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Switching mechanism investigation for the supersonic jet element: Deflection, attachment and adjustment stages
Accepted Manuscript Switching mechanism investigation for the supersonic jet element: Deflection, attachment and adjustment stages Y. Xu, G.Q. Zhang PII:
S0094-5765(19)30741-6
DOI:
https://doi.org/10.1016/j.actaastro.2019.05.027
Reference:
AA 7513
To appear in:
Acta Astronautica
Received Date: 25 April 2019 Revised Date:
13 May 2019
Accepted Date: 16 May 2019
Please cite this article as: Y. Xu, G.Q. Zhang, Switching mechanism investigation for the supersonic jet element: Deflection, attachment and adjustment stages, Acta Astronautica (2019), doi: https:// doi.org/10.1016/j.actaastro.2019.05.027. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Switching Mechanism Investigation for the Supersonic Jet Element: Deflection, attachment and adjustment stages Y. Xu1 and G. Q. Zhang1,* 1
Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering,
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Beijing Institute of Technology, Beijing, 100081, China * Corresponding author: [email protected]
ABSTRACT: Based on the unsteady viscous flow simulation, the flow characteristics inside the supersonic jet
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Element have been investigated numerically. The results have revealed that once the specific structures of the supersonic jet element are finalized, even if the boundary conditions remain unchanged, the corresponding internal
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flow will also shows strong unsteady in a certain range of primary gas source pressure. The instability of the main vortex center is a main reason to make the output thrust of the supersonic jet components fluctuate all the time in the attached wall condition. At the deflection of the jet stage, once the static pressure for the right side of the wedge exceeds the left side, the transverse expansion for the stripping zone near the right output channel entrance can play a significant role in making the primary jet deflect successfully. When the jet starts to attach the wall layer, due to the “Coanda” effect, the jet also can attach the layer successfully even though the corresponding control flow is
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totally closed. When the jet enters the adjustment stage, the thrust changes process for the left and right outputs will experience two typical stages: adjusting and adjusted stages. The corresponding vortex structures at different
analyzed in details.
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switching time together with the force variations etc. inside the jet Element have been obtained computationally and
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Keywords: Supersonic jet element, layer, shock, vortex, switching time
F
=
Total thrust (N)
e
=
k
Nomenclature T
=
Temperature (K)
Dissipation rate (m2/s3)
t
=
Switching time (s)
=
Kinetic energy (m2/s2)
fl
=
Left outlet thrust (N)
m
=
Mass flow rate (K)
fr
=
Right outlet thrust (N)
Ma
=
Mach number
lout
=
Pressure outlet (Pa)
Ps
=
Main gas source pressure(bar)
P01
=
Point from main jet inlet (-0.0062,0.042)
∆t
=
=
Point from main jet inlet (0.0062,0.042)
Time step (s)
P02
ACCEPTED MANUSCRIPT I.
T
Introduction
he supersonic jet element (SJE) is the execution unit in the attitude control system. Its performance (such as the
control force, switching time, energy efficiency and reliability etc.) can play an extremely important role in controlling
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the accuracy of the whole system. Therefore, the research on the jet flow has drawn a lot of attentions [1-3]. The SJE is also called fluidic amplifier. On one hand, it could be used as the logic control system components to achieve a variety of logic functions, on the other hand, through a small energy control signals, it also could be served as an
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implementation part of the control system and offer much large energy or torque directly [4-7]. Nowadays the
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supersonic jet element is applied to the Multiple Launch Rocket System project as the implemental components of the control system, such as the “tornado” weapon system of Russia [8-11]. Raja [12] had numerically investigated the mixed convection flow and heat transfer characteristics in a two-dimensional plane, laminar offset jet issuing parallel to an isothermal flat plate. The results had shown that the reattachment length is strongly dependent on both the Reynolds
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number and the Grashof number for the range considered.
Draksler [13] had analyzed numerically the highly turbulent flow of the air impinging jets in hexagonal arrangement is by the means of Large Eddy Simulation. All important flow phenomena, i.e. the formation of the fountain flow as
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well as the negative production of normal stresses near target wall are successfully predicted by the simulation[14].
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Smirnov[15-18] had studied parametrically the variation of the key parameters of this problem (e.g., the geometrical curvature of oblique shock emanating from the nozzle edge) based on the jet flow parameters. The results had proved that differential parameters of the flow field crucially depend not only of the key parameters, but on the symmetry type as well.
Our previous research mainly focus on the early two stages [19]: i) The switching oblique shock wave formation and boundary layer separation (Stage I); ii) The jet starts to separate from the right side of the wall layer (Stage II). The results have shown that the presence of the switching oblique shock near the control port is not the necessary condition
ACCEPTED MANUSCRIPT to make the jet deflect, but its formation is good for the early transverse extension of the stripping vortex zone. A new concept named minimum control mass flow rate has also been proposed and emphasized. However, the switching mechanism of the supersonic jet components still lacks of a deep understanding. The corresponding switching reasons
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are always simply attributed to the oblique shock and the Coanda effect. Based on this, our present research will further analyze the flow characteristics of the switching process and then reveal the specific switching mechanism for the jet supersonic components. The specific contents are listed as: i) The
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deflection of the jet (Stage III); ii) The jet starts to attach the wall layer (Stage IV) and iii) Adjustment stage (Stage
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V).
The paper is organized as follows. Section II describes the governing equations including a brief description of the adopted numerical algorithm. The validation method and results will be also presented and discussed. Section III will specifically discuss the flow field when the jet starts to deflect, attach the wall layer and adjust to stable eventually. The
II.
Simulation setup and comparison with the experimental data
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A. Simulation setup
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brief concluding remarks.
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changes for different parameters in the whole switching process are also investigated in details. The paper will end with
The flow field of supersonic jet elements always includes the strong separation flow, compression corner flow and shock wave-boundary layer interference flow. Therefore, the present research will adopt the RANS equation as the control equation, combined with the Realizable k-epsilon turbulence model, near-wall area processing, closed equations, boundary conditions, and initial conditions, as a mathematical model for numerical calculation of supersonic jet elements.According to the flow characteristics of supersonic jet elements, the volume force and heat source are
ACCEPTED MANUSCRIPT ignored, and the wall surface is assumed to be the insulation wall surface. The Continuous equation and Momentum equation in the Cartesian coordinate system can be rewritten as:
∂ρ ∂ + ( ρ ui ) = 0 ∂t ∂xi
(1)
where
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∂ ∂ ∂P ∂τ ij ∂ ( ρ ui ) + ( ρ ui u j ) = − + + ( − ρ ui′u j′ ) ∂t ∂x j ∂xi ∂x j ∂x j
(2)
ui is the pulsing value, τ ij is a viscous stress Tensor, which can be expressed as: 2 3
(3)
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τ ij = 2 µ Sij + ( µ ′ − µ ) S kk δ ij
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Sij is the strain rate of fluid motion:
1 ∂u ∂u j Sij = i + 2 ∂x j ∂xi
(4)
The turbulent energy(k) and turbulence dissipation(Ɛ) can be defined as:
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1 k = ui′u j′ 2
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∂u ′∂u ′ ε =ν i j ∂x j ∂x j
(5)
(6)
Transport equation of turbulent energy can be written as:
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∂ ∂ ∂ (ρ k ) + ( ρ ku j ) = ∂t ∂xi ∂x j
µt µ + σk
∂k + Gk − ρε − YM ∂x j
(7)
and the Transport equation of turbulence dissipation can be written as:
∂ ∂ ∂ ( ρε ) + ( ρε u j ) = ∂t ∂x j ∂x j
µt µ + σε
∂ε ε2 + ρ C1S ε − ρ C2 k + νε ∂x j
(8)
As shown in Fig.1(a), the supersonic jet element (SJE) is a kind of amplifying Element without moving parts and it uses gas as the working medium. Fig.1(b) shows the work mechanism of the SJE. Compared with the other flexible parts of the amplifier (such as pneumatic or hydraulic slide valve, rotary valve or diaphragm valve), the supersonic jet
ACCEPTED MANUSCRIPT element has many advantages, such as the low accuracy manufacture and low cost, which also has no moving parts, so the work process does not appear "stuck" in. As implemental components of the control system, the SJE’s work performance (control, switching time, energy efficiency, working reliability et al.) has an important role in the control
left spout
right spout
right sloping wall
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left sloping wall
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accuracy of the system.
right control gas
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left control gas
Main chamber pressure
(a) The physical model
(b) Work mechanism
(c) Computational grids
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Fig.1 The work mechanism and grids for the supersonic jet element
Fig.1(c) shows the surface girds of the SJE. In modeling the whole SJE, the quality and size of the mesh generation would often directly affect the accuracy of the experimental results. So during meshing the structural grids of the SJE,
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the simulation took the method of size function control grid size (It doesn’t change the entire structure of the network
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topology). And the parts of structural grid were refined for the whole Element, and the structural grid had been smoothed and swapped, enhancing the accuracy of the experiment. Due to the flow symmetry in x-y plane for the jet element, the simulation is two-dimensional. And the simulation is conducted using the unsteady ANSYS/FLUENT V.6.3.26. The corresponding unsteady mass flow rate can be changed by adopting the UDF Function code. The GaussSeidel method is used to calculate the inner fluid field of the supersonic jet element. The governing equations of flow are Reynolds Averaging Navier-Stokes equation. The model of turbulence is Realizable k-epsilon model, and Twolayer zonal model is used near the wall.
ACCEPTED MANUSCRIPT B. The comparison with the experimental data
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(a) The Validation of the Thrust in the Attached Wall Condition
Fig.2 The schematic diagram of jet components test system ①
Air Compressors Force Sensor
Throttle
Filter SJE
Air Cylinder
Decompression Pressure Valve
Three Links Directional Electromagnetic Control Valve
Cutoff Value
Pressure Table Adjustable
Both of the validation of the thrust in the attached wall condition and the verification of the switching time in
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switching process are completed by the single-channel air test system. This test system consists of high pressure and large flow of the pressure regulator system, data acquisition and processing system, moving frame of the jet components as well as the control components. The Fig.2 has shown the schematic diagram of jet components test system. The high
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pressure and large flow of the pressure regulator system includes a high-pressure compressor(WZ2.3/450) and a high pressure cylinder group (TGP-50L/35Mpa), which mainly provide the high pressure gas source and control flow for the
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jet components. The moving frame is used to fix the jet components, on which we would install two high frequency high-precision force sensors to detect the size of a combined thrust and thrust switch changes in the course of joint of the mobile frame. The data acquisition and processing system are used to collect the voltage signal coming from the collection force sensor.
