Proof of concept study on a self-driven pulsed jet on a compressor stator blade by numerical simulation

Abstract

This study presents a new concept of self-driven pulsed jet flow control on a compressor stator blade. This passive unsteady flow control method has the advantage that neither external flow nor electrical source is needed. This study’s preliminary proof-of-concept study on a low-speed compressor stator blade is performed using numerical simulation. When the pulsed jet frequency is 100 Hz, the optimum control performance and control efficiency are reached, and the total pressure loss coefficient is reduced by 8.9%. As the valve’s rotational speed increases, the pulsed jet’s momentum coefficient decreases gradually. The analysis of the unsteady characteristics of the self-driven pulsed jet shows that the jet velocity is close to a periodic square wave signal, and the typical reduced jet velocity ranges from approximately 0.15 to 0.8. Moreover, the time-averaged driving torque on the valve depends on the rotational speed and is relevant to the self-starting and self-driven characteristics of this passive flow control method. Under different bearing resistance torque, the self-driven valve behaves differently in three cases that can self-start and be self-driven, cannot self-start but can be self-driven, and cannot self-start nor be self-driven.

Keywords

Passive flow control unsteady flow control flow separation self-driven pulsed jet

Introduction

Nowadays, the development trend of aero-engines is to pursue a higher thrust-weight ratio and lower fuel consumption. To realize this goal, state-of-the-art compressors are designed to be highly loaded while maintaining high efficiency and stall margin. However, highly loaded compressors will likely encounter flow separation because the flow is under a high adverse pressure gradient. Compared with the flow attached over the compressor stator blade, flow separation tends to reduce the pressure ratio, isentropic efficiency, and stall margin of the compressor. Thus, mitigating the unwanted effects of flow separation is necessary and introducing flow control is one effective way.

Steady or unsteady flow control techniques unusually realize flow control on flow separation. Unlike steady flow control methods, such as steady blowing or suction, unsteady flow control methods use periodic excitation that exploits natural flow instability.¹ Moreover, unsteady flow control methods were shown to be much more effective than traditional steady flow control methods and could attain a saving of up to two orders of magnitude in the momentum coefficient required to achieve a similar gain in performance.² Effectiveness and efficiency drive unsteady flow control to develop rapidly in the last decade.

Unsteady flow control is usually realized by active flow control (AFC). In AFC, actuators are used to convert a controllable electrical signal to a desired physical quantity to modify a flow.³ The field of AFC has witnessed explosive growth in the variety of actuators, and popular actuator types can be typically classified (based on function) as fluidic, moving surface and plasma.³ Fluidic actuators use fluid injection or suction, including synthetic jet,^4,5 pulsed jet,⁶ pulsed suction,⁷ and fluidic oscillators.⁸ Moving surface actuators mainly include wall oscillation,⁹ oscillating wire,¹⁰ rotating surfaces,¹¹ and traveling wave walls.¹² The last class, plasma actuators, includes single dielectric barrier discharge (SDBD) plasma,¹³ localized-arc-filament plasma actuators (LAFPA),¹⁴ and plasma synthetic jet actuators (PSJA).^15,16 Active unsteady flow control has the advantage in that it can adjust control parameters such as frequency and momentum injection, making active unsteady flow control function well at off-design conditions and suitable for open-loop and closed-loop (feedback) control.

However, there are two main disadvantages to active unsteady flow control. One is related to the periodic excitation’s flow or power source, and the other is related to the modulation of excitation unsteadiness. Taking the fluidic actuator as an example, to provide the energy for excitation, the pulsed jet or suction requires an external flow source (either a fluid source or sink),³ whereas the synthetic jet requires an electrical source instead of an external fluid source. Furthermore, to modulate a small amount of fluid into an unsteady jet, pulsed jet or suction actuation typically uses an electricity-driven valve that opens and closes at a certain frequency. In contrast, synthetic jet actuation uses a piezoelectric or electrodynamic configuration to convert electrical power into an oscillatory motion of the fluid. For unsteady active flow control, as shown in this example, either an external flow source or an electrical source is required to provide energy for periodic excitation, and an electrical source is necessary to power the actuator to generate excitation unsteadiness. The existence of external flow or electrical sources and resultant electric and air circuits dramatically increases the complexity of the active unsteady flow control system. These disadvantages greatly limit active unsteady flow control applications from laboratory to engineering, especially in aero-compressors with thin blades and installation space. If the external flow or electrical source can be spared, active unsteady flow control then turns into a passive one, and the shortcomings can be avoided.

