3. Case two—effect of plasma on cavity

#### 3.1. Plasma model

On account of the direction of the quasi-DC discharge path for the inflow, the discharge modes include two types: longitudinal mode and transversal mode [12]. In case one, the quasi-dc discharge operates under transversal mode, while in case two, a longitudinal mode is adopted for the configuration. As shown in Figure 9, there are five anodes set upstream of the backward wall of cavity, while in the bottom wall of cavity the five cathodes are set. The five pairs of identical electrodes are arranged parallel and symmetrically. Besides, the flow will not be disturbed directly resulted from all the electrodes flush mounted in the wall. A filament plasma forms when applied a high voltage (generally 150–1200 V) between anode and cathode, which

Figure 9. The schematic of plasma filaments in the cavity.

acts as strongly oscillating and bright and looks like an inverted "L" crossing the backward wall of cavity. Since the major heat energy of the quasi-DC discharge plasma focuses on the bright filament domain [10, 25], the filament plasma domain can be established in such a simplified shape as given in Figure 9.

Just like the way given in case one, every quasi-DC plasma filament is dealt as a volumetric heat source. To represent the plasma heat strength reasonably, the mean temperature of the plasma domain Tpl is specified in this simulation too. Considering the real size of the plasma filament, all the heat source is modeled with a section of dimensions 3 3 mm. the distance between the backward wall and left side of plasma filament upstream of cavity is 6 mm as depicted in Figure 9. While the distance between the backward wall and the right side of plasma filament downstream of cavity is 25.5 mm. Based on the previous research, Tpl = 3000 K is specified as the actuating strength, and five actuators work together in a pulsed mode with plasma actuation frequency F<sup>c</sup> = 5 kHz and duty cycle D = 1/5. In this case, all configurations (e.g., combustor, cavity) and simulation conditions are identical to those used in case one, except for the plasma. Therefore, the numerical methods, including physical models, numerical schemes, computational zone, etc., could be found in Section 2.1.

### 3.2. Results and discussion

resulting in the cross-section shape of the fuel jet varying from a narrow and long profile to a circular profile. Because of the variation of fuel mixing, in the upper space, more water forms while less appears near the wall compared with the case without plasma. (3) The stagnation pressure loss of combustor increases a little as actuator works, but the combustion efficiency in the combustor rises obviously. These above can be summarized as the comprehensive effects of flow structure changes caused by the plasma, including waves induced and heat transfer. Since it is negligible for the relative change of stagnation pressure recovery coefficient in the actuator working cases, and the ratio of deposited plasma energy to the increased combustion heat release is very little, it can be obtained that the quasi-DC discharge plasma can make more benefits than penalties for the scramjet combustor, when proper adopting control parameters

On account of the direction of the quasi-DC discharge path for the inflow, the discharge modes include two types: longitudinal mode and transversal mode [12]. In case one, the quasi-dc discharge operates under transversal mode, while in case two, a longitudinal mode is adopted for the configuration. As shown in Figure 9, there are five anodes set upstream of the backward wall of cavity, while in the bottom wall of cavity the five cathodes are set. The five pairs of identical electrodes are arranged parallel and symmetrically. Besides, the flow will not be disturbed directly resulted from all the electrodes flush mounted in the wall. A filament plasma forms when applied a high voltage (generally 150–1200 V) between anode and cathode, which

of the plasma actuator.

3.1. Plasma model

3. Case two—effect of plasma on cavity

Figure 8. Distribution of combustion efficiency downstream of fuel orifice.

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Three representative times are chosen from one plasma cycle, which consists of the later actuator free duration and the actuator working duration, for comparison after computations converged. In the simulation, the duration of one plasma cycle is T<sup>c</sup> = 200 μs. The end of an actuator working duration t = 1/5T<sup>c</sup> , t = 3/5T<sup>c</sup> , and the end of a cycle t = T<sup>c</sup> are named A, B, and C, respectively.

### 3.2.1. Typical parameters distribution of cavity flowfield

The Mach number distribution of local cavity flowfield on the symmetrical xy plane is shown in Figure 10. A distinct shear layer forms upon the cavity mouth and develops toward the

Figure 10. Mach number distribution: (a) no plasma; (b) with plasma, time A; (c) with plasma, time B; (d) with plasma, time C.

to minimum value. But at time C, its pressure peak value is nearly 2.8 times as the no plasma case. Results about show that the original waves around the rear edge are altered which results from local combustion zone induced by the periodic disturbance of plasma and the variation of

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Figure 11. Wall pressure distribution near the rear edge of cavity. (a) z = 0 mm. (b) z = 16 mm.

