the mixing of the air-fuel and the stable combustion, more studies have demonstrated the investigation of supersonic combustion of gaseous hydrocarbon fuel [43]. Based on the experiment results, the turbulent intensity of 10% was appropriate to the inlet of fully development of turbulence. The hydraulic diameter (to estimate the turbulent length scale) at the entrance and the fuel injection ports were specified as 40mm and 0.05mm. All the variables of the supersonic flow at the combustor

exit were determined from the interior of the domain by extrapolation. Non-slip and adiabatic wall conditions were employed for the solid boundary with standard wall functions. The species concentrations of air stream and hydrocarbon jet were shown in Table 2. Table 2 Species concentrations of air stream and hydrocarbon jet

3)

Parameter of entrance

Air

Hydrocarbon jet

YO2

0.21

0

YN2

0.79

0

YC 12 H 23

0

1

Grid Independence and Validation The computational mesh was established using the ICEM software. The computational grids were

optimized to capture the physical and chemical relevant features. Three computational grids of 0.32million (coarse) nodes, 0.75 million (medium) nodes, and 1.1 million (fine) nodes were used to study the grid independence. The grids approaching the wall were scattered using geometric proportion way in order to capture the boundary layer. The height of the first cell center above the test-section floor was 0.01mm. As shown in Fig.2, the static pressure distribution was obtained from three computational grids. The detailed definition can be seen from good agreement between the curves of static pressures. The method of error estimates was from Smirnov [44-45]. The numerical calculation error estimates were shown in Table 3. Table 3 Numerical calculation error estimates Allowable

Grid

Number of

Accumulated

Allowable number of

Reliability

error

resolution

iterations (n)

error

time steps(nmax)

Rs=nmax/n

5%

160·2000

150000

2.443·10-7

4.189·1010

2.79·104

5%

250·3000

150000

6.403·10-8

6.096·1011

4.06·105

5%

320·3450

150000

3.054·10-8

2.68·1012

1.78·106

Fig.2 Certification of grids independence 4)

Numerical Methods of Validation ξ = 1.3, φ = 0.6 ξ = 1.3, φ = 0.6

a. ξ = 1.3, φ = 0.6 ξ = 1.6, φ = 0.8 ξ = 1.6, φ = 0.8

b. ξ = 1.6, φ = 0.8 Fig.3 Comparison of experimental and numerical static pressure distributions For validation of numerical methods, two wall static pressure distributions by comparing numerical

simulation with experimental results were obtained in Fig.3. The current set of experimental results were obtained by the variable geometry dual-mode combustor using the supersonic combustion research facility of Harbin Institute of Technology in China. The experimental results were represented using different symbols with different conditions. Numerical simulation results were represented by straight line with different colors. The wall pressure distribution was obtained on the center line of side wall. It could been seen from Fig.3 that the pressure increased from the vicinity of isolator entrance along the airflow direction and then reached the peak pressure with the increasing of heat release. The wall pressure progressively decreased. By comparing numerical simulation with experimental results, the static pressure distributions remained almost coincidence in the combustor.

3. Results and Discussion 3.1 Effect of hysteresis on wall static pressure As presented in Fig.4, several typical wall static pressure distributions were obtained in the variable geometry dual-mode combustor. The geometry path variation was considered as the control variable by adjusting the wedge. The divergent ratio of the combustor was repeatedly varied with ξ = 1.6 ↔ 1.85 and ξ = 2.0 ↔ 2.54 . It can be seen from Fig.4a that when the divergent ratio of the combustor is not great enough to accommodate great heat release, the wall static pressure starts to increase, until a maximum peak pressure is reached. As the divergent ratio continues to rise until ξ = 1.85 , the effect of increasing the divergent ratio overcomes the effect of increasing the total temperature generated by the heat addition. So the peak pressure progressively falls down. Moreover, it can also be seen in Fig.4a that there is an obvious hysteresis phenomenon to be found with ξ = 1.6 _1 and ξ = 1.6 _ 2 . The static pressure with ξ = 1.6 _ 2 is obviously less than the static pressure with ξ = 1.6 _1 . For example, the peak pressure with ξ = 1.6 _1 is 13.6% higher than that with ξ = 1.6 _ 2 . Correspondingly, as shown in