Existing passive unsteady flow control methods are few in the literature. As discussed before, a passive unsteady flow control method must solve two major problems: the provision of excitation energy and the generation of excitation unsteadiness without using an external flow or electrical source. Huang et al.¹⁷ proposed a passive unsteady flow control method with equal-circumferential-spacing through-holes on the casing of a centrifugal compressor. They found that it can increase the stall margin by 19.3% at the cost of a 0.8% decrease in efficiency. In this method, the energy of periodic excitation comes from the pressure difference between the inner and outer sides of the casing where the holes are located, whereas the periodicity of excitation is caused by the rotor-stator interaction of the holes and rotational blades. Liu et al. successfully applied a similar concept to an axial compressor using axial slots instead of holes.¹⁸ Ayed et al.¹⁹ studied a passive unsteady flow control method, which is a flexible vibrating membrane forced by the flow field. They found that it can reduce the extreme pressure coefficients on the roof of a surface-mounted prism by approximately 25%. In this study, the energy and periodicity of excitation come from the unsteady flow field due to fluid-structure interaction or flow–membrane coupling. A similar concept is membrane wings, in which separated flows are the main sources of unsteadiness to drive the membrane to vibrate via fluid-structure interaction. Rojratsirikul et al.¹⁹ made an experimental comparison between rigid and flexible membrane airfoils. They found a strong coupling of unsteady flow with the membrane oscillations and a smaller separation region for the flexible membrane. Furthermore, Rosti et al.²⁰ numerically studied a passive unsteady flow control, which uses a self-activated flap to control the flow around airfoils. They found that a flap measuring 10% of the chord, with a natural frequency matching the shedding frequency of the baseline foil, delivers the best performance in delaying the dynamic stall process. To conclude, for the aforementioned passive unsteady flow control methods, excitation energy can be realized by an interior flow source or fluid-structure interaction, while the generation of excitation unsteadiness can be realized by fluid-structure or rotor-stator interaction.

The slotted blade is a kind of steady flow separation control method and has attracted attention from many researchers. Wang et al²¹ proposed a new double-slotted configuration and found this configuration can achieve an average reduction of 43.7% in the total pressure loss, compared with the single-slotted configuration. Also, Wang et al²² adopted the blade slotting technology to control the shockwave-induced corner separation and found that the low-energy fluid accumulated in the corner region can be re-energized by the slot jet with high momentum, thus effectively reducing its mutual interference with the shock wave and suppressing its separation. However, the slotted blade is a steady control method and cannot utilize the advantages of unsteady flow excitation. Therefore, on the basis of the slotted blade, this study proposes a new concept of passive unsteady flow control to suppress flow separation on compressor stator blades. This method can produce a self-excited pulsed jet without using any external flow or electrical source because the energy of the pulsed jet and the energy to produce unsteadiness are all from the flow field itself.

In the following sections, we introduces the concept of the self-driven pulsed jet on a compressor stator blade and discusses the basic theory of the self-driven pulsed jet. Furthermore, the concept of the self-driven pulsed jet is proven preliminarily by numerical simulation with its characteristics explored.

Concept of the self-driven pulsed jet on a compressor stator blade

To avoid the disadvantages of active unsteady flow control discussed in the Introduction, a new concept of the self-driven pulsed jet installed on a compressor stator blade, which is a passive unsteady flow control method, is proposed in this study. This concept, without using any external flow or electrical source, can reduce the structural complexity of the flow control system and improve engineering practicability. The diagram of this concept is illustrated in Figure 1.

Figure 1.

Diagram of the self-driven pulsed jet on a compressor stator blade.

Figure 1 shows a typical highly loaded compressor stator blade with flow separation on the suction surface. Like the slotted blade, a jet slot that bridges the pressure and suction surfaces to form a flow path is built into the blade. Utilizing the pressure difference between the pressure and suction surfaces of the blade, the jet slot can inhale air at the pressure surface and generate a jet at the suction surface. Furthermore, a rotational cylindrical self-driven valve with slots is installed on the path of the jet slot. The self-driven valve has two functions: a valve and an air turbine. As an air turbine, via the aerodynamic design of the jet slot, the self-driven valve can extract energy from the pressure difference between the pressure and suction surfaces and overcome the air and bearing resistance to keep rotating. The self-driven valve rotates; thus, the rotational slots on it open and close the flow path formed by the jet slot at a certain frequency. As a result, a pulsed jet is formed at the outlet of the jet slot and its frequency depends on the rotational speed of the self-driven valve. In this passive unsteady flow control method, the pressure difference between the pressure and suction surfaces of the blade is very important and plays two roles. One is as an interior flow source to provide energy for the jet, and the other is as an energy source to power the rotational valve to generate the unsteadiness of the jet. By doing so, the flow or electrical source, as that in active unsteady flow control, is avoided. The concept of the self-driven pulsed jet on a compressor stator blade is among the few passive unsteady flow control methods and has the following advantages.

a. Passivity and self-driven. The jet energy comes from the compressor flow field without an external flow source; the compressor flow field also drives the actuator and no electrical source is needed to generate the unsteadiness of the jet.

b. Unsteadiness. The pulsed jet is an unsteady flow control which is more efficient than steady jet flow control.

c. Simple structure. Without using any external flow or electrical source and resultant flow and electric circuits, the structure of the flow control system is relatively simpler, compared with active flow control.