As presented in Figure 11b, the wall pressure of cavity rises distinctly on z = 16 mm, which illuminates that the temperature goes up in the cavity. Besides, the pressure peak value at time A equals the value of the no plasma case and is lower than the value at time C. This suggests clearly that the plasma filaments can weaken the waves around the cavity rear edge during plasma actuators free period (means the actuators shutdown.), which is the same to the study of nonreaction flow [19]. Moreover, because of the effect of expansion waves (as it is known that the abrupt pressure reduction near a cavity rear edge is induced by expansion waves), the normalized pressure reduces to around0.8 and then drops in a relative gentle way until x ≈ 140.0 mm at time C, which is due to the change of unsteady local waves and combustion. Because the distributions of combustion products may be impacted by the change of cavity shear layer and waves, the mass fraction iso-surface of water (YH2O = 0.05) is shown in Figure 12. Firstly, around the fuel orifice for the no plasma case some water forms, but there is no water forming during the plasma actuator working duration. Secondly, compared with the no plasma case having a smooth iso-surface, the iso-surface looks wrinkled distinctly due to the influence of plasma filaments, especially in the shear layer zone. Several symmetry structures shaped like "branch pipes" can be found obviously because of plasma. The "branch pipes" generate from the front or middle part of the cavity, and then move downstream. And "branch pipes" also move upward in the process mentioned above. Hence, a "branch pipes" produced cycle is established from time A to time C. The wide extent of product water

shortens in the z direction, while more water forms in the y direction at the same time.

The above phenomena can be attributed to the reasons as follows: (1) Plasma filaments existed upstream of the cavity front edge release a large amount of heat, so the local static temperature increases and then the movement of fuel jet is promoted toward upper space. Because of the

cavity shear layer.

main flow, which is well known as the typical combustion flowfield of cavity. Contrasted with the flowfield of the no plasma case, the cavity shear layer fluctuates abruptly, particularly in the y direction. And it develops unsteadily resulted in the division of cavity recirculation zone at time B, as depicted in Figure 10c. Resulted from the plasma observed the Mach number distribution from time A to C, the main flow velocity downstream of the cavity decreases to a certain degree. On the whole, the oscillation phenomenon of cavity shear layer strengthens firstly and then weakens in pace with the plasma cycle.

Owing to the periodic heat release from the plasma filaments and the thermal blocking functions, the plasma filaments behave as five knives cut the cavity shear layer. Hence, the mass transportation process will be disturbed by this "cutting" effect, and the moving direction of original shear layer has to be changed also. And then, the turbulence intensity and vorticity magnitude both are increased, so the fuel and air existing around the original shear layer can exchange through the cavity mouth more easily. Furthermore, the combustion enhancement downstream or over the cavity causes the decrease of local flow velocity, which results in the rise of local static pressure and blocks the incoming flow.

Figure 11 presents the distributions of wall pressure near the rear edge of cavity on z = 0 and z = 16 mm plane. The pressure value on y-axis is normalized by the inflow static pressure. Being the same as the Mach number in Figure 10, the plasma influence on the wall pressure is strong and unsteady. On z = 0 mm plane, the pressure peaks of both time A and C move upstream compared with the no plasma case, which are shown in Figure 11a). At time A, its peak value is nearly equal to the case without plasma, and decreases more gently from its peak

Figure 11. Wall pressure distribution near the rear edge of cavity. (a) z = 0 mm. (b) z = 16 mm.

main flow, which is well known as the typical combustion flowfield of cavity. Contrasted with the flowfield of the no plasma case, the cavity shear layer fluctuates abruptly, particularly in the y direction. And it develops unsteadily resulted in the division of cavity recirculation zone at time B, as depicted in Figure 10c. Resulted from the plasma observed the Mach number distribution from time A to C, the main flow velocity downstream of the cavity decreases to a certain degree. On the whole, the oscillation phenomenon of cavity shear layer strengthens

Figure 10. Mach number distribution: (a) no plasma; (b) with plasma, time A; (c) with plasma, time B; (d) with plasma,