Fig.4b, the geometry path was repeatedly increased and decreased with ξ = 2.0 ↔ 2.54 for φ = 0.72 .The trend of the wall static pressure is similar to Fig.4a. The hysteresis phenomena can also be repeated in Fig.4b. ξ = 1.6 _ 1, φ = 0.6 ξ = 1.85, φ = 0.6 ξ = 1.6 _ 2, φ = 0.6

a)

ξ = 1.6 ↔ 1.85, φ = 0.6 ξ = 2.0 _ 1, φ = 0.72 ξ = 2.54, φ = 0.72 ξ = 2.0 _ 2, φ = 0.72

b)

ξ = 2.0 ↔ 2.54, φ = 0.72

Fig.4 Variations of wall static pressure with path chosen of different divergent ratio Hysteresis is an inherent property of bistable systems, in which the response of a system to an external influence depends not only on the present magnitude of that influence but also on the previous history of the system. It indicates that the external influence can exceed a threshold to switch the system to another steady state, at which it may remain, when the external influence falls. For example, in case of ξ = 1.6 ↔ 1.85 , when the combustion begins with ξ = 1.6 _1 , the adverse pressure generated

by the heat addition is great enough in the section II that the strong oblique shock train is formed in the isolator. The isolator is used to avoid the effect of the high back pressure on the combustor performance [46]. Then the air flow velocity gradually increases in the section II with the rising of the divergent ratio, which reduces the residence time for the progress of fuel evaporating and mixing. Meanwhile, the combustion heat release is not great enough that the length of the oblique shock train reduces in the section I. As also presented in Fig. 4, the combustion of the former in ξ = 1.6 _1 is obviously stronger than the latter with ξ = 1.6 _ 2 . The hysteresis phenomenon may be formed due to the interaction between the oblique shock train motion and combustion heat release distribution. 3.2 Hysteresis characteristics of feature points The static pressure of feature points was selected as the state variable, which was used to describe the hysteresis characteristic of combustion performance. The reason which picked feature points 1 and 2 was as follows: a pre-combustion shock train was formed in the isolator. The wall static pressure distribution has a discontinuous sudden change near the strut as the divergent ratio varies. Thus, the two feature points, which are located near the exit of the isolator, can effectively monitor the hysteresis process of the combustion-induced adverse pressure transmission upstream. The two feature points are disturbed earlier by the leading edge of the shock train than other measuring points. So the two points were selected to investigate the hysteresis characteristics of combustion performance for the combustor. As shown in Fig.5, the time response of the feature points about the static pressure was obtained with

ξ = 1.6 ↔ 1.85 . When the combustion heat release begins to rise near the strut, the static pressure is added, then reaches the maximum peak pressure. The wall static pressure progressively decreases with the increasing of the divergent ratio, and the minimum pressure is reached until ξ = 1.85 .It can be seen

in Fig.5 that the static pressure of the feature point 1 is much higher than the pressure of the feature point 2. Moreover, for a given feature point, the pressure with ξ = 1.6 _1 is 1.12 times larger than the pressure with ξ = 1.6 _ 2 due to the effect of the hysteresis. With the increasing of the divergent ratio until ξ = 2.0 ↔ 2.54 , there is still a hysteresis phenomenon to be found in Fig.6. The static pressure of the feature point drops off due to accommodating great heat release by increasing the divergent ratio. For a given feature point 2, the static pressure with ξ = 2.0 _1 is 1.04 times higher than the pressure

with ξ = 2.0 _ 2 .