There are various kinds of noise in the test system of jet element, in which the vibration of the jet element platform can strongly affect the accuracy of the test data. In order to eliminate noise and make the obtained thrust value meet the error requirements of the test, the system identification must be firstly carried out to obtain the mathematical model of the platform frame, and then according to the results of the system identification, the transient response analysis method will be adopted, calculating the rising time or falling time of thrust when the jet element is dynamically switched and the combined thrust value in the working state near the attached wall. By adjusting the pressure relief valve, the
ACCEPTED MANUSCRIPT different main gas source pressure needed for the experiment can be obtained, and the performance test of the jet element at different main gas source pressure can be carried out. The control flow into the supersonic jet element can be controlled by controlling the throttle valve, and the performance test of the jet element in different control flow can be carried out. The thrust in the attached wall condition is one important measurement to indicate the performance of supersonic jet components good or bad. Therefore, we have calculated the pressure thrust created by a variety of
the main gas source pressure [19].
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primary gas source pressure, and then under the same conditions, we have obtained the thrust value corresponding to
Ta ble 1 The comparison of the calculated and the experimental thrust fr(N) 49.65 84.34 108.91 142.09 164.59 170.27 186.87 220.68 263.05 296.36 346.18
fl(N) 0.42 1.95 4.43 4.58 5.66 5.46 6.43 8.55 18.04 21.78 27.89
F(N) 49.23 82.40 104.48 137.51 158.93 164.81 180.44 212.12 245.01 274.58 318.29
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Test(N) 44.0 79.9 106.0 126.3 142.6 161.4 179.3 202.8 239.9 275.4 313.2
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Ps (bar) 20 30 40 45 50 55 60 70 80 90 100
Error(%) 11.89 3.13 -1.43 8.87 11.45 2.11 0.64 4.6 2.1 -0.3 1.63
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Tab.1 has showed the thrust corresponding to the main gas source pressure (Ps) and test stands for the total thrust in the X direction of the jet components. fl and fr stand for their own thrusts of two outlets,
r r r F = f l + f r , Error stands
for the relative error between the calculation ( F ) and test. The maximum and minimum difference is within 12% and
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0.3%, respectively.
(b) Accumulation of errors estimate
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Estimating precision and errors accumulation is extremely necessary for large simulations of the complex fluid problem. And accumulation of error is proportional to the square rote of the number of time steps. It should be evaluated for each numerical simulation, especially for the unsteady flow state, which has been investigated by Smirnov in details[15-18] The corresponding research have revealed that the relative error of integration in 1D case S1 is proportional to the mean ratio of cell size ∆L to the computational domain size
L1 in the direction of integration
depending on scheme accuracy:
∆L S1 = L1
k +1
(9)
ACCEPTED MANUSCRIPT As referred by Smirnov[15-18], it can be simplified as: S1 ≈ (1 N1 )
k +1
in the uniform grid, where N1 is the
number of cells in the direction of integration. k is the order of accuracy of numerical scheme. Then the Eq.9 can be summed up as: 2
Serr ≈ ∑ Si
(10)
The allowable value of total error
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i =1
S max can be presumed from 1% to 5%[15-18], then the following inequality
should be satisfied as:
Serr ⋅ n ≤ S max
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(11)
where n is the number of time steps in Navier-Stokes equations integration. Then the maximum allowable number of
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time steps can be determined according to the Eq.11:
nmax = ( S max Serr ) 2
(12)
The ratio of maximal allowable number of time steps is introduced by Smirnov[15-18], which can be used to characterize the reliability of results. the higher is the value of nmax, the lower is the errors.
Rs = nmax n
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(13)
For this, we have conducted a representative case in the current research. The grid sensitivity test is performed subsequently. X stands for the nodes in the streamwise direction of the walls, and Y stands for the nodes in the transverse direction of the walls. i and j respectively stands for the nodes on the whole edges of the throat and control
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flow inlet(left and right). As shown in Fig.4, four different grid sizes (X×Y, i× j) do not deviate from each other within
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5%.
Allowable error ( %) 5 5 5 5 5 5 5 5
Grid resolution (X×Y, i) 260×50, 60×100 500×100, 100×150 1000×200, 200×260 2000×400, 400×500 260×50, 60×100 500×100, 100×150 1000×200, 200×260 2000×400, 400×500
Table.2 Error estimate
Physical time simulated (s) 0.06 0.06 0.06 0.06 0.6 0.6 0.6 0.6
Number of time steps 620 1312 2823 5800 7128 12600 31000 65000
Accumulated error 0.000197 0.000036 0.0000066 0.0000012 0.000680 0.000113 0.000022 0.000004
Allowable number of time steps 3.85107 2.46109 1.571011 1.011013 3.85107 2.46109 1.571011 1.011013
Reliability Rs=nmax/n 62097 1.86106 5.56107 1.74109 5401 1.95105 5.06106 1.55108
The detailed analysis for the accumulation of error on different combinations (iteration time step and grid resolution) have also been shown in Table.2. The accumulation of error for different grid resolution and physical time simulations have been studied in details. The allowable error is assumed to 5%. The accumulation of error will begin in
ACCEPTED MANUSCRIPT a fast manner for coarse grid and decreases on the increasing grid resolution and scheme accuracy, but decreases when the physical time increases. This phenomena has also been discussed by Smirnov[15-18]. For present simulations all the results demonstrate high reliability of the used program. but the number of time steps and physical time should not be
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chosen too large in order to avoiding the accumulation of error in the whole simulation process. Therefore, keeping the certain iteration accuracy and simultaneously avoiding too long simulated period should be the top priority. After the
subsequent computations.
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(c) The Verification of the Switching Time in Switching Process
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carefully consideration, we will adopt the relatively moderate computation domain size: [1000×200, 200×260 ] in the
The switching time is another important measurement to indicate the performance of supersonic jet components good or bad. We defined the switching time as from the beginning of the control flow changes to the reverse thrust on output became to be stable. In the experiment, the control signal is changed by changing the role of the control signals
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on the implementation of the electromagnet. Since the change from the control solenoid to control the flow signal changes in response between the solenoid armatures, pneumatic amplification mechanism, air flow through the control of channel and other areas of the delay, which is difficult to determine the starting of the switching time. Therefore, this
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research would be carried out by comparing the thrust rise time and thrust falling time obtained by calculating and
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experiment, which could verify the correctness of the unsteady calculation.
(a) Simulated thrust change
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(b) Experimental thrust change
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Fig.3 Thrust change during switching waveforms (simulation VS experiment) For the simulated result (shown in Fig.3(a)), Ps 100bar
ml=0.1kg/s
mr=0.003kg/s, and the thrust rise time is
1.11ms, the thrust falling time is 1.23ms. For the experimental result (shown in Fig.3(b)), it is the thrust change during switching waveforms in experimental value. Among them, the top of the square wave stand for the input control signal,
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the below purple curve is the combined thrust on output measured by two force sensors, the green curve is to eliminate noise and smooth thrust after the output. The thrust rise time is 1.18ms, and the thrust falling time is 0.85ms.
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Considering the accuracy of the measurement method and the simulation errors, the comparisons are showing
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reasonable overall quantitative agreement, showing the accuracy of the current simulation method.
C. Setup for the inlet control flow
Due to the fact that the switching time for the supersonic jet element operation is extremely short, the corresponding control signals will always show the typical quati-step functions, so the mass flow rate and static pressure for the both left and right inlet control flows can be assumed as a ramp function within a very short period (∆t) at the beginning of the switch, which can be written as:
ACCEPTED MANUSCRIPT m2 t<0 t mr = m2 + (m1 − m2 ) 0 ≤ t ≤ ∆t ∆t m1 t > ∆t
(14)
P1 t<0 t Pl = P1 + ( P2 − P1 ) 0 ≤ t ≤ ∆t ∆t P2 t > ∆t
P2 t<0 t Pr = P2 + ( P1 − P2 ) 0 ≤ t ≤ ∆t ∆t P1 t > ∆t
(15)
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m1 t<0 t ml = m1 + (m2 − m1 ) 0 ≤ t ≤ ∆t ∆t m2 t > ∆t
As shown in Table.3, ml and mr stands for the mass flow rate on left and right inlet control flow respectively, pl and
2×10 5s.
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decided by the ratio of p1 and p2, here ∆t 1×10-5
m2=0.003kg/s
p1=210000Pa
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pr will stand for the corresponding static pressure. m1=0.08kg/s
p2=57000Pa. ∆t is
Table 3 Setup for the parameters of the control flow Case
Boundary
Total pressure or
type
mass flow rate
Pressure
100×105
inlet
(Pa)
Static
5
pressure(10 Pa)
Temperature
Turbulence
(K)
parameters
Remarks
Turbulence intensity
ml
Mass-flow
0.08(kg/s)
inlet mr
0.57
—
1.0144
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outlet
4% —
Hydraulic diameter 0.0384(m)
Turbulence intensity
2.1
0.003(kg/s) Pressure
300
300
6% Hydraulic diameter 0.012(m)
300
Positive X-axial
k=1(m2/s2) e=1(m2/s3)
Negative X-axial reflux conditions
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lout
99.8
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Ps
Fig.4 The specific sizes of SJE The specific sizes of geometry have been marked in Fig.4. The inlet condition for the main gas source will be set as pressure inlet condition, left and right outlet condition will be set as pressure outlet condition. The left and right control flow inlet will be set as the mass flow inlet condition. Other surfaces will be set as the non-sliding insulation wall
ACCEPTED MANUSCRIPT condition. The air parameters are the following: temperature T=300K, the local heat flux
−λ (
∂T ) w determined based ∂y
on near wall turbulence model incorporating local flow characteristics. And the total pressure of the main gas source is
& =0.08 and 0.003kg/s, respectively. Left and right 100×105(Pa), the mass flux of left and right control inlet will be set m
Both the turbulent energy(k) and turbulence dissipation(Ɛ) are set as 1.0.
Results and discussion
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III.
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outlets are set as free pressure outlet condition. Turbulence intensity is 6% and the Hydraulic diameter is 0.012(m).
A. The changes of the different parameters
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F i g.5 s ho ws the location change of the spiral vortex center i n o ne p er io d ( t = 3.56ms ~ 6.32ms). Moreover, it can be seen that the main vortex center of supersonic jet element always moves within a small range in the centerline (X=0) during the cycle time. At the initial and end stage, the vortex center is very close to centerline, and we also find
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that when the primary vortex moves near the center line, the left and right thrust would become smaller.