Owing to the periodic heat release from the plasma filaments and the thermal blocking functions, the plasma filaments behave as five knives cut the cavity shear layer. Hence, the mass transportation process will be disturbed by this "cutting" effect, and the moving direction of original shear layer has to be changed also. And then, the turbulence intensity and vorticity magnitude both are increased, so the fuel and air existing around the original shear layer can exchange through the cavity mouth more easily. Furthermore, the combustion enhancement downstream or over the cavity causes the decrease of local flow velocity, which

Figure 11 presents the distributions of wall pressure near the rear edge of cavity on z = 0 and z = 16 mm plane. The pressure value on y-axis is normalized by the inflow static pressure. Being the same as the Mach number in Figure 10, the plasma influence on the wall pressure is strong and unsteady. On z = 0 mm plane, the pressure peaks of both time A and C move upstream compared with the no plasma case, which are shown in Figure 11a). At time A, its peak value is nearly equal to the case without plasma, and decreases more gently from its peak

firstly and then weakens in pace with the plasma cycle.

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time C.

results in the rise of local static pressure and blocks the incoming flow.

to minimum value. But at time C, its pressure peak value is nearly 2.8 times as the no plasma case. Results about show that the original waves around the rear edge are altered which results from local combustion zone induced by the periodic disturbance of plasma and the variation of cavity shear layer.

As presented in Figure 11b, the wall pressure of cavity rises distinctly on z = 16 mm, which illuminates that the temperature goes up in the cavity. Besides, the pressure peak value at time A equals the value of the no plasma case and is lower than the value at time C. This suggests clearly that the plasma filaments can weaken the waves around the cavity rear edge during plasma actuators free period (means the actuators shutdown.), which is the same to the study of nonreaction flow [19]. Moreover, because of the effect of expansion waves (as it is known that the abrupt pressure reduction near a cavity rear edge is induced by expansion waves), the normalized pressure reduces to around0.8 and then drops in a relative gentle way until x ≈ 140.0 mm at time C, which is due to the change of unsteady local waves and combustion.

Because the distributions of combustion products may be impacted by the change of cavity shear layer and waves, the mass fraction iso-surface of water (YH2O = 0.05) is shown in Figure 12. Firstly, around the fuel orifice for the no plasma case some water forms, but there is no water forming during the plasma actuator working duration. Secondly, compared with the no plasma case having a smooth iso-surface, the iso-surface looks wrinkled distinctly due to the influence of plasma filaments, especially in the shear layer zone. Several symmetry structures shaped like "branch pipes" can be found obviously because of plasma. The "branch pipes" generate from the front or middle part of the cavity, and then move downstream. And "branch pipes" also move upward in the process mentioned above. Hence, a "branch pipes" produced cycle is established from time A to time C. The wide extent of product water shortens in the z direction, while more water forms in the y direction at the same time.

The above phenomena can be attributed to the reasons as follows: (1) Plasma filaments existed upstream of the cavity front edge release a large amount of heat, so the local static temperature increases and then the movement of fuel jet is promoted toward upper space. Because of the

Figure 12. Iso-surface of product water, YH2O = 0.05: (a) no plasma; (b) with plasma, time A; (c) with plasma, time B; (d) with plasma, time C.

larger velocity far from the bottom wall (i.e., flow moves faster in the middle height of the combustor), the original product water will surely move downstream quickly. Therefore, the little water is found around the orifice when plasma actuators work, and the symmetry structures are formed in higher space downstream. It can be verified from the "branch pipes" presented in Figure 12b–d. (2) As stated above, the periodic fluctuation of cavity shear layer is related to the plasma filaments "cutting" effect on it, and then the mass transportation process is blocked which leads to the obvious change of combustion over the cavity. Hence, it affects the distribution of product water. Meanwhile, notice that the "branch pipes" curve structures match with the typical spanwise reverse vortex pairs in shape. Generally speaking, the spanwise reverse vortex pairs can extend the contact area between air and fuel by the methods of entraining air into its fuel core, which makes the mixing between fuel and air better and also the local combustion efficiency arisen. Whereas, the periodic disturbance from the plasma filaments impels the air and fuel transporting in the y direction. Hence, at certain periods, the efficiency of vortex pairs rises in several zones, which leads to the unsteady variation of product water, as shown in Figure 12. (3) The phenomena that more water forms in higher space and the less exists near the cavity mouth and combustor wall, which can be attributed to the heating effect on the whole cavity by the plasma filaments.