ξ

ξ

Fig.5 Time response of feature points pressure with ξ = 1.6 ↔ 1.85

ξ

ξ

Fig.6 Time response of feature points pressure with ξ = 2.0 ↔ 2.54 As presented in Fig.7, an obvious hysteresis and nonlinear variation about the static pressure of the feature point 2 were obtained through the path response. The static pressure begins to reduce slightly

with the increasing of the divergent ratio until 1.85. When the divergent ratio progressively decreases from ξ = 1.85 to ξ = 1.6 by adjusting the wedge, there is a small fluctuation to produce about the static pressure. The phenomenon may be explained that the adverse pressure is not great enough that the disturbance is not propagated into the isolator. The combustion heat release starts to add with the reduction of the divergent ratio. Then there is an oblique shock train to form in the isolator due to the augment of the adverse pressure. It can also be seen from Fig.7 that the hysteresis is not closed with

ξ = 1.6 . As showed in Fig.8, a similar hysteresis phenomenon appears with ξ = 2.0 ↔ 2.54 . The trend of the static pressure at feature point 2 over path is similar to Fig.7. However, it can be obviously seen from Fig.8 that the hysteresis is closed to the closed state. The result indicates that the hysteresis effect is gradually weaken due to the reduction of the interaction between the oblique shock train motion and the combustion heat release with the increasing of the divergent ratio.

ξ = 1.6 ↔ 1.85, φ = 0.6

ξ

Fig.7 Path response of wall static pressure with ξ = 1.6 ↔ 1.85

ξ = 2.0 ↔ 2.54, φ = 0.72

ξ

Fig.8 Path response of wall static pressure with ξ = 2.0 ↔ 2.54 3.3 Hysteresis formation mechanism The hysteresis behavior is characterized by an abrupt transition from one steady state to another steady state. The hysteresis is leaded by an unstable positive feedback effect. In order to further explore the hysteresis formation mechanism in response to changes in the static pressure, the computational fluid dynamics simulation was offered utilizing to analyze the hysteresis response of the variable geometry dual mode combustor. As shown in Fig.9, the axial distribution of mass-weighted average Mach number was obtained with ξ = 1.6 ↔ 1.85 . As the combustion heat release begins to increase near the strut, the Mach number decreases abruptly in the section I and II. The incoming supersonic flow falls down to the sonic speed at the section II. A shock wave with sonic upstream speed is created in the section II and then propagates into the isolator. Once the divergent ratio of the combustor is sufficient to accommodate the great heat release, the Mach number is added to the supersonic flow and the combustion heat release distribution moves backwards along the combustor. The external perturbation expels the oblique shock train motion upstream into the isolator. The positive feedback effect is eliminated. When the divergent ratio is reduced from ξ = 1.85 to ξ = 1.6 _ 2 , the adverse pressure gradient arises because of the thermal occlusion resulted by the heat addition. The supersonic airflow is slowed down by the oblique shock train in the isolator.

ξ = 1.6 _1, φ = 0.6 ξ = 1.85, φ = 0.6 ξ = 1.6 _ 2, φ = 0.6

Fig.9 Mass-weighted average Mach number distribution with ξ = 1.6 ↔ 1.85 It can be seen through the analysis above that the hysteresis behavior of the static pressure originates from the positive feedback mechanism of an unstable shock train motion in the isolator. In order to further interpret the hysteresis formation mechanism, as shown in Fig.10, the Mach number contour was shown for the significant differences about flow separation and pre-combustion oblique shock train information in detail. As presented in Fig.10a, both the speed of propagation and the downstream location, at which the shock train originates, depend strongly on the adverse pressure resulted by heat addition. The adverse pressure induces the boundary layer separation on the upper and lower wall. Then it forms an effective compression ramp from compression wavelets coalesce into the oblique shock waves. The oblique shock waves intersect and improve the refracted shock waves that bound a primary compression region. The shock refractions subsequently interact with the separated boundary layer and reflect into additional shock structures, which causes further thickening of the boundary. The localised boundary layer separation is observed to accompany the shock system as it moves upstream in the isolator. The pressure-disturbance generated through time-dependent heat release propagate upstream through the subsonic portions of the boundary layer. This causes in local thickening and separation of the viscous layer within the section I, leading to changes in the oblique shock pattern. Under the critical condition, a small perturbation may lead an abrupt oblique shock train motion due to