Basic theory of the self-driven pulsed jet on a compressor stator blade

This section briefly discusses the basic theory of the proposed self-driven pulsed jet in Section 2. When the self-driven valve is on, as illustrated in Figure 2, it plays the same role as an air turbine, and the inlet and outlet of this turbine are denoted by cross Sections 1 and 2. For simplicity, incompressible and inviscid flow is considered. According to the incompressible Bernoulli equation in Sections 1 and 2,

\frac{P_{1}}{ρ} + \frac{C_{1}^{2}}{2} = \frac{P_{2}}{ρ} + \frac{C_{2}^{2}}{2} + H,

(1)

where

P

C

ρ

, and

H

are the static pressure, absolute velocity, density, and isentropic specific work of the air turbine, respectively. Subscripts 1 and 2 denote cross sections 1 and 2, respectively. Assuming that the static pressure at the bleeding inlet and jet outlet of the stationary slot are

P_{b}

and

P_{j}

(neglecting the flow losses),

{\begin{cases} P_{b} \approx P_{1} + ρ \frac{C_{1}^{2}}{2} \\ P_{j} + ρ \frac{C_{j}^{2}}{2} \approx P_{2} + ρ \frac{C_{2}^{2}}{2} \end{cases},

(2)

where

C_{j}

is the jet velocity when the self-driven valve is on. According to flow continuity,

ρ C_{2} h = ρ C_{1} h,

(3)

where

h

is the width of the stationary slot. Assuming the stationary slot's width is invariant, then it can be estimated that

C_{j} = C_{2} = C_{1} = C_{b}

. Given that the bleeding inlet and jet outlet of the stationary slot are located at the pressure and suction surfaces of the compressor stator blade,

P_{b}

and

P_{j}

are related to the static pressure coefficient of the blade, and the relation is as follows:

P_{b} - P_{j} = ρ \frac{C_{0}^{2}}{2} Δ C_{p},

(4)

where

C_{0}^{2}

is the freestream velocity of the compressor stator blade and

Δ C_{p}

is the difference between the static pressure coefficient at the inlet and outlet of the stationary slot. Combing equations (1), (2), and (4), the following can be obtained:

\frac{C_{j}^{2}}{2} = \frac{C_{0}^{2}}{2} Δ C_{p} - H .

(5)

Figure 2.

Schematic diagram of the self-driven pulsed jet on a compressor stator blade and corresponding velocity triangles.

According to the Euler turbine equation and the velocity triangle illustrated in Figure 2,

H = U (C_{1 u} - C_{2 u}) = U (C_{1} \sin α_{1} - U),

(6)

where

U

is the linear velocity at the surface of the self-driven valve,

α_{1}

is the absolute flow angle, and the subscript u denotes the circumferential component of velocity. Furthermore,

U

has the following relation:

U = ω r,

(7)

where

ω

and

r

are the angular velocity and radius of the self-driven valve, respectively. Air friction in the clearance and bearing resistance are factors that resist the rotation of the self-driven valve. Thus, the resistance torque per unit height

M_{r}

of the self-driven valve has the following expression:

M_{r} \approx M_{f} + M_{b},

(8)

where

M_{f}

and

M_{b}

are the resistance torque per unit height caused by air friction in the clearance and the bearing, respectively. To be explicit,

M_{f}

and

M_{b}

has the following approximations:

{\begin{cases} M_{f} = \frac{2 π μ U r^{2}}{w} \\ M_{b} = u F_{b} r_{b} \end{cases},

(9)

where

μ

is the coefficient of viscosity,

u

is the friction coefficient of the bearing (typically equals 0.001–0.0015 for a deep groove ball bearing),

F_{b}

is the bearing load per unit height (thus,

M_{b}

can be adjusted by changing

F_{b}

artificially), and

r_{b}

is the internal radius of a bearing. Thus, when a self-sustaining rotational speed is reached, a power equilibrium of the self-driven valve must be attained.