Fig.5 Location change of main vortex center
Fig.6 Total pressure change of main vortex center
Figs.6 shows the total pressure change trend. There are two peaks of total pressure. Before the peak of total pressure coming, some of the control air enters the main vortex zone. With the air in the main vortex area increase, the pressure in the main vortex zone will rise. When the total pressure in the vortex zone became great enough, they will prevent the control air from entering the vortex zone, and compel the control air to eject from the left output. As the air
ACCEPTED MANUSCRIPT in the main vortex zone is ceaselessly swept away by the main jet and the left control flow, the total pressure in the main vortex zone begin to decrease. But when the total pressure dropped enough, part of the main jet will be inhaled into the main vortex zone, the total pressure of the main vortex would rise again. Thus, the total pressure of the main
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supersonic jet element at different time in one period.
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vortex area always continues to experience increase-decrease-increase cycle. Fig.7 shows the vortex distributing in
main vortex
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main vortex
Figure 7 The stream function contour at different time
Figure 8 Static pressure changes of right wall
Figure 9 Static pressure changes on concave wall
ACCEPTED MANUSCRIPT As shown in Fig.8, affected by the main vortex, the two peak static pressure on the wall is changing. The location of the first peak is moving upstream with the time increases, and the location of the second peak is moving more obvious in the first three times. However, at t = 4.08ms, the moving position is very small, and the static pressure increases larger. As shown in Fig.9, disturbed by the upstream flow, the static pressure value of the stagnation point keeps
stagnation point moves to the right output.
B. The deflection of the jet (Stage III)
(b) t=9.2×10-4s
(c) t=1.48×10-3s
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(a) t=7.2×10-4s
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Separated vortex zone
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(a) The changes for the stripping and attaching vortex zone
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oscillating between 8.6×105 ~ 8.7 × 105Pa. In the right thrust changes from the minimum to maximum process,
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Separated vortex zone
(d) t=1.64×10-3s
(e) t=1.84×10-3s
(f) t=2.04×10-3s
Figure 10 The changes for the stripping vortex zone (Stage III) As shown in Fig.10, before the stage III, the stripping vortex zone is reducing and expanding in the in the longitudinal and transverse directions respectively, the vortex center also has moved to the right output. When t=9.2×103
s, the vortex center has reached near the right output channel, the corresponding position also tends to be stable. Due to
the expanding stripping vortex zone, a small and narrow flow channel has been generated on the right side of the wedge
ACCEPTED MANUSCRIPT (shown in Fig.10(b)). When the supersonic air flows into this small channel, it can make the corresponding Mach number become smaller and increase the pressure. Once the static pressure for the right side of the wedge exceeds the left side, due to the effect of the pressure difference, part of the primary jet will flow into the left output channel. Thus, the transverse expansion for the stripping zone near the right output channel entrance can play a significant role in
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making the primary jet deflect successfully.
(b) t=9.2×10-4s
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(a) t=7.2×10-4s
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Attached vortex zone
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(d) t=1.64×10-3s
(e) t=1.84×10-3s
(c) t=1.48×10-3s
Attached Vortex Zone
(f) t=2.04×10-3s
Figure 11 The changes of the attached vortex zone (Stage III)
As shown in Fig.11, there are no obvious changes at the early time of the stage III. When t>1.48×10-3s, the attached vortex zone will shrink in the transverse direction. At mean time, the corresponding vortex center is also moving up and then flowing into the left output channel. (b) The changes of the Mach numbers As shown in Figs.12(a) and (b), at the early time of the stage III, the detached oblique shock wave near the wedge (GH) has gradually transformed into the detached bow shock wave. The strength of the oblique shock (IJ) near the right output channel has also been increased, and the corresponding position has also been moved up. With the stripped
ACCEPTED MANUSCRIPT vortex area continues to shrink in the longitudinal direction, the shock (EF) will gradually move up and the shock (CD)
(b) t=9.2×10-4s
(c) t=1.48×10-3s
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(a) t=7.2×10-4s
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will begin to extend to the right upper direction.
(d) t=1.64×10-3s
(e) t=1.84×10-3s
(f) t=2.04×10-3s
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Figure 12 The contour of the Mach number at different time (Stage III) When t=1.48×10-3s, the strength of the detached shock(GH) will further increases, the original oblique shock(IJ)
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will begin to transform into a normal shock. With the stripping vortex zone continuously expands, the oblique shock (EF) will move down and the shock (CD) will begin to reduce to the left lower direction. Due to the fact that the strength of the shock (IJ) gradually decreases, while the shock (KL) increases, when t=1.64×10-3s, both the strength of the shock (KL) and (IJ) are similar in the left and right output channel. After t>1.64×10-3s, the detached bow shock (GH) will be gradually transformed into the detached oblique shock, the shock (IJ) is also moving down and the corresponding strength is also weakening. The strength of the shock (KL) will experience decreasing then increasing trends. When t=2.04×10-3s, the shock (CD) and (IJ) will totally disappear (shown in Fig.10(f)).
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(c) The thrust changes for the left and right outputs
(a) left output
(b) Right output
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Figure 13 The thrust changes for the left and right outputs at different time (Stage III) As shown in Fig.13, before the stage III, due to the fact that the primary jet in the left output channel increases extremely slowly, so the left thrust has little changes. Except this, the increasing trend for the left thrust has two oscillations. When t> 1.48×10-3s, as the primary jet flowing into the left output increases gradually, the left thrust has increased sharply. This phenomenon also has shown on the right thrust. The direction of the combined thrust has
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changed at this stage. The thrust has shown an approximately linear relationship with the time.
C. The jet attaches wall layer again (Stage IV)
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(a) The shear stress distributions for the different wall layers
Y(m)
(a) t=2.24~2.8ms
Y(m)
(b) t=3.04~3.2ms
Figure 14 The shear stress distributions for the left wall layers at different time (Stage IV)
ACCEPTED MANUSCRIPT As shown in Fig.14, due to the flow pattern near the left attached vortex zone is the reverse flow, so the corresponding shear stress always shows negative. As the switching time increases, the lower end of the attached vortex zone will move to the positive Y direction. It indicates that the attached vortex zone is reduced gradually while the jet layer zone is expanded. When t=3.04×10-3s, the shear stress for two zones shows negative value shown in Fig.12(b). The first zone stands for the position of the separated bubble, which is generated by the interference between the
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switching oblique shock and layer boundary. The second zone will stand for the position of the attached vortex zone. When t =3.2×10-3s, the shear stress on the left layer will be negative, and the corresponding attached vortex zone has also disappeared.
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(b) The changes in the attached vortex zone
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Attached vortex zone
(b) t=2.4×10-3s
(c) t=2.6×10-3s
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(a) t=2.24×10-3s
(d) t=2.8×10-3s
Attached vortex zone
Separated bubble
(e) t=3.04×10-3s
Separated Bubble
(f) t=3.2×10-3s
Figure 15 The changes in the attached vortex zone at different time (Stage IV)
ACCEPTED MANUSCRIPT As shown in Fig.15, affected by the suction effect from the jet, the air in the attached vortex zone has been swept away gradually. This can make the corresponding pressure decreased, and the pressure balance in the two sides of the jet will also be broke. The jet will move to the left layer affected by the pressure difference, which can make the jet keep balance. However, as the air in the attached vortex zone continuously swept away, which can be reduced to a small vortex zone at the lower left output position (shown in Fig.15(f)). At this stage, if the input of the control flow is totally
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closed, the vortex zone at right side of the primary jet will have a higher static pressure. Due to the “Coanda” effect, the jet also can attach the layer successfully even though the corresponding control flow is totally closed.
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(c) The changes for the Mach numbers
(b) t=2.4×10-3s
(c) t=2.6×10-3s
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(a) t=2.24×10-3s
(d) t=2.8×10-3s
(e) t=3.04×10-3s
(f) t=3.2×10-3s
Figure 16 Contour of the Mach numbers at different time (Stage IV) As shown in Fig.16, when t=2.6×10-3s, the oblique shock near the left side of the wedge has disappeared, the strength of the switching oblique shock formed on the free shear layer at left and right primary jets has also been weakened. When t=3.04×10-3s, a new oblique shock has been generated on the free shear layer at the right side of the primary jet, whose strength is weaker than the switching oblique shock. Except this, another new oblique shock is also
ACCEPTED MANUSCRIPT generated between the attached vortex zone and separated bubble. As this moment, the oblique from the left output channel has been transformed into a stronger detached normal shock. In addition, the shear layer evolution in the primary jet flow field can also be observed in Fig.16. Due to the fact that the attached vortex zone is reducing and the wall layer zone is expanding, the free shear layer at left side of the primary jet has been shortening gradually. The free shear layer at right side will deflect to the left together with the primary jet.
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At the end of the stage IV, squeezed by the right side of the vortex zone, the free shear layer on the right side will show the expansion - compression - expansion condition (shown in Figs.16(d), (e) and (f)).
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(d) The changes for the left and right output
(a) Left output
(b) Right output
Figure 17 The changes for the left and right output (Stage IV)
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As shown in Fig.17, at the initial time of the this stage, the thrust for the left output has increased with the increasing switching time. When t=2.4×10-3s, the corresponding thrust has exceeded 452.5N. After that, the
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corresponding thrust will begin to decrease with the strength of shock from left output increases. When t> 2.92×10-3s, due to the right control flow and a lot of primary jet flow into the vortex zone through the left side of the wedge, the air ejected from the left output channel has been reduced sharply. Except this, a stronger detached normal shock has been generated in the left output channel, and the corresponding thrust is decreasing sharply. When t=3.2×10-3s, the thrust has decreased to less than 281N. At the very beginning time of the stage IV, due to some parts of the right control flow will be ejected from the right output, the corresponding thrust should be positive. However, at the later time, the outside air has been inhaled into the supersonic jet Element from the right output, the right output will generate a reverse thrust compared with the left output. As the inhaled air continuously increases, the corresponding reverse thrust will increase gradually, the maximum value can exceed 37.5N. From t=3.04×10-3s to 3.2×10-3s, due to the outside air flowing into the jet Element
ACCEPTED MANUSCRIPT has been decreased sharply, as a result, the reverse thrust generated by the right output also has been reduced greatly. At the end of the stage IV, the right thrust has recovered to be positive.