the existence of unstable positive feedback effect. As can be seen in Fig.10b, the adverse pressure begins to reduce because of the increasing of the divergent ratio, so that the length of the oblique shock train falls down progressively in the isolator. The combustor operates continuously from one stable state to another stable state. Similarly, when the combustor operates continuously from ξ = 1.85 to ξ = 1.6 _ 2 , there is an obvious increment about the length of the oblique shock train in the isolator in Fig.10c. The change of the divergent ratio causes the additional pressure gradients to thicken the boundary layer, accelerating the supersonic flow down to subsonic velocity. Then the oblique shock train is again formed in the isolator. The combustor operates in the critical state.

a. ξ = 1.6 _1, φ = 0.6

b. ξ = 1.85, φ = 0.6

c. ξ = 1.6 _ 2, φ = 0.6

Fig.10 Mach number contour with ξ = 1.6 ↔ 1.85 The hysteresis effect also implies that the state of the combustion heat release distribution depends on the previous history of the combustion in the geometry path variation. To further explain the hysteresis formation mechanism based on the combustion heat release analysis, as presented in Fig.11, the total temperature contour shows the development and distribution of the combustion heat release. The addition in more heat release is accompanied by an increase in total temperature. As shown in Fig.11a, the divergent ratio is not great enough that the boundary layer separation does occur in the isolator due to the great heat addition rate. With the increasing of the divergent ratio until ξ = 1.85 in

Fig.10b, the effect of increasing the divergent ratio overcomes the effect of increasing total temperature resulted by the heat addition. So there is no high temperature area to be found in the isolator. Moreover, as can also be seen from Fig.11c, the high temperature area in the section I once again begins to improve due to the reduction of the divergent ratio. However, the combustion heat release with

ξ = 1.6 _ 2 is weaker than the initial combustion heat release with ξ = 1.6 _1 .Thus, Based on the above discussion, the effect of the hysteresis on the flow field parameters can be analyzed by the interaction between the oblique shock train motion and the combustion heat release distribution.

a. ξ = 1.6 _1, φ = 0.6

b. ξ = 1.85, φ = 0.6

c. ξ = 1.6 _ 2, φ = 0.6

Fig.11 Total temperature contour with ξ = 1.6 ↔ 1.85 3.4 Effect of hysteresis on combustion performance In general, the total pressure recovery coefficient associated with combustion performance of a combustor is always considered and analyzed for the variable geometry dual mode combustor. As shown in Fig.12a, an evident great total pressure loss can be observed in the section I and section II. The drop-off in the total pressure is accompanied by an irreversible entropy loss. The boundary layer separation does occur due to the intense adverse pressure gradient resulted by heat addition. Then, as can be seen from Fig.12b, the total pressure loss in the isolator starts to reduce with the growth of the divergent ratio. The result can be analyzed that the heat release distribution progressively moves backwards along the combustor due to the growth of the air flow velocity in the section II, which leads

the adequately mixing between the air and fuel to occur in the section III. In addition, as also shown in Fig.12c, as the wedge moves repeatedly backwards the initial location with ξ = 1.6 _1 , the total pressure loss is added. However, the total pressure loss is less than the total pressure loss with

ξ = 1.6 _1 due to the effect of hysteresis. An entropy increase associated with irreversibility is an important performance parameter utilizing to evaluate the combustion performance. Any entropy increase associated with total pressure decrease is considered as irreversibility. The entropy loss is generated by the boundary layer friction loss in the section I and combustion heat loss resulted by heat addition in the section II and section III. As presented in Fig.13a, the airflow of the combustor has a cumulative entropy increase due to irreversibility. The entropy loss is generated by the friction and oblique shock train due to the boundary layer separation in the section I. Fig.13b indicates that the entropy loss in the section I and section II gradually drops with the increasing of the divergent ratio. The divergent ratio overcomes the effect of the heat release resulted by combustion. Moreover, as presented in Fig.13c, the irreversible entropy loss is added due to the interaction between great heat release and the back and forth movement of the wedge. It indicates that the heat addition dominates the combustion system from ξ = 1.85 to ξ = 1.6 _ 2 . However, the entropy increase is smaller than the entropy increase with ξ = 1.6 _1 in the section I due to the effect of the hysteresis. Very significantly, the effect of the hysteresis actually has a great influence on the overall irreversible entropy loss.