#### 3.2.2. Analysis of cavity drag and mass exchange rate

The drag and mass exchange rate of cavity are two important cavity performance parameters [26]. The combustor's drag is mainly generated by the cavity as it has a nearly constant cross section. The cavity drag includes two types: pressure drag and friction drag. Pressure drag is defined as the difference value between the force on cavity front wall and rear wall. Usually, the friction drag is too small compared with pressure drag so it always can be ignored. So the pressure drag is regarded as the cavity drag here. The drag coefficient of flame holding cavity is defined as:

$$\mathcal{C}\_D = \frac{2F\_D}{A\rho V^2} \tag{9}$$

and C but larger than time A. In view of the change of cavity shear layer in Figure 10, it should be due to the moving upward of the shear layer as actuator works. As the shear layer moves upward, it deflects to the main flow and no more impacts on the cavity rear wall so the drag decreases. However, at time B and C, because of the variation of the shear layer fluctuation, stronger impact may appear the rear wall of cavity, and the combustion may be boosted in the cavity rear part, which can both lead to the increase of back pressure in the cavity. Considering the drag coefficient values at the three typical times, the time weighted average cavity drag coefficient

No plasma �19.8 34.3 14.5 0.060 Plasma, A �19.6 33.1 13.5 0.056 Plasma, B �19.6 37.6 18.0 0.075 Plasma, C �18.1 33.5 15.4 0.064

Front wall Rear wall Drag Drag coefficient

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As we know, the cavity drag in reaction flow is closely related to the combustion heat release zone of a cavity. So only reducing the cavity drag may not bring benefit for the whole combustor. It can be seen that the further detail analysis about the combustion distribution in flowfield should be done to understand the effect of plasma on the cavity performance.

The mass exchange rate of cavity m<sup>0</sup> is another important parameter, which stands for the fluid mass that is inhaled into cavity from the main flow every second. The shedding vortex from cavity shear layer primary conducts the mass exchange process which can effectively inhale air and fuel into cavity and then takes them away. More shedding vortices and quicker movement can advance the flame holding ability of cavity with efficient mixing. The mass exchange rate can be analyzed by monitoring the mass flux that passes through the cavity mouth when the shear layer just covers the cavity. The mixing gas in the cavity is marked as "Y" in the unsteady flowfield at a time with stable result, and the dynamic process until all the gas "Y" leave the cavity can be observed and then the whole time involved "τ" in this process (i.e., residence time in cavity) has been recorded easily. Afterward, the mass exchange rate is m<sup>0</sup> ¼ m=τ, where m is the total mass of fluid in a cavity. Basically, the fuel exchange between internal and external of cavity is mostly affected by the interaction between the back wall and shear layer

The mass exchange rate of cavity is given in Table 5. Because the species transportation is achieved through the vortices in the cavity shear layer, it is very significant for enhancing the diffusion ability of cavity shear layer. Owing to the break on the original stable structure of the shear layer resulted from the plasma filaments "cutting" effect, the mass exchange rates at plasma existing cases at times A, B, and C are 9.2, 197.2, and 107.8 times, respectively, than the "no plasma" case. In addition, the time-weighted average mass exchange rate is 8.217 g/s, which is 122.6 times than the "no plasma" case. And the variation of the mass exchange rate magnitude from time A to C is in related to fluctuation degree of cavity shear layer. As a result, the quasi-DC plasma does obviously promote the species transportation between external and

is calculated to be 0.0668 which means the cavity drag usually increases by the plasma.

of cavity and the fuel distribution in cavity shear layer.

Table 4. Calculation results about the cavity drag, unit: N.

internal cavity.

where F<sup>D</sup> is the cavity drag; A is the section area of combustor inlet; r is mass density; and V is velocity of inflow.

In Table 4, the calculation results of the cavity drag and its drag coefficient are listed. It can be observed that the drag coefficient is 0.060 at the no plasma case, which is smaller than time B


Table 4. Calculation results about the cavity drag, unit: N.

larger velocity far from the bottom wall (i.e., flow moves faster in the middle height of the combustor), the original product water will surely move downstream quickly. Therefore, the little water is found around the orifice when plasma actuators work, and the symmetry structures are formed in higher space downstream. It can be verified from the "branch pipes" presented in Figure 12b–d. (2) As stated above, the periodic fluctuation of cavity shear layer is related to the plasma filaments "cutting" effect on it, and then the mass transportation process is blocked which leads to the obvious change of combustion over the cavity. Hence, it affects the distribution of product water. Meanwhile, notice that the "branch pipes" curve structures match with the typical spanwise reverse vortex pairs in shape. Generally speaking, the spanwise reverse vortex pairs can extend the contact area between air and fuel by the methods of entraining air into its fuel core, which makes the mixing between fuel and air better and also the local combustion efficiency arisen. Whereas, the periodic disturbance from the plasma filaments impels the air and fuel transporting in the y direction. Hence, at certain periods, the efficiency of vortex pairs rises in several zones, which leads to the unsteady variation of product water, as shown in Figure 12. (3) The phenomena that more water forms in higher space and the less exists near the cavity mouth and combustor wall, which can be attributed to the heating