D. Adjustment stage (Stage V)
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(a) The thrust changes for the left and right outputs
(a) Left output
(b) Right output
Figure 18 The thrust changes for the left and right outputs Figs.18(a) and (b) have shown the thrust changes process for the left and right outputs including the adjusting and
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adjusted stages. In the adjusting stage, due to the airflow in the jet Element is extremely unstable, as a result, the corresponding thrust has shown a very substantial oscillation state. After a period of self adjustment, the corresponding flow will become relatively stable, and the oscillating amplitude has also greatly reduced. (b) The formation of the primary vortex zone
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Core 2
Vortex Core 1
Vortex Core 2
(a) t=3.36×10-3s
Vortex Core 3
Vortex Core 1
Vortex Core 3
Vortex Core 4
(b) t=3.64×10-3s
Vortex Core 1 Vortex Core 4
(c) t=3.84×10-3s
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Vortex Core 3
(d) t=4.04×10-3s
Main Vortex Zone
Vortex Core 5
(e) t=4.48×10-3s
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Vortex Core 4
(f) t=5.00×10-3s
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Figure 19 The total pressure distributions at different time (Stage V)
As shown in Fig.19, when t=3.36×10-3s, squeezed by the left concave wall, some parts of the primary jet have
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flown back the jet Element center, where has generated two vortex zone (zone 1 and 2) on the left and right sides. When t=3.64×10-3s, the domain of the zone 1 has expanded and the corresponding vortex center has also moved. However, the zone 2 has left the Element center and entered the right output channel. At this time, a long narrow vortex zone (zone 3) has been generated near the left side of the wedge and zone 4 has been generated at the upper right control flow position. When t=3.84×10-3s, the domain of the zone 1 has been reduced, and the corresponding vortex center has moved to the
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right side of the wall. At the same time, the domains for the zone 3 and 4 have also expanded. When t=5.00×10-3s, the vortex center of the zone 5 will move to the lower right position, a new larger primary vortex zone has been generated in the jet Element center (shown in Fig.19(f)). The formation of the primary vortex zone means that the flow inside the
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jet Element has tended to be stable. Which also means that the end of the switching process.
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(c) The static pressure changes for the left and right sides
(a) Left attached wall
(b) Right attached wall
Figure 20 Static pressure distributions for the left and right attached wall at different time
ACCEPTED MANUSCRIPT As shown in Fig.20(a), for the left attached wall, the static pressure has always firstly increased and then decreased. It has indicated that the jet flowing along the layer has experienced a strong compressing and expanding process. At the early time of the adjusting stage, the static pressure peak always changes obviously (dropped from 8.2×105Pa to 3.7×105Pa). At the later time of the adjusting stage, the corresponding peak has been greatly reduced (oscillating from 4.6×105Pa to 5.7×105Pa). In addition, the positions on the left side for the static pressure peak are also distinct at
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different time. The nearest and farthest position away from the left control flow is at t=3.64×10-3s and 4.04×10-3s respectively. Fig.20(b) has shown the phenomena on the right attached wall. Compared with the left, the static pressure peak is not very obvious, especially at t=3.84×10-3s and 4.04×10-3s, the corresponding change is extremely flat.
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(d) The shear stress distribution for the wall r1 in Y direction
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Figure 21 The shear stress distribution for the wall r1 and left attached wall As shown in Fig.21, due to the presence of the switching oblique shock, the flow has separated on the wall r1, and the separation point can be decided by the shear stress in Y direction. At the early adjusting stage, the position for the
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separation point always changes greatly. The farthest away from the right control flow input is happened at t=3.64×10-3s. However, at the end time of the adjusting stage, the position changes very little. With the airflow inside the jet Element
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becomes stable, the strength and position for the switching oblique shock will be fixed. (e) The shear stress distribution for the left attached wall
Figure 22 The shear stress distribution for the left attached wall
ACCEPTED MANUSCRIPT Fig.22 has shown the shear stress distribution for the left attached wall in Y direction at different time. In general, due to the interference between the switching oblique shock and left attached layer, there will exist a separating bubble near the left layer. The negative shear stress part means the specific position of the bubble. When t=3.64×10-3s, the strength of the switching oblique shock is very strong, and the corresponding angle is also very large. It can connect with the separated bubble and low pressure vortex zone near the left control flow input. When t=4.04×10-3s, the strength
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of the switching oblique shock however will becomes weaker, and the separating bubble is also very far away from the left control flow. Except this, there is another new separating bubble has been generated at the downstream of the first separated bubble.
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(f) The changes of the Mach numbers
Separated bubble 1
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Separated bubble 2
(b) t=3.64×10-3s
(c) t=3.84×10-3s
(e) t=4.48×10-3s
(f) t=5.00×10-3s
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(a) t=3.36×10-3s
(d) t=4.04×10-3s
Figure 23 Contour of the Mach numbers at different time (Stage V) As shown in Fig.23, there are two aspects have more obvious changes, one is the free shear layer at the right side of the primary jet, the shape and thin will be changed with the changing domain, strength and the position of the vortex center. The other one is the changes of the shock in the left output channel. At t=3.64×10-3s and 3.84×10-3s, the corresponding shock is a weak oblique shock. However, at t=4.04×10-3s and 4.48×10-3s, it has been changed to a normal shock.
ACCEPTED MANUSCRIPT When t=4.04×10-3s, due to the strength of the switching oblique shock is extremely weak, the increasing extent for the pressure is very small and the flow direction also has no changes. When the jet with the low pressure and high speed flows into the free shear layer near the zone 3 at a larger angle, a stronger oblique shock will be generated near the free shear layer, which can make the boundary layer begin to separate, and another separating bubble will be formed at the
(a) P s=2 0 ×1 0 5 P a
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downstream of the previous separated bubble (shown in Fig.23(d)).
(b) P s=6 0 ×1 0 5 P a
(c) Ps=1 2 0 ×1 0 5 P a
Fig.24 Shock wave distribution on the right convex wall Fig.24(a),(b) and(c) are the shock wave distribution near the right convex wall in the case of Ps=20×105 Pa, Ps=60×105Pa and Ps=120×105Pa, respectively. It can be seen that when the jet flows through the right convex wall,
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different degrees of expansion have occurred. Afterwards, due to the effect of external reflux or internal vortex, a clear shock wave has been generated at the right convex wall. When the jet flow through the shock wave, the higher back pressure will be formed and cause the flow separation eventually."
Conclusion
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IV.
Based on the unsteady viscous flow simulation, the flow characteristics inside the supersonic jet Element have been
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investigated numerically. The corresponding switching mechanism and conclusions have been given as follows: (1) The instability of the main vortex center is a main reason to make the output thrust of the supersonic jet components fluctuate all the time in attached wall condition. (2) At the deflection of the jet stage (Stage III), Once the static pressure for the right side of the wedge exceeds the left side, the transverse expansion for the stripping zone near the right output channel entrance can play a significant role in making the primary jet deflect successfully. The direction of the combined thrust has changed at this stage. The thrust has shown an approximately linear relationship with the time. (3) When the jet starts to attach the wall layer (Stage IV), if the input of the control flow is totally closed, the vortex zone at right side of the primary jet will have a higher static pressure. Due to the “Coanda” effect, the jet also can attach the layer successfully even though the corresponding control flow is totally closed.
ACCEPTED MANUSCRIPT (4) When the jet enters the adjustment stage (Stage V), the thrust changes process for the left and right outputs including the adjusting and adjusted stages. In the adjusting stage, due to the airflow in the jet Element is extremely unstable, as a result, the corresponding thrust has shown a very substantial oscillation state. After a period of self adjustment, the corresponding flow will become relatively stable, and the oscillating amplitude has also greatly reduced.
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Acknowledgments Financial supports from National Science Foundation of China [Grant number 11602023 and U1430113] are
References
New, T. H. and Tsioli, E., “Effects of area-ratio on the near-field flow characteristics and deflection of circular inclined coaxial
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[1]
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gratefully acknowledged. The authors also would like to thank the reviewers for their helpful suggestions.
jets,” Experimental Thermal and Fluid Science, Vol. 54, No.4 , 2014, pp. 225-236. [2]
Hsu, C. M., Huang, R. F. and Loretero, M. E., “Unsteady flow motions of an oscillating jet in crossflow,” Experimental Thermal and Fluid Science, Vol. 55, 2014, pp. 77-85.
[3]
Cadirci, S., Gunes, H. and Rist, U., “Numerical investigation of a jet and vortex actuator in a cross flow boundary layer,” European Journal of Mechanics B/Fluids, , Vol. 44, 2014, pp. 42-59.
El-Amin, M. F., Sun, S. Y. and Salam, A., “Simulation of buoyancy-induced turbulent flow from a hot horizontal jet,” Journal
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[4]
of Hydrodynamics, Vol. 26, No. 1, 2014, pp. 104-113. [5]
Keerthi, M. C. and Kushari, A., “Effectiveness of vortex generator jets and wall suction on separated flows in serpentine-duct diffuser,” Aerospace Science and Technology, Vol. 34, 2014, pp. 12-19. Bühler, S., Kleiser, L. and Bogey, C., “Simulation of Subsonic Turbulent Nozzle Jet Flow and Its Near-Field Sound,” AIAA
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[6]
Journal, Vol. 52, No. 8, 2014, pp. 1653-1669. Yang, S., He, X.H., Yuan, S.Q., Zhu, J.W., Deng, Z.D. "A valveless piezoelectric micropump with a Coanda jet element,"
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[7]
Sensors and Actuators A-Physical, Vol. 230, 2015, pp. 74-82.. [8]
Johnson, D. A. and Wood, P. E., “Self-sustainable oscillations in opposed impinging jets in an enclosure,” Can J Chem Eng, Vol. 78, No. 5, 2000, pp. 867-875.
[9]
Li, W. F., Huang, G. F., Tu, G. Y., Liu, H. F. and Wang, F. C., “Experimental study of oscillation of axisymmetric turbulent opposed jets with modulated airflow,” AIChE J, Vol. 59, No. 12, 2013, pp. 4828-4838.
[10] O’Neill, P., Soria, J., Honnery, D., “The stability of low Reynolds number round jets,” Exp Fluids, Vol. 36, No.3, 2004, pp. 473-483. [11] Crow, S. C. and Champagne, F. H., “Orderly structure in jet turbulence,” J Fluid Mech, Vol. 48, No. 3, 1971, pp. 547-591. [12] Raja,
K.
K.,
Das,
M.