ω M_{r} = η {\dot{m}}_{j} H,

(10)

where

{\dot{m}}_{j}

is the jet mass flow rate and

η

is the isentropic turbine efficiency of the self-driven valve. The time-averaged mass flow rate of the self-driven valve can be expressed as follows:

{\dot{m}}_{j} \approx ρ C_{j} h \frac{2 N s}{2 π r} = \frac{ρ N h s C_{j}}{π r},

(11)

where

N

h

, and

s

are the number of the rotational slots, the width of the stationary, and rotational slot, respectively. The pulsed jet frequency can be calculated using the following relation:

f_{j} = \frac{ω N}{π} .

(12)

The momentum coefficient of the pulsed jet is defined as follows:

{C_{μ}}_{j} = \frac{{\dot{m}}_{j} C_{j}}{{\dot{m}}_{0} C_{0}},

(13)

where

{\dot{m}}_{0}

is the mass flow rate of the cascade for one single blade. Combing equations (3) and (6)–(13), the main parameters of the pulsed jet, such as jet frequency, jet velocity, jet mass flow rate, and self-sustained rotational speed, can be theoretically estimated.

Numerical simulation of a self-driven pulsed jet on a compressor cascade

Numerical methods

In an attempt to prove the concept of the self-driven pulsed jet on a compressor stator blade preliminarily, this study conducts numerical simulations based on a low-speed plane cascade and a prototype self-driven pulsed jet generator. Main parameters and the model of the cascade and the prototype self-driven pulsed jet generator are presented in Table 1 and Figure 3, respectively. As shown in Figure 3, the self-driven valve is a 2-mm-radius cylinder with two 1.5-mm-width rotational slots forming the shape of a cross. The jet slot is located at approximately 45% chord length from the leading edge, and the jet angle is 15 degrees. The clearance between the self-driven valve and the blade is designed as a low value of 0.05 mm to reduce the leakage when the valve is off.

Table 1.

Main parameters of the compressor cascade and the prototype self-driven pulsed jet generator.

Part	Parameter	Symbol	Value
Plane cascade	Chord length	b	60 mm
	Blade pitch	t	45 mm
	Maximum thickness	c _m	6 mm
	Leading edge blade angle	β ₁	46°
	Trailing edge blade angle	β ₂	−10°
	Angle of attack	i	9°
	Inlet Mach number	Ma	0.1
Self-driven pulsed jet generator	Radius of the valve	r	2 mm
	Number of the rotational slots	N	2
	Clearance	w	0.05 mm
	Absolute bleeding flow angle	α ₁	10°
	Stationary slot width	h	0.75 mm
	Rotational slot width	s	1.5 mm
	Jet location	x/b	0.45
	Jet angle	θ	15°

Figure 3.

Model of the self-driven pulsed jet on a compressor stator blade.

Three-dimensional (3D) large eddy simulation (LES) is performed using the flow solver Ansys Fluent to prove the concept preliminarily and obtain the characteristics of the self-driven pulsed jet on a compressor stator blade. In LES, a steady flow field computed by the shear stress transfer (SST) k-ω turbulence model is used as the initial field, whereas, for the unsteady simulation, the wall-adapting local eddy-viscosity (WALE) sub-grid model is used. The pressure boundary conditions at the inlet and outlet of the cascade are given to ensure the inlet Mach number approximately equals 0.1. In the pitchwise direction, the computational domain stretches one pitch (45 mm) of the cascade and transitional periodic boundary conditions are applied on the pitchwise boundaries. While in the staggering (spanwise) direction, only 10-mm height (1/6 the chord length) is covered by the mesh, and transitional periodic boundary conditions are also applied to reduce the computational cost. This study uses dual time stepping, and the physical time step is set as 1 × 10⁻⁵ s so that the convective Courant–Friedrichs–Lewy (CFL) number is less than 1. Concerning the grids, a structured grid is used, and near-wall layers are densified to ensure $y^{+} \approx 1$ . The grid dependence results, as illustrated in Figure 4(a), shows that the dominant frequency and the total pressure loss coefficient of the baseline cascade (without flow control) change slightly when the grid number is larger than 3.5 million. Also, the grid independence verification for the self-driven device is illustrated in Figure 4(b) (the total pressure loss coefficient is much larger than 1 due to the energy extraction of the self-driven valve). The total pressure loss coefficient is often used to evaluate the flow losses of compressor cascades, and it has the following definition:

ω_{L} = \frac{P_{i}^{*} - P_{o}^{*}}{P_{i}^{*} - P_{i}},

(14)

where

P_{i}^{*}

and

P_{i}

are the time-averaged total and static pressure at the inlet, respectively;

P_{o}^{*}

is the time-averaged total pressure at the outlet. Furthermore, as illustrated in Figure 5, the time-averaged loading distribution of the cascade blade acquired by simulation is compared with that from the experiment.²³ The time-averaged loading distribution is evaluated by the static pressure coefficient, which is defined as follows:

