Numerical Analysis and Optimization of Three-Dimensional
MHD Controlled Inlets
Haoyu Lu ∗ , Chun-Hian Lee ** , Haitao Dong***, and Deyi Cao*
National Lab. for Computational Fluid Dynamics