K.
and
Kanna,
P.
R.,
“Numerical
study
of
mixed
convection
in
a two-
dimensional laminar incompressible offset jet flow,” International Journal of Heat and Mass Transfer. Vol. 52, No. 3, 2008, pp. 1023-1035.
ACCEPTED MANUSCRIPT [13] Draksler, M.,
Niceno, B., Koncar, B. and Cizelj, L., “Large eddy simulation of multiple impinging jets in hexagonal
configuration-Mean flow characteristics,” International Journal of Heat and Fluid Flow, Vol. 46, 2014, pp. 147-157. [14] Silnikov, M.V., Chernyshov, M.V. and Uskov, V. N., “Two-dimensional over-expanded jet flow parameters in supersonic nozzle lip vicinity,” Acta Astronautica, Vol. 97, 2014, pp. 38-41. [15] Smirnov, N.N., Betelin, V.B., Shagaliev, R.M., Nikitin, V.F., Belyakov, I.M., Deryuguin, Yu.N. Aksenov, S.V. Korchazhkin,
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D.A. "Hydrogen fuel rocket engines simulation using LOGOS code," International Journal of Hydrogen Energy, Vol.39, 2014, pp.10748-10756.
[16] Smirnov, N.N., Betelin V.B., Nikitin, V.F., Stamov, L.I., Altoukhov, D.I. "Accumulation of Errors in Numerical Simulations of Chemically Reacting Gas Dynamics," Acta Astronautica, Vol. 117, 2015, pp.338-355.
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[17] Betelin, V. B., Kushnirenko, A. G., Smirnov, N. N., Nikitin, V. F., Tyurenkova, V. V., Stamov, L. I. "Numerical investigations of hybrid rocket engines," Acta Astronautica, Vol. 144, 2018, pp. 363–370.
[18] Smirnov, N. N., Penyazkov, O. G., Sevrouk, K. L., Nikitin, V. F., Stamov, L. I., Tyurenkova, V. V. "Onset of detonation in
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hydrogen-air mixtures due to shock wave reflection inside a combustion chamber," Acta Astronautica, Vol. 149, 2018, pp.77– 92.
[19] Xu, Y. and Zhang, G. Q., "Numerical investigation of switching mechanism for the supersonic jet element," Acta Astronautica,
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Vol. 136, 2017, pp. 342-353.
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Highlights The instability of main vortex center is a main reason to make the output thrust fluctuate.
•
The transverse expansion for the stripping zone can make the primary jet deflect successfully.
•
Jet can attach the layer successfully due to “Coanda” effect even though control flow is totally closed.
•
The thrust changes process will experience two typical stages: adjusting and adjusted stages.
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•
S0094-5765(19)30741-6
DOI:
https://doi.org/10.1016/j.actaastro.2019.05.027
Reference:
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To appear in:
Acta Astronautica
Received Date: 25 April 2019 Revised Date:
13 May 2019
Accepted Date: 16 May 2019
Please cite this article as: Y. Xu, G.Q. Zhang, Switching mechanism investigation for the supersonic jet element: Deflection, attachment and adjustment stages, Acta Astronautica (2019), doi: https:// doi.org/10.1016/j.actaastro.2019.05.027. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Switching Mechanism Investigation for the Supersonic Jet Element: Deflection, attachment and adjustment stages Y. Xu1 and G. Q. Zhang1,* 1
Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering,
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Beijing Institute of Technology, Beijing, 100081, China * Corresponding author: [email protected]
ABSTRACT: Based on the unsteady viscous flow simulation, the flow characteristics inside the supersonic jet
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Element have been investigated numerically. The results have revealed that once the specific structures of the supersonic jet element are finalized, even if the boundary conditions remain unchanged, the corresponding internal
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flow will also shows strong unsteady in a certain range of primary gas source pressure. The instability of the main vortex center is a main reason to make the output thrust of the supersonic jet components fluctuate all the time in the attached wall condition. At the deflection of the jet stage, once the static pressure for the right side of the wedge exceeds the left side, the transverse expansion for the stripping zone near the right output channel entrance can play a significant role in making the primary jet deflect successfully. When the jet starts to attach the wall layer, due to the “Coanda” effect, the jet also can attach the layer successfully even though the corresponding control flow is
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totally closed. When the jet enters the adjustment stage, the thrust changes process for the left and right outputs will experience two typical stages: adjusting and adjusted stages. The corresponding vortex structures at different
analyzed in details.
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switching time together with the force variations etc. inside the jet Element have been obtained computationally and
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Keywords: Supersonic jet element, layer, shock, vortex, switching time
F
=
Total thrust (N)
e
=
k
Nomenclature T
=
Temperature (K)
Dissipation rate (m2/s3)
t
=
Switching time (s)
=
Kinetic energy (m2/s2)
fl
=
Left outlet thrust (N)
m
=
Mass flow rate (K)
fr
=
Right outlet thrust (N)
Ma
=
Mach number
lout
=
Pressure outlet (Pa)
Ps
=
Main gas source pressure(bar)
P01
=
Point from main jet inlet (-0.0062,0.042)
∆t
=
=
Point from main jet inlet (0.0062,0.042)
Time step (s)
P02
ACCEPTED MANUSCRIPT I.
T
Introduction
he supersonic jet element (SJE) is the execution unit in the attitude control system. Its performance (such as the
control force, switching time, energy efficiency and reliability etc.) can play an extremely important role in controlling
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the accuracy of the whole system. Therefore, the research on the jet flow has drawn a lot of attentions [1-3]. The SJE is also called fluidic amplifier. On one hand, it could be used as the logic control system components to achieve a variety of logic functions, on the other hand, through a small energy control signals, it also could be served as an
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implementation part of the control system and offer much large energy or torque directly [4-7]. Nowadays the
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supersonic jet element is applied to the Multiple Launch Rocket System project as the implemental components of the control system, such as the “tornado” weapon system of Russia [8-11]. Raja [12] had numerically investigated the mixed convection flow and heat transfer characteristics in a two-dimensional plane, laminar offset jet issuing parallel to an isothermal flat plate. The results had shown that the reattachment length is strongly dependent on both the Reynolds
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number and the Grashof number for the range considered.
Draksler [13] had analyzed numerically the highly turbulent flow of the air impinging jets in hexagonal arrangement is by the means of Large Eddy Simulation. All important flow phenomena, i.e. the formation of the fountain flow as
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well as the negative production of normal stresses near target wall are successfully predicted by the simulation[14].
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Smirnov[15-18] had studied parametrically the variation of the key parameters of this problem (e.g., the geometrical curvature of oblique shock emanating from the nozzle edge) based on the jet flow parameters. The results had proved that differential parameters of the flow field crucially depend not only of the key parameters, but on the symmetry type as well.
Our previous research mainly focus on the early two stages [19]: i) The switching oblique shock wave formation and boundary layer separation (Stage I); ii) The jet starts to separate from the right side of the wall layer (Stage II). The results have shown that the presence of the switching oblique shock near the control port is not the necessary condition
ACCEPTED MANUSCRIPT to make the jet deflect, but its formation is good for the early transverse extension of the stripping vortex zone. A new concept named minimum control mass flow rate has also been proposed and emphasized. However, the switching mechanism of the supersonic jet components still lacks of a deep understanding. The corresponding switching reasons
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are always simply attributed to the oblique shock and the Coanda effect. Based on this, our present research will further analyze the flow characteristics of the switching process and then reveal the specific switching mechanism for the jet supersonic components. The specific contents are listed as: i) The
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deflection of the jet (Stage III); ii) The jet starts to attach the wall layer (Stage IV) and iii) Adjustment stage (Stage
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V).
The paper is organized as follows. Section II describes the governing equations including a brief description of the adopted numerical algorithm. The validation method and results will be also presented and discussed. Section III will specifically discuss the flow field when the jet starts to deflect, attach the wall layer and adjust to stable eventually. The
II.
Simulation setup and comparison with the experimental data
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A. Simulation setup
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brief concluding remarks.
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changes for different parameters in the whole switching process are also investigated in details. The paper will end with
The flow field of supersonic jet elements always includes the strong separation flow, compression corner flow and shock wave-boundary layer interference flow. Therefore, the present research will adopt the RANS equation as the control equation, combined with the Realizable k-epsilon turbulence model, near-wall area processing, closed equations, boundary conditions, and initial conditions, as a mathematical model for numerical calculation of supersonic jet elements.According to the flow characteristics of supersonic jet elements, the volume force and heat source are
ACCEPTED MANUSCRIPT ignored, and the wall surface is assumed to be the insulation wall surface. The Continuous equation and Momentum equation in the Cartesian coordinate system can be rewritten as:
∂ρ ∂ + ( ρ ui ) = 0 ∂t ∂xi
(1)
where
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∂ ∂ ∂P ∂τ ij ∂ ( ρ ui ) + ( ρ ui u j ) = − + + ( − ρ ui′u j′ ) ∂t ∂x j ∂xi ∂x j ∂x j
(2)
ui is the pulsing value, τ ij is a viscous stress Tensor, which can be expressed as: 2 3
(3)
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τ ij = 2 µ Sij + ( µ ′ − µ ) S kk δ ij
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Sij is the strain rate of fluid motion:
1 ∂u ∂u j Sij = i + 2 ∂x j ∂xi
(4)
The turbulent energy(k) and turbulence dissipation(Ɛ) can be defined as:
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1 k = ui′u j′ 2
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∂u ′∂u ′ ε =ν i j ∂x j ∂x j
(5)
(6)
Transport equation of turbulent energy can be written as:
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∂ ∂ ∂ (ρ k ) + ( ρ ku j ) = ∂t ∂xi ∂x j
µt µ + σk
∂k + Gk − ρε − YM ∂x j
(7)
and the Transport equation of turbulence dissipation can be written as:
∂ ∂ ∂ ( ρε ) + ( ρε u j ) = ∂t ∂x j ∂x j
µt µ + σε
∂ε ε2 + ρ C1S ε − ρ C2 k + νε ∂x j
(8)
As shown in Fig.1(a), the supersonic jet element (SJE) is a kind of amplifying Element without moving parts and it uses gas as the working medium. Fig.1(b) shows the work mechanism of the SJE. Compared with the other flexible parts of the amplifier (such as pneumatic or hydraulic slide valve, rotary valve or diaphragm valve), the supersonic jet
ACCEPTED MANUSCRIPT element has many advantages, such as the low accuracy manufacture and low cost, which also has no moving parts, so the work process does not appear "stuck" in. As implemental components of the control system, the SJE’s work performance (control, switching time, energy efficiency, working reliability et al.) has an important role in the control
left spout
right spout
right sloping wall
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left sloping wall
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accuracy of the system.
right control gas
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left control gas
Main chamber pressure
(a) The physical model
(b) Work mechanism
(c) Computational grids
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Fig.1 The work mechanism and grids for the supersonic jet element
Fig.1(c) shows the surface girds of the SJE. In modeling the whole SJE, the quality and size of the mesh generation would often directly affect the accuracy of the experimental results. So during meshing the structural grids of the SJE,
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the simulation took the method of size function control grid size (It doesn’t change the entire structure of the network
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topology). And the parts of structural grid were refined for the whole Element, and the structural grid had been smoothed and swapped, enhancing the accuracy of the experiment. Due to the flow symmetry in x-y plane for the jet element, the simulation is two-dimensional. And the simulation is conducted using the unsteady ANSYS/FLUENT V.6.3.26. The corresponding unsteady mass flow rate can be changed by adopting the UDF Function code. The GaussSeidel method is used to calculate the inner fluid field of the supersonic jet element. The governing equations of flow are Reynolds Averaging Navier-Stokes equation. The model of turbulence is Realizable k-epsilon model, and Twolayer zonal model is used near the wall.