C_{p} = \frac{P}{P_{i}^{*} - P_{i}},

(15)

where

P

is the time-averaged static pressure on the blade surface. Moreover, the dominant frequency of shedding vortices in the experiment is 478 Hz, which is close to 505 Hz obtained by the simulation. Thus, the numerical results agree with the experimental results. The results indicate that this numerical method can be used to preliminarily predict the time-averaged and transient characteristics of the low-speed compressor cascade in this proof-of-concept study. Considering computational cost and accuracy, the total grid number used for the compressor cascade with the self-driven pulsed jet generator is approximately 4.13 million (0.58 million for self-driven valve and stationary slot domain and 3.55 million for the blade passage domain). As illustrated in Figure 6, a moving mesh is used for the self-driven valve with an adjustable rotational speed, and an interface is employed to connect the moving mesh with the normal stationary mesh.

Figure 4.

Grid dependence results. (a) The baseline cascade; (b) the self-driven device.

Figure 5.

Comparison between experimental (data from Ref.²³) and numerical results (Ma = 0.1). (a) Blade loading distribution; (b) pitchwise total pressure coefficient (at the trailing edge); (c) pitchwise total pressure coefficient (One chord length downstream the trailing edge).

Figure 6.

Mesh of the cascade and self-driven pulsed jet generator.

Numerical results and discussion

Comparison between the theoretical and simulated results

Based on the equations in the basic theory section and the parameters of the self-driven pulsed jet generator, some main parameters can be theoretically estimated. The input parameters and output estimations, which are also compared with the simulated results, are listed in Table 2. In this case, the rotational speed of the self-driven valve is 157 rad/s, and the difference in static pressure coefficients between the inlet and outlet of the stationary slot is approximately 0.7. Predicting the efficiency of the valve, which serves as an air turbine, is challenging. Thus, an ideal or isentropic situation where efficiency equals 1 was considered. As shown in this table, the jet frequency, maximum jet velocity, and jet mass flow rate predicted by the theory agree well with those by simulation, while jet momentum coefficient and driving torque do not. The error of the jet momentum coefficient was attributed to the hypothesis of pulsed jet velocity (pure square wave) made in the basic theory section. Furthermore, the simulated value of the valve's driving torque is lower than the estimated one because the actual flow is not isentropic and the efficiency of the valve is lower than 1. Thus, the theory introduced in the basic theory section can be used to predict jet frequency, jet maximum velocity, and jet mass flow rate, only helping the rough estimation of the jet momentum coefficient and driving torque of the valve.

Table 2.

Comparison between theoretical estimations and CFD results.

	Parameter	Symbol	Theory	Simulation	Unit
Input	Radius of valve	r	2	/	mm
	Stationary slot width	h	0.75	/	mm
	Rotational slot width	s	1.5	/	mm
	Number of the rotational slots	N	2	/	1
	Difference of static pressure coefficients	$Δ C_{p}$	0.7	/	1
	Efficiency of valve	η	1	/	1
	Rotational speed of valve	ω	157	/	rad/s
	Absolute bleeding flow angle	α ₁	10	/	degree
Output	Jet frequency	f _j	100	100	Hz
	Reduced maximum jet velocity	(C _j ) _max /C ₀	0.84	≈0.8	1
	Jet mass flow rate	${\dot{m}}_{j}$	0.012	0.011	kgs⁻¹m⁻¹
	Jet momentum coefficient	${C_{μ}}_{j}$	0.97%	0.64%	1
	Driving torque of valve	M _r	1.1 × 10⁻⁴	0.42 × 10⁻⁴	N

Analysis of time-averaged characteristics

The time-averaged characteristics of the self-driven pulsed jet when the valve is under different rotational speeds are analyzed in this section. Equation (12) indicates that the pulsed jet frequency is proportional to the rotational speed of the valve. Figure 7 shows the variations of the total pressure loss and momentum coefficients with the rotational speed of the valve. In the figure, the reduced frequency is defined as follows:

F^{+} = \frac{f_{j}}{f_{0}},

(16)

where

f_{0}

is the dominant frequency of the separated flow without flow control. Furthermore, to reflect the control performance, the relative total pressure loss coefficient is defined as follows:

ω_{R L} = \frac{ω_{L, o n} - ω_{L, o f f}}{ω_{L, o f f}},

(17)

where the second subscripts on and off denote the self-driven pulsed jet is turned on (with flow control) and off (without flow control), respectively. Thus, as indicated by this definition, the lower less-than-zero relative total pressure loss coefficient reflects better control performance. As shown in Figure 7, as the reduced jet frequency or the rotational speed of the valve increases, the total pressure loss coefficient reduces at first and then increases. When

F^{+} < 1.7

ω < 1300 r a d / s

, the control performance is positive; otherwise, the control performance is negative. In the positive control performance interval, the optimum control performance is at

F^{+} \approx 0.2

(

f_{j} = 100 H z

), with a total pressure loss coefficient of −8.9%. Figure 8 illustrates the time-averaged velocity contour of cases without and with the optimum (

f_{j} = 100 H z

) self-driven pulsed jet flow control. This figure shows that the pulsed jet slightly postpones the separation point and minimizes the low-velocity zone near the jet location. The frequency-dependent phenomenon can also be reflected by the index of control efficiency, which is defined as follows:

η_{e} = \frac{ω_{L, o f f} - ω_{L, o n}}{C_{μ}} .