a. ξ = 1.6 _1, φ = 0.6

b. ξ = 1.85, φ = 0.6

c. ξ = 1.6 _ 2, φ = 0.6

Fig.12 Total pressure recovery coefficient contour with ξ = 1.6 ↔ 1.85

a. ξ = 1.6 _1, φ = 0.6

b. ξ = 1.85, φ = 0.6

c. ξ = 1.6 _ 2, φ = 0.6

Fig.13 Entropy contour with ξ = 1.6 ↔ 1.85 According to the above analysis, there will be a hysteresis phenomenon in the wall static pressure by the continuous movement of the wedge. As shown in Table 4, a summary of the thrust coefficients for the variable geometry dual mode combustor were obtained through a series of hysteretic experiment data with different geometry path variations and fuel equivalence ratios. The thrust can be obtained by the one-dimensional wall static pressure integral. The thrust coefficient was defined as 2F/ȡAV2. For a given fuel equivalence ratio, as can be seen in Table 4, there is a hysteresis phenomenon about the thrust coefficient to be found. And the thrust coefficient progressively drops off with the increasing of the divergent ratio. Similarly, the effect of the hysteresis can also be observed with the increasing of the fuel equivalence ratio. Thus, the hysteresis can play an important role in improving the thrust performance of the variable geometry dual mode combustor.

Table 4 Thrust coefficient with hysteretic experiment results

φ = 0.6

φ = 0.72

φ = 1.0

ξ

Fd

ξ

Fd

ξ

Fd

1.6_1

0.255

2.0_1

0.196

2.2_1

0.226

1.6_2

0.251

2.0_2

0.190

2.2_2

0.215

1.7_1

0.242

2.1_1

0.186

2.4_1

0.199

1.7_2

0.233

2.1_2

0.174

2.4_2

0.195

1.85_1

0.184

2.3_1

0.162

2.5_1

0.181

1.85_2

0.181

2.3_2

0.153

2.5_2

0.178

4.

Conclusion

Numerical and experimental investigation about hysteretic characteristic of combustion performance were conducted by a series of geometry path continuous adjustment experiment results and numerical simulation results in the variable geometry dual mode combustor with a Mach number of 3, a divergent ratio ranging from 1.6 to 2.54 and a fuel equivalence ratio ranging from 0.6 to 1.0. It could be seen from the presentation above that for a given fuel equivalence ratio, the path-domain hysteretic characteristic of the wall static pressure was found and analyzed through the static pressure variation of the feature points. There was a very conspicuous hysteretic phenomenon to be obtained in the feature points. For a given feature point, the hysteresis is closed to the closed state with the increasing of the divergent ratio. Moreover, the hysteresis formation mechanism was analyzed numerically by the interaction between the oblique shock train motion and the combustion heat release distribution. It was found that the hysteresis phenomenon was originated from the unstable positive feedback influence of the oblique shock train in the isolator. The effect of hysteretic on the total pressure loss, irreversible entropy loss and thrust performance of the variable geometry dual mode combustor were also obtained numerically and experimentally. Very significantly, it was therefore strongly believed that the study of the hysteresis characteristic and formation mechanism on the thrust performance could be very

significant to improve combustion performance for the variable geometry dual mode combustor, especially for a wide range of flight Mach numbers. Acknowledgments This research work is supported by National Natural Science Foundation of China (Grants No. 91441204, and No. 51421063, and No. 51676204).

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