ACCEPTED MANUSCRIPT B. The comparison with the experimental data
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(a) The Validation of the Thrust in the Attached Wall Condition
Fig.2 The schematic diagram of jet components test system ①
Air Compressors Force Sensor
Throttle
Filter SJE
Air Cylinder
Decompression Pressure Valve
Three Links Directional Electromagnetic Control Valve
Cutoff Value
Pressure Table Adjustable
Both of the validation of the thrust in the attached wall condition and the verification of the switching time in
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switching process are completed by the single-channel air test system. This test system consists of high pressure and large flow of the pressure regulator system, data acquisition and processing system, moving frame of the jet components as well as the control components. The Fig.2 has shown the schematic diagram of jet components test system. The high
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pressure and large flow of the pressure regulator system includes a high-pressure compressor(WZ2.3/450) and a high pressure cylinder group (TGP-50L/35Mpa), which mainly provide the high pressure gas source and control flow for the
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jet components. The moving frame is used to fix the jet components, on which we would install two high frequency high-precision force sensors to detect the size of a combined thrust and thrust switch changes in the course of joint of the mobile frame. The data acquisition and processing system are used to collect the voltage signal coming from the collection force sensor.
There are various kinds of noise in the test system of jet element, in which the vibration of the jet element platform can strongly affect the accuracy of the test data. In order to eliminate noise and make the obtained thrust value meet the error requirements of the test, the system identification must be firstly carried out to obtain the mathematical model of the platform frame, and then according to the results of the system identification, the transient response analysis method will be adopted, calculating the rising time or falling time of thrust when the jet element is dynamically switched and the combined thrust value in the working state near the attached wall. By adjusting the pressure relief valve, the
ACCEPTED MANUSCRIPT different main gas source pressure needed for the experiment can be obtained, and the performance test of the jet element at different main gas source pressure can be carried out. The control flow into the supersonic jet element can be controlled by controlling the throttle valve, and the performance test of the jet element in different control flow can be carried out. The thrust in the attached wall condition is one important measurement to indicate the performance of supersonic jet components good or bad. Therefore, we have calculated the pressure thrust created by a variety of
the main gas source pressure [19].
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primary gas source pressure, and then under the same conditions, we have obtained the thrust value corresponding to
Ta ble 1 The comparison of the calculated and the experimental thrust fr(N) 49.65 84.34 108.91 142.09 164.59 170.27 186.87 220.68 263.05 296.36 346.18
fl(N) 0.42 1.95 4.43 4.58 5.66 5.46 6.43 8.55 18.04 21.78 27.89
F(N) 49.23 82.40 104.48 137.51 158.93 164.81 180.44 212.12 245.01 274.58 318.29
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Test(N) 44.0 79.9 106.0 126.3 142.6 161.4 179.3 202.8 239.9 275.4 313.2
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Ps (bar) 20 30 40 45 50 55 60 70 80 90 100
Error(%) 11.89 3.13 -1.43 8.87 11.45 2.11 0.64 4.6 2.1 -0.3 1.63
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Tab.1 has showed the thrust corresponding to the main gas source pressure (Ps) and test stands for the total thrust in the X direction of the jet components. fl and fr stand for their own thrusts of two outlets,
r r r F = f l + f r , Error stands
for the relative error between the calculation ( F ) and test. The maximum and minimum difference is within 12% and
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0.3%, respectively.
(b) Accumulation of errors estimate
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Estimating precision and errors accumulation is extremely necessary for large simulations of the complex fluid problem. And accumulation of error is proportional to the square rote of the number of time steps. It should be evaluated for each numerical simulation, especially for the unsteady flow state, which has been investigated by Smirnov in details[15-18] The corresponding research have revealed that the relative error of integration in 1D case S1 is proportional to the mean ratio of cell size ∆L to the computational domain size
L1 in the direction of integration
depending on scheme accuracy:
∆L S1 = L1
k +1
(9)
ACCEPTED MANUSCRIPT As referred by Smirnov[15-18], it can be simplified as: S1 ≈ (1 N1 )
k +1
in the uniform grid, where N1 is the
number of cells in the direction of integration. k is the order of accuracy of numerical scheme. Then the Eq.9 can be summed up as: 2
Serr ≈ ∑ Si
(10)
The allowable value of total error
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i =1
S max can be presumed from 1% to 5%[15-18], then the following inequality
should be satisfied as:
Serr ⋅ n ≤ S max
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(11)
where n is the number of time steps in Navier-Stokes equations integration. Then the maximum allowable number of
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time steps can be determined according to the Eq.11:
nmax = ( S max Serr ) 2
(12)
The ratio of maximal allowable number of time steps is introduced by Smirnov[15-18], which can be used to characterize the reliability of results. the higher is the value of nmax, the lower is the errors.
Rs = nmax n
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(13)
For this, we have conducted a representative case in the current research. The grid sensitivity test is performed subsequently. X stands for the nodes in the streamwise direction of the walls, and Y stands for the nodes in the transverse direction of the walls. i and j respectively stands for the nodes on the whole edges of the throat and control
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flow inlet(left and right). As shown in Fig.4, four different grid sizes (X×Y, i× j) do not deviate from each other within
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5%.
Allowable error ( %) 5 5 5 5 5 5 5 5
Grid resolution (X×Y, i) 260×50, 60×100 500×100, 100×150 1000×200, 200×260 2000×400, 400×500 260×50, 60×100 500×100, 100×150 1000×200, 200×260 2000×400, 400×500
Table.2 Error estimate
Physical time simulated (s) 0.06 0.06 0.06 0.06 0.6 0.6 0.6 0.6
Number of time steps 620 1312 2823 5800 7128 12600 31000 65000
Accumulated error 0.000197 0.000036 0.0000066 0.0000012 0.000680 0.000113 0.000022 0.000004
Allowable number of time steps 3.85107 2.46109 1.571011 1.011013 3.85107 2.46109 1.571011 1.011013
Reliability Rs=nmax/n 62097 1.86106 5.56107 1.74109 5401 1.95105 5.06106 1.55108
The detailed analysis for the accumulation of error on different combinations (iteration time step and grid resolution) have also been shown in Table.2. The accumulation of error for different grid resolution and physical time simulations have been studied in details. The allowable error is assumed to 5%. The accumulation of error will begin in
ACCEPTED MANUSCRIPT a fast manner for coarse grid and decreases on the increasing grid resolution and scheme accuracy, but decreases when the physical time increases. This phenomena has also been discussed by Smirnov[15-18]. For present simulations all the results demonstrate high reliability of the used program. but the number of time steps and physical time should not be
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chosen too large in order to avoiding the accumulation of error in the whole simulation process. Therefore, keeping the certain iteration accuracy and simultaneously avoiding too long simulated period should be the top priority. After the
subsequent computations.
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(c) The Verification of the Switching Time in Switching Process
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carefully consideration, we will adopt the relatively moderate computation domain size: [1000×200, 200×260 ] in the
The switching time is another important measurement to indicate the performance of supersonic jet components good or bad. We defined the switching time as from the beginning of the control flow changes to the reverse thrust on output became to be stable. In the experiment, the control signal is changed by changing the role of the control signals
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on the implementation of the electromagnet. Since the change from the control solenoid to control the flow signal changes in response between the solenoid armatures, pneumatic amplification mechanism, air flow through the control of channel and other areas of the delay, which is difficult to determine the starting of the switching time. Therefore, this
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research would be carried out by comparing the thrust rise time and thrust falling time obtained by calculating and
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experiment, which could verify the correctness of the unsteady calculation.
(a) Simulated thrust change
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(b) Experimental thrust change
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Fig.3 Thrust change during switching waveforms (simulation VS experiment) For the simulated result (shown in Fig.3(a)), Ps 100bar
ml=0.1kg/s
mr=0.003kg/s, and the thrust rise time is
1.11ms, the thrust falling time is 1.23ms. For the experimental result (shown in Fig.3(b)), it is the thrust change during switching waveforms in experimental value. Among them, the top of the square wave stand for the input control signal,
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the below purple curve is the combined thrust on output measured by two force sensors, the green curve is to eliminate noise and smooth thrust after the output. The thrust rise time is 1.18ms, and the thrust falling time is 0.85ms.
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Considering the accuracy of the measurement method and the simulation errors, the comparisons are showing
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reasonable overall quantitative agreement, showing the accuracy of the current simulation method.
C. Setup for the inlet control flow
Due to the fact that the switching time for the supersonic jet element operation is extremely short, the corresponding control signals will always show the typical quati-step functions, so the mass flow rate and static pressure for the both left and right inlet control flows can be assumed as a ramp function within a very short period (∆t) at the beginning of the switch, which can be written as:
ACCEPTED MANUSCRIPT m2 t<0 t mr = m2 + (m1 − m2 ) 0 ≤ t ≤ ∆t ∆t m1 t > ∆t
(14)
P1 t<0 t Pl = P1 + ( P2 − P1 ) 0 ≤ t ≤ ∆t ∆t P2 t > ∆t
P2 t<0 t Pr = P2 + ( P1 − P2 ) 0 ≤ t ≤ ∆t ∆t P1 t > ∆t
(15)
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m1 t<0 t ml = m1 + (m2 − m1 ) 0 ≤ t ≤ ∆t ∆t m2 t > ∆t
As shown in Table.3, ml and mr stands for the mass flow rate on left and right inlet control flow respectively, pl and
2×10 5s.