(18)

Figure 7.

Total pressure loss and momentum coefficient variations with the angular velocity or reduced jet frequency of the self-driven valve.

Figure 8.

Time-averaged velocity contour: (a) self-driven pulsed jet in off status (baseline); (b) self-driven pulsed jet in on status ( $F^{+} \approx 0.2$ or $f_{j} = 100 H z$ ).

This index has the physical meaning of the saved momentum (benefit) divided by the momentum injection of flow control (cost). $η_{e} > 1$ indicates a good economy of flow control, especially for passive flow control, because the momentum injection comes from the flow rather than an external source. $η_{e} < 1$ means poor efficiency of flow control, but does not necessarily mean the use of flow control is meaningless. The reason is that, in practical application, flow separation may lead to serious consequences, such as aircraft stall or engine surge, and the economy of flow control is subordinate to its effectiveness. As illustrated in Figure 9, when $F^{+} < 0.9$ , $η_{e} > 1$ , and the maximum $η_{e}$ is at $F^{+} \approx 0.2$ , which is the same optimum value of $F^{+}$ for achieving the best control performance in Figure 7. Thus, in this study, the most effective and efficient self-driven pulsed jet flow control coincide. Furthermore, when $F^{+} > 0.9$ , $η_{e} < 1$ , and insufficient momentum coefficient is the main reason for this inefficiency, as discussed before.

Figure 9.

Control efficiency variation with reduced jet frequency.

This study's optimum is different from the previous finding that the optimum control performance is at $F^{+} \approx 1$ ²³under the same momentum coefficient. This phenomenon is attributed to the decrease of the momentum coefficient with $F^{+}$ , also illustrated in Figure 7. As $F^{+}$ increases from 0.04 to 3.0, the momentum coefficient of the pulsed jet decreases gradually from 0.64% to 0.15%. Given that the unsteady flow control performance is normally positively correlated with the momentum injection of the external excitation,² the optimum case ( $F^{+} \approx 0.2$ and $C_{μ} \approx 0.64 %$ ) in this study is a result of the integrated effects of frequency and momentum injection of external excitation. The momentum coefficient drops with the rotational speed of the valve mainly because of the relaxation time of the valve. When the rotational valve opens and closes at a certain frequency, the unsteady flow inside the rotational slot also switches between low-velocity (C_j≈0) and high-velocity (C_j≈C_jmax) modes, and the time needed for this transition is the relaxation time. However, if the valve rotates so fast that the opening time for the valve is less than the relaxation time, the unsteady flow cannot respond in time, resulting in the decrease in the mass flow rate or the momentum coefficient of the pulsed jet.

Analysis of unsteady characteristics

The previous section analyzes the characteristics of the self-driven pulsed jet in a time-averaged sense and focuses on its unsteady characteristics. Figure 10 shows the time domain and frequency spectrum of the pulsed jet velocity monitored at the outlet of the jet slot when the jet frequency equals 100 Hz. The time domain curve of the jet velocity is close to a periodic square wave signal, and the velocity ranges from approximately 5 m/s (C_j/C₀ ≈ 0.15) to 28 m/s (C_j/C₀ ≈ 0.8). The lowest value of the pulsed jet is not zero because of the clearance between the self-driven valve and the compressor stator blade; thus, the valve cannot be completely closed. The periodicity of the pulsed jet velocity can be reflected by the frequency spectrum obtained by fast Fourier transformation (FFT), as illustrated in Figure 10(b). The peak of the dominant frequency of 100 Hz, along with some of its harmonics, can be found in this spectrum. Furthermore, the odd harmonics outstand the nearby even ones in amplitude because an ideal square wave only consists of odd harmonics, which can be expressed mathematically as follows:

\begin{array}{l} v (t) = \frac{sgn (\sin ω t) + 1}{2} \\ = \frac{1}{2} + \frac{2}{π} \sum_{k = 1}^{\infty} \frac{1}{2 k - 1} \sin (2 k - 1) ω t \end{array} .

(19)

Figure 10.