Figure 12. Iso-surface of product water, YH2O = 0.05: (a) no plasma; (b) with plasma, time A; (c) with plasma, time B;

The drag and mass exchange rate of cavity are two important cavity performance parameters [26]. The combustor's drag is mainly generated by the cavity as it has a nearly constant cross section. The cavity drag includes two types: pressure drag and friction drag. Pressure drag is defined as the difference value between the force on cavity front wall and rear wall. Usually, the friction drag is too small compared with pressure drag so it always can be ignored. So the pressure drag is regarded as the cavity drag here. The drag coefficient of flame holding cavity

CD <sup>¼</sup> <sup>2</sup>FD

where F<sup>D</sup> is the cavity drag; A is the section area of combustor inlet; r is mass density; and V is

In Table 4, the calculation results of the cavity drag and its drag coefficient are listed. It can be observed that the drag coefficient is 0.060 at the no plasma case, which is smaller than time B

<sup>A</sup>rV<sup>2</sup> (9)

effect on the whole cavity by the plasma filaments.

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3.2.2. Analysis of cavity drag and mass exchange rate

is defined as:

(d) with plasma, time C.

velocity of inflow.

and C but larger than time A. In view of the change of cavity shear layer in Figure 10, it should be due to the moving upward of the shear layer as actuator works. As the shear layer moves upward, it deflects to the main flow and no more impacts on the cavity rear wall so the drag decreases. However, at time B and C, because of the variation of the shear layer fluctuation, stronger impact may appear the rear wall of cavity, and the combustion may be boosted in the cavity rear part, which can both lead to the increase of back pressure in the cavity. Considering the drag coefficient values at the three typical times, the time weighted average cavity drag coefficient is calculated to be 0.0668 which means the cavity drag usually increases by the plasma.

As we know, the cavity drag in reaction flow is closely related to the combustion heat release zone of a cavity. So only reducing the cavity drag may not bring benefit for the whole combustor. It can be seen that the further detail analysis about the combustion distribution in flowfield should be done to understand the effect of plasma on the cavity performance.

The mass exchange rate of cavity m<sup>0</sup> is another important parameter, which stands for the fluid mass that is inhaled into cavity from the main flow every second. The shedding vortex from cavity shear layer primary conducts the mass exchange process which can effectively inhale air and fuel into cavity and then takes them away. More shedding vortices and quicker movement can advance the flame holding ability of cavity with efficient mixing. The mass exchange rate can be analyzed by monitoring the mass flux that passes through the cavity mouth when the shear layer just covers the cavity. The mixing gas in the cavity is marked as "Y" in the unsteady flowfield at a time with stable result, and the dynamic process until all the gas "Y" leave the cavity can be observed and then the whole time involved "τ" in this process (i.e., residence time in cavity) has been recorded easily. Afterward, the mass exchange rate is m<sup>0</sup> ¼ m=τ, where m is the total mass of fluid in a cavity. Basically, the fuel exchange between internal and external of cavity is mostly affected by the interaction between the back wall and shear layer of cavity and the fuel distribution in cavity shear layer.

The mass exchange rate of cavity is given in Table 5. Because the species transportation is achieved through the vortices in the cavity shear layer, it is very significant for enhancing the diffusion ability of cavity shear layer. Owing to the break on the original stable structure of the shear layer resulted from the plasma filaments "cutting" effect, the mass exchange rates at plasma existing cases at times A, B, and C are 9.2, 197.2, and 107.8 times, respectively, than the "no plasma" case. In addition, the time-weighted average mass exchange rate is 8.217 g/s, which is 122.6 times than the "no plasma" case. And the variation of the mass exchange rate magnitude from time A to C is in related to fluctuation degree of cavity shear layer. As a result, the quasi-DC plasma does obviously promote the species transportation between external and internal cavity.


Table 5. Calculation results of cavity mass exchange rate, unit: g/s.