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decided by the ratio of p1 and p2, here ∆t 1×10-5
m2=0.003kg/s
p1=210000Pa
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pr will stand for the corresponding static pressure. m1=0.08kg/s
p2=57000Pa. ∆t is
Table 3 Setup for the parameters of the control flow Case
Boundary
Total pressure or
type
mass flow rate
Pressure
100×105
inlet
(Pa)
Static
5
pressure(10 Pa)
Temperature
Turbulence
(K)
parameters
Remarks
Turbulence intensity
ml
Mass-flow
0.08(kg/s)
inlet mr
0.57
—
1.0144
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outlet
4% —
Hydraulic diameter 0.0384(m)
Turbulence intensity
2.1
0.003(kg/s) Pressure
300
300
6% Hydraulic diameter 0.012(m)
300
Positive X-axial
k=1(m2/s2) e=1(m2/s3)
Negative X-axial reflux conditions
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lout
99.8
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Ps
Fig.4 The specific sizes of SJE The specific sizes of geometry have been marked in Fig.4. The inlet condition for the main gas source will be set as pressure inlet condition, left and right outlet condition will be set as pressure outlet condition. The left and right control flow inlet will be set as the mass flow inlet condition. Other surfaces will be set as the non-sliding insulation wall
ACCEPTED MANUSCRIPT condition. The air parameters are the following: temperature T=300K, the local heat flux
−λ (
∂T ) w determined based ∂y
on near wall turbulence model incorporating local flow characteristics. And the total pressure of the main gas source is
& =0.08 and 0.003kg/s, respectively. Left and right 100×105(Pa), the mass flux of left and right control inlet will be set m
Both the turbulent energy(k) and turbulence dissipation(Ɛ) are set as 1.0.
Results and discussion
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III.
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outlets are set as free pressure outlet condition. Turbulence intensity is 6% and the Hydraulic diameter is 0.012(m).
A. The changes of the different parameters
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F i g.5 s ho ws the location change of the spiral vortex center i n o ne p er io d ( t = 3.56ms ~ 6.32ms). Moreover, it can be seen that the main vortex center of supersonic jet element always moves within a small range in the centerline (X=0) during the cycle time. At the initial and end stage, the vortex center is very close to centerline, and we also find
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that when the primary vortex moves near the center line, the left and right thrust would become smaller.
Fig.5 Location change of main vortex center
Fig.6 Total pressure change of main vortex center
Figs.6 shows the total pressure change trend. There are two peaks of total pressure. Before the peak of total pressure coming, some of the control air enters the main vortex zone. With the air in the main vortex area increase, the pressure in the main vortex zone will rise. When the total pressure in the vortex zone became great enough, they will prevent the control air from entering the vortex zone, and compel the control air to eject from the left output. As the air
ACCEPTED MANUSCRIPT in the main vortex zone is ceaselessly swept away by the main jet and the left control flow, the total pressure in the main vortex zone begin to decrease. But when the total pressure dropped enough, part of the main jet will be inhaled into the main vortex zone, the total pressure of the main vortex would rise again. Thus, the total pressure of the main
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supersonic jet element at different time in one period.
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vortex area always continues to experience increase-decrease-increase cycle. Fig.7 shows the vortex distributing in
main vortex
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main vortex
Figure 7 The stream function contour at different time
Figure 8 Static pressure changes of right wall
Figure 9 Static pressure changes on concave wall
ACCEPTED MANUSCRIPT As shown in Fig.8, affected by the main vortex, the two peak static pressure on the wall is changing. The location of the first peak is moving upstream with the time increases, and the location of the second peak is moving more obvious in the first three times. However, at t = 4.08ms, the moving position is very small, and the static pressure increases larger. As shown in Fig.9, disturbed by the upstream flow, the static pressure value of the stagnation point keeps
stagnation point moves to the right output.
B. The deflection of the jet (Stage III)
(b) t=9.2×10-4s
(c) t=1.48×10-3s
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(a) t=7.2×10-4s
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Separated vortex zone
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(a) The changes for the stripping and attaching vortex zone
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oscillating between 8.6×105 ~ 8.7 × 105Pa. In the right thrust changes from the minimum to maximum process,
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Separated vortex zone
(d) t=1.64×10-3s
(e) t=1.84×10-3s
(f) t=2.04×10-3s
Figure 10 The changes for the stripping vortex zone (Stage III) As shown in Fig.10, before the stage III, the stripping vortex zone is reducing and expanding in the in the longitudinal and transverse directions respectively, the vortex center also has moved to the right output. When t=9.2×103
s, the vortex center has reached near the right output channel, the corresponding position also tends to be stable. Due to
the expanding stripping vortex zone, a small and narrow flow channel has been generated on the right side of the wedge
ACCEPTED MANUSCRIPT (shown in Fig.10(b)). When the supersonic air flows into this small channel, it can make the corresponding Mach number become smaller and increase the pressure. Once the static pressure for the right side of the wedge exceeds the left side, due to the effect of the pressure difference, part of the primary jet will flow into the left output channel. Thus, the transverse expansion for the stripping zone near the right output channel entrance can play a significant role in
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making the primary jet deflect successfully.
(b) t=9.2×10-4s
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(a) t=7.2×10-4s
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Attached vortex zone
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(d) t=1.64×10-3s
(e) t=1.84×10-3s
(c) t=1.48×10-3s
Attached Vortex Zone
(f) t=2.04×10-3s
Figure 11 The changes of the attached vortex zone (Stage III)
As shown in Fig.11, there are no obvious changes at the early time of the stage III. When t>1.48×10-3s, the attached vortex zone will shrink in the transverse direction. At mean time, the corresponding vortex center is also moving up and then flowing into the left output channel. (b) The changes of the Mach numbers As shown in Figs.12(a) and (b), at the early time of the stage III, the detached oblique shock wave near the wedge (GH) has gradually transformed into the detached bow shock wave. The strength of the oblique shock (IJ) near the right output channel has also been increased, and the corresponding position has also been moved up. With the stripped
ACCEPTED MANUSCRIPT vortex area continues to shrink in the longitudinal direction, the shock (EF) will gradually move up and the shock (CD)
(b) t=9.2×10-4s
(c) t=1.48×10-3s
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(a) t=7.2×10-4s
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will begin to extend to the right upper direction.
(d) t=1.64×10-3s
(e) t=1.84×10-3s
(f) t=2.04×10-3s
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Figure 12 The contour of the Mach number at different time (Stage III) When t=1.48×10-3s, the strength of the detached shock(GH) will further increases, the original oblique shock(IJ)
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will begin to transform into a normal shock. With the stripping vortex zone continuously expands, the oblique shock (EF) will move down and the shock (CD) will begin to reduce to the left lower direction. Due to the fact that the strength of the shock (IJ) gradually decreases, while the shock (KL) increases, when t=1.64×10-3s, both the strength of the shock (KL) and (IJ) are similar in the left and right output channel. After t>1.64×10-3s, the detached bow shock (GH) will be gradually transformed into the detached oblique shock, the shock (IJ) is also moving down and the corresponding strength is also weakening. The strength of the shock (KL) will experience decreasing then increasing trends. When t=2.04×10-3s, the shock (CD) and (IJ) will totally disappear (shown in Fig.10(f)).
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(c) The thrust changes for the left and right outputs
(a) left output
(b) Right output
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Figure 13 The thrust changes for the left and right outputs at different time (Stage III) As shown in Fig.13, before the stage III, due to the fact that the primary jet in the left output channel increases extremely slowly, so the left thrust has little changes. Except this, the increasing trend for the left thrust has two oscillations. When t> 1.48×10-3s, as the primary jet flowing into the left output increases gradually, the left thrust has increased sharply. This phenomenon also has shown on the right thrust. The direction of the combined thrust has
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changed at this stage. The thrust has shown an approximately linear relationship with the time.
C. The jet attaches wall layer again (Stage IV)
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(a) The shear stress distributions for the different wall layers
Y(m)
(a) t=2.24~2.8ms
Y(m)
(b) t=3.04~3.2ms
Figure 14 The shear stress distributions for the left wall layers at different time (Stage IV)
ACCEPTED MANUSCRIPT As shown in Fig.14, due to the flow pattern near the left attached vortex zone is the reverse flow, so the corresponding shear stress always shows negative. As the switching time increases, the lower end of the attached vortex zone will move to the positive Y direction. It indicates that the attached vortex zone is reduced gradually while the jet layer zone is expanded. When t=3.04×10-3s, the shear stress for two zones shows negative value shown in Fig.12(b). The first zone stands for the position of the separated bubble, which is generated by the interference between the
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switching oblique shock and layer boundary. The second zone will stand for the position of the attached vortex zone. When t =3.2×10-3s, the shear stress on the left layer will be negative, and the corresponding attached vortex zone has also disappeared.
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(b) The changes in the attached vortex zone
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Attached vortex zone
(b) t=2.4×10-3s
(c) t=2.6×10-3s
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(a) t=2.24×10-3s
(d) t=2.8×10-3s
Attached vortex zone
Separated bubble
(e) t=3.04×10-3s
Separated Bubble
(f) t=3.2×10-3s
Figure 15 The changes in the attached vortex zone at different time (Stage IV)
ACCEPTED MANUSCRIPT As shown in Fig.15, affected by the suction effect from the jet, the air in the attached vortex zone has been swept away gradually. This can make the corresponding pressure decreased, and the pressure balance in the two sides of the jet will also be broke. The jet will move to the left layer affected by the pressure difference, which can make the jet keep balance. However, as the air in the attached vortex zone continuously swept away, which can be reduced to a small vortex zone at the lower left output position (shown in Fig.15(f)). At this stage, if the input of the control flow is totally
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closed, the vortex zone at right side of the primary jet will have a higher static pressure. Due to the “Coanda” effect, the jet also can attach the layer successfully even though the corresponding control flow is totally closed.