Unsteady characteristics of the self-driven pulsed jet velocity ( $f_{j} = 100 H z$ or $F^{+} \approx 0.2$ ): (a) time domain; (b) frequency spectrum.

The driving torque on the self-driven valve is also unsteady. In Figure 11, the non-dimensional driving torque of the self-driven valve is defined as follows

{\tilde{M}}_{r} = \frac{M_{r}}{0.5 ρ C_{0}^{2} r^{2}} .

(20)

Figure 11.

Driving torque and jet velocity over two periods ( $f_{j} = 100 H z$ or $F^{+} \approx 0.2$ ).

It is illustrated over two periods (the period is denoted by T). The driving torque has the same frequency as the jet velocity. However, its phase is ahead of the jet velocity for about a quarter of the period. In the present design, the driving torque can be either more than or less than zero, indicating that the driving torque's direction can be the same and the opposite of the rotational direction. To explain this phenomenon further, typical snapshots of velocity contour at t = 1T, 1.3 T, and 1.6 T in Figure 11 are shown in Figure 12. At t = 1T, the rotational valve is open, and the jet pushes the right wall of the valve, serving as a driving force, so the driving torque is positive. At t = 1.6 T, the jet pushes the right wall of the valve, serving as a resistance force. Thus, the driving torque is negative. However, at t = 1.3 T, the rotational valve is closed; thus, the driving torque is close to zero. Given that the rotational valve has inertia. The time-averaged driving torque is more important than its unsteady characteristics when the torque on the valve is in equilibrium. Thus, the time-averaged driving torque, which is closely relevant to the self-driven and self-starting characteristics of valve, will be discussed in the next section.

Figure 12.

Typical snapshots of the transient velocity contour within one period ( $f_{j} = 100 H z$ or $F^{+} \approx 0.2$ ) (a) t = 1T; (b) t = 1.3 T; (c) t = 1.6 T. The period is denoted by T.

Analysis of self-driven and self-starting characteristics

An important advantage of the self-driven pulsed jet is being passive, which means the actuator should self-start and be self-driven. Indicated by unsteady characteristics in Figure 12, if the initial position of the valve (when its rotational speed is zero) is at an off status, the driving torque of the valve is close to zero, then the valve cannot self-start and rotate to the designed rotational speed. This problem can be solved by substituting the aligned rotational slots in this study with the staggered rotational slots, as illustrated in Figure 13, so that a part of the valve always stays at an open status and produces a non-zero driving torque to help the valve to start.

Figure 13.

Different types of self-driven valves: (a) with aligned rotational slots; (b) with staggered rotational slots.

Another issue is whether the valve could be self-driven or self-sustained at a certain rotational speed. This issue is related to the time-averaged driving torque on the valve. Figure 14 shows the time-averaged driving torque variation with the angular velocity of rotation in the simulation. The driving torque keeps positive. It increases when the rotational speed is between 30 and 700 rad/s and decreases when rotational speed is between 700 and 2500 rad/s. It is negative when the rotational speed is above 2200 rad/s. According to Figure 2 and equation (10), the increasing driving torque between 30 and 700 rad/s is caused by the reduction in the angle of attack of the relative flow velocity W₁ to the rotational slot and resultant higher efficiency η of the valve as an air turbine. The driving torque decreases when the rotational speed is approximately 700 rad/s because isentropic-specific work H drops with the increasing linear velocity at the surface of the self-driven valve U, as indicated by equation (6).

Figure 14.

Time-averaged driving torque variation with the angular velocity of rotation.

With the time-averaged driving torque characteristics illustrated in Figure 14, the self-driven and self-starting characteristics of the self-driven pulsed jet can be analyzed. Three circumstances were considered that are distinguished by whether the rotational valve can self-start or be self-driven, as illustrated in Figure 15. As shown in Figure 15(a), if the bearing resistance torque is lower than the driving torque M_r at $ω$ = 0, then the self-driven valve can self-start because the driving torque is larger than the bearing resistance torque to force the rotational speed to rise until $ω = ω_{1}$ , where the equilibrium of torque is reached and the rotational speed stabilizes and does not increase. As shown in Figure 15(b), if the bearing resistance torque is between M_r ( $ω$ = 0) and (M_r)_max, then the self-driven valve cannot self-start but can be self-driven. Before $ω = ω_{1}$ , the driving torque is less than the bearing resistance torque; thus, additional external torque is needed to start the valve and bring it to $ω = ω_{1}$ . However, $ω = ω_{1}$ , an unstable equilibrium in torque means a disturbance in rotational speed will either bring the rotational speed to $ω = 0$ or $ω = ω_{2}$ , which is a stable equilibrium in torque. Thus, in this circumstance, the valve can be self-driven at $ω = ω_{2}$ , but cannot start without an external torque to pull it through the unstable equilibrium point in torque. The last situation that the valve cannot self-start nor be self-driven is shown in Figure 15(c). In this situation, the bearing resistance torque is higher than (M_r)_max. Thus, no equilibrium point in torque can make the valve self-sustained in rotation.

Figure 15.

Self-starting characteristics of the self-driven valve: (a) can self-start and be self-driven; (b) cannot self-start but can be self-driven; (c) cannot self-start nor be self-driven.

In this section, the proof-of-concept study of the self-driven pulsed jet is preliminarily carried out by numerical simulation. The numerical results confirm that the passive unsteady flow control method can produce an unsteady jet and a positive control performance without employing any external flow or electrical source. Moreover, the self-driven and self-starting characteristics of the valve are discussed. This study is only a preliminary attempt to prove the concept; thus, many aspects can still be improved. Apart from the general unsteady flow control parameters, including jet location, momentum coefficient, and angle, which can be further optimized, the geometry of the self-driven pulsed jet generator, along with the stationary and rotational slots, needs to be optimized to reduce flow losses. The reason is that no aerodynamic shape is used for them in this proof-of-concept study. Furthermore, the control performance and the self-driven and self-starting characteristics are based on the present design. Thus, the results may be different if a future state-of-the-art design (e.g., the design presented in Figure 13(b)) is employed.

It is worth noting that only one major working condition, for example, the blade working under the design point, is considered in this paper. It is thought by us that the performance will definitely change under off-design points. The self-driven valve is essentially an air turbine, and when the angle of attack or Mach number of the compressor blade increases, the pressure difference between the two sides of the self-driven valve will become larger, thus generating more driving power and making the rotational speed faster. As a result, the pulsed jet frequency and jet velocity also increase. Also, when the angle of attack or Mach number of the compressor blade changes, the natural frequency of the separation vortex and the required momentum for the pulsed jet also change. Therefore, as the angle of attack or Mach number changes, flow control performance of the blade will change, but the specific flow control characteristic needs to be determined by further numerical simulations.

Conclusions

This study proposes a new passive unsteady flow control concept of the self-driven pulsed jet method to avoid any external flow or electrical source common in active flow control methods. The self-driven valve serves as both a valve and an air turbine. The pressure difference between the pressure and suction surfaces of the blade has the function of providing energy for the jet and the valve to rotate to generate a jet with unsteadiness. The control effectiveness of the self-driven pulsed jet is preliminarily proven via numerical simulation. Also, the unsteady and self-driven characteristics are discussed. The main conclusions are listed as follows:

1. The numerical simulations preliminarily verify the effectiveness of the self-driven pulsed jet on a typical low-speed compressor stator blade. When the pulsed jet frequency is 100 Hz (or the reduced frequency $F^{+} \approx 0.2$ ), the optimum control performance and control efficiency is reached, and the total pressure loss coefficient is reduced by 8.9%. It is also found that as the valve's rotational speed increases, the pulsed jet’s momentum coefficient decreases gradually.

2. Analysis of the unsteady characteristics of the self-driven pulsed jet shows that the jet velocity is close to a periodic square wave signal and the typical velocity ranges from approximately 5 m/s (C_j/C₀ ≈ 0.15) to 28 m/s (C_j/C₀ ≈ 0.8). The driving torque on the valve has the same frequency as the jet velocity. In the present design of the self-driven valve, the torque can be larger than, equal to, and less than zero when the valve rotates in different directions.

3. The time-averaged driving torque on the valve is dependent on the rotational speed. In this study, the driving torque keeps positive, increases when the rotational speed is between 30 and 700 rad/s, and decreases when the rotational speed is between 700 and 2500 rad/s. However, the driving torque is negative when the rotational speed is above 2200 rad/s. This feature makes the self-driven valve behave differently under different bearing resistance torque. If the bearing resistance torque is lower than the driving torque M_r ( $ω$ = 0), then the self-driven valve can self-start and be self-driven. If the bearing resistance torque is between M_r ( $ω$ = 0) and (M_r)_max, the self-driven valve cannot self-start but can be self-driven; and the bearing resistance torque is higher than (M_r)_max, then the valve cannot self-start nor be self-driven.

Footnotes

Acknowledgments

This research was funded by the National Natural Science Foundation of China (grant number 52106046 and 52106246) and the Natural Science Foundation of Jiangsu Province (grant number BK20200680). The authors wish to express their gratitude to Department of Engineering Mechanics (affiliated with School of Physical and Mathematical Sciences, Nanjing Tech University) for technical support.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number 52106046 and 52106246) and the Natural Science Foundation of Jiangsu Province (grant number BK20200680).

ORCID iD

Lu Weiyu

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