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(c) The changes for the Mach numbers
(b) t=2.4×10-3s
(c) t=2.6×10-3s
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(a) t=2.24×10-3s
(d) t=2.8×10-3s
(e) t=3.04×10-3s
(f) t=3.2×10-3s
Figure 16 Contour of the Mach numbers at different time (Stage IV) As shown in Fig.16, when t=2.6×10-3s, the oblique shock near the left side of the wedge has disappeared, the strength of the switching oblique shock formed on the free shear layer at left and right primary jets has also been weakened. When t=3.04×10-3s, a new oblique shock has been generated on the free shear layer at the right side of the primary jet, whose strength is weaker than the switching oblique shock. Except this, another new oblique shock is also
ACCEPTED MANUSCRIPT generated between the attached vortex zone and separated bubble. As this moment, the oblique from the left output channel has been transformed into a stronger detached normal shock. In addition, the shear layer evolution in the primary jet flow field can also be observed in Fig.16. Due to the fact that the attached vortex zone is reducing and the wall layer zone is expanding, the free shear layer at left side of the primary jet has been shortening gradually. The free shear layer at right side will deflect to the left together with the primary jet.
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At the end of the stage IV, squeezed by the right side of the vortex zone, the free shear layer on the right side will show the expansion - compression - expansion condition (shown in Figs.16(d), (e) and (f)).
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(d) The changes for the left and right output
(a) Left output
(b) Right output
Figure 17 The changes for the left and right output (Stage IV)
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As shown in Fig.17, at the initial time of the this stage, the thrust for the left output has increased with the increasing switching time. When t=2.4×10-3s, the corresponding thrust has exceeded 452.5N. After that, the
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corresponding thrust will begin to decrease with the strength of shock from left output increases. When t> 2.92×10-3s, due to the right control flow and a lot of primary jet flow into the vortex zone through the left side of the wedge, the air ejected from the left output channel has been reduced sharply. Except this, a stronger detached normal shock has been generated in the left output channel, and the corresponding thrust is decreasing sharply. When t=3.2×10-3s, the thrust has decreased to less than 281N. At the very beginning time of the stage IV, due to some parts of the right control flow will be ejected from the right output, the corresponding thrust should be positive. However, at the later time, the outside air has been inhaled into the supersonic jet Element from the right output, the right output will generate a reverse thrust compared with the left output. As the inhaled air continuously increases, the corresponding reverse thrust will increase gradually, the maximum value can exceed 37.5N. From t=3.04×10-3s to 3.2×10-3s, due to the outside air flowing into the jet Element
ACCEPTED MANUSCRIPT has been decreased sharply, as a result, the reverse thrust generated by the right output also has been reduced greatly. At the end of the stage IV, the right thrust has recovered to be positive.
D. Adjustment stage (Stage V)
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(a) The thrust changes for the left and right outputs
(a) Left output
(b) Right output
Figure 18 The thrust changes for the left and right outputs Figs.18(a) and (b) have shown the thrust changes process for the left and right outputs including the adjusting and
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adjusted stages. In the adjusting stage, due to the airflow in the jet Element is extremely unstable, as a result, the corresponding thrust has shown a very substantial oscillation state. After a period of self adjustment, the corresponding flow will become relatively stable, and the oscillating amplitude has also greatly reduced. (b) The formation of the primary vortex zone
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Core 2
Vortex Core 1
Vortex Core 2
(a) t=3.36×10-3s
Vortex Core 3
Vortex Core 1
Vortex Core 3
Vortex Core 4
(b) t=3.64×10-3s
Vortex Core 1 Vortex Core 4
(c) t=3.84×10-3s
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Vortex Core 3
(d) t=4.04×10-3s
Main Vortex Zone
Vortex Core 5
(e) t=4.48×10-3s
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Vortex Core 4
(f) t=5.00×10-3s
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Figure 19 The total pressure distributions at different time (Stage V)
As shown in Fig.19, when t=3.36×10-3s, squeezed by the left concave wall, some parts of the primary jet have
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flown back the jet Element center, where has generated two vortex zone (zone 1 and 2) on the left and right sides. When t=3.64×10-3s, the domain of the zone 1 has expanded and the corresponding vortex center has also moved. However, the zone 2 has left the Element center and entered the right output channel. At this time, a long narrow vortex zone (zone 3) has been generated near the left side of the wedge and zone 4 has been generated at the upper right control flow position. When t=3.84×10-3s, the domain of the zone 1 has been reduced, and the corresponding vortex center has moved to the
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right side of the wall. At the same time, the domains for the zone 3 and 4 have also expanded. When t=5.00×10-3s, the vortex center of the zone 5 will move to the lower right position, a new larger primary vortex zone has been generated in the jet Element center (shown in Fig.19(f)). The formation of the primary vortex zone means that the flow inside the
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jet Element has tended to be stable. Which also means that the end of the switching process.
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(c) The static pressure changes for the left and right sides
(a) Left attached wall
(b) Right attached wall
Figure 20 Static pressure distributions for the left and right attached wall at different time
ACCEPTED MANUSCRIPT As shown in Fig.20(a), for the left attached wall, the static pressure has always firstly increased and then decreased. It has indicated that the jet flowing along the layer has experienced a strong compressing and expanding process. At the early time of the adjusting stage, the static pressure peak always changes obviously (dropped from 8.2×105Pa to 3.7×105Pa). At the later time of the adjusting stage, the corresponding peak has been greatly reduced (oscillating from 4.6×105Pa to 5.7×105Pa). In addition, the positions on the left side for the static pressure peak are also distinct at
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different time. The nearest and farthest position away from the left control flow is at t=3.64×10-3s and 4.04×10-3s respectively. Fig.20(b) has shown the phenomena on the right attached wall. Compared with the left, the static pressure peak is not very obvious, especially at t=3.84×10-3s and 4.04×10-3s, the corresponding change is extremely flat.
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(d) The shear stress distribution for the wall r1 in Y direction
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Figure 21 The shear stress distribution for the wall r1 and left attached wall As shown in Fig.21, due to the presence of the switching oblique shock, the flow has separated on the wall r1, and the separation point can be decided by the shear stress in Y direction. At the early adjusting stage, the position for the
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separation point always changes greatly. The farthest away from the right control flow input is happened at t=3.64×10-3s. However, at the end time of the adjusting stage, the position changes very little. With the airflow inside the jet Element
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becomes stable, the strength and position for the switching oblique shock will be fixed. (e) The shear stress distribution for the left attached wall
Figure 22 The shear stress distribution for the left attached wall
ACCEPTED MANUSCRIPT Fig.22 has shown the shear stress distribution for the left attached wall in Y direction at different time. In general, due to the interference between the switching oblique shock and left attached layer, there will exist a separating bubble near the left layer. The negative shear stress part means the specific position of the bubble. When t=3.64×10-3s, the strength of the switching oblique shock is very strong, and the corresponding angle is also very large. It can connect with the separated bubble and low pressure vortex zone near the left control flow input. When t=4.04×10-3s, the strength
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of the switching oblique shock however will becomes weaker, and the separating bubble is also very far away from the left control flow. Except this, there is another new separating bubble has been generated at the downstream of the first separated bubble.
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(f) The changes of the Mach numbers
Separated bubble 1
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Separated bubble 2
(b) t=3.64×10-3s
(c) t=3.84×10-3s
(e) t=4.48×10-3s
(f) t=5.00×10-3s
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(a) t=3.36×10-3s
(d) t=4.04×10-3s
Figure 23 Contour of the Mach numbers at different time (Stage V) As shown in Fig.23, there are two aspects have more obvious changes, one is the free shear layer at the right side of the primary jet, the shape and thin will be changed with the changing domain, strength and the position of the vortex center. The other one is the changes of the shock in the left output channel. At t=3.64×10-3s and 3.84×10-3s, the corresponding shock is a weak oblique shock. However, at t=4.04×10-3s and 4.48×10-3s, it has been changed to a normal shock.
ACCEPTED MANUSCRIPT When t=4.04×10-3s, due to the strength of the switching oblique shock is extremely weak, the increasing extent for the pressure is very small and the flow direction also has no changes. When the jet with the low pressure and high speed flows into the free shear layer near the zone 3 at a larger angle, a stronger oblique shock will be generated near the free shear layer, which can make the boundary layer begin to separate, and another separating bubble will be formed at the
(a) P s=2 0 ×1 0 5 P a
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downstream of the previous separated bubble (shown in Fig.23(d)).
(b) P s=6 0 ×1 0 5 P a
(c) Ps=1 2 0 ×1 0 5 P a
Fig.24 Shock wave distribution on the right convex wall Fig.24(a),(b) and(c) are the shock wave distribution near the right convex wall in the case of Ps=20×105 Pa, Ps=60×105Pa and Ps=120×105Pa, respectively. It can be seen that when the jet flows through the right convex wall,
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different degrees of expansion have occurred. Afterwards, due to the effect of external reflux or internal vortex, a clear shock wave has been generated at the right convex wall. When the jet flow through the shock wave, the higher back pressure will be formed and cause the flow separation eventually."
Conclusion
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Based on the unsteady viscous flow simulation, the flow characteristics inside the supersonic jet Element have been
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investigated numerically. The corresponding switching mechanism and conclusions have been given as follows: (1) The instability of the main vortex center is a main reason to make the output thrust of the supersonic jet components fluctuate all the time in attached wall condition. (2) At the deflection of the jet stage (Stage III), Once the static pressure for the right side of the wedge exceeds the left side, the transverse expansion for the stripping zone near the right output channel entrance can play a significant role in making the primary jet deflect successfully. The direction of the combined thrust has changed at this stage. The thrust has shown an approximately linear relationship with the time. (3) When the jet starts to attach the wall layer (Stage IV), if the input of the control flow is totally closed, the vortex zone at right side of the primary jet will have a higher static pressure. Due to the “Coanda” effect, the jet also can attach the layer successfully even though the corresponding control flow is totally closed.
ACCEPTED MANUSCRIPT (4) When the jet enters the adjustment stage (Stage V), the thrust changes process for the left and right outputs including the adjusting and adjusted stages. In the adjusting stage, due to the airflow in the jet Element is extremely unstable, as a result, the corresponding thrust has shown a very substantial oscillation state. After a period of self adjustment, the corresponding flow will become relatively stable, and the oscillating amplitude has also greatly reduced.
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Acknowledgments Financial supports from National Science Foundation of China [Grant number 11602023 and U1430113] are
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Vol. 136, 2017, pp. 342-353.
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Highlights The instability of main vortex center is a main reason to make the output thrust fluctuate.
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The transverse expansion for the stripping zone can make the primary jet deflect successfully.
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Jet can attach the layer successfully due to “Coanda” effect even though control flow is totally closed.
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The thrust changes process will experience two typical stages: adjusting and adjusted stages.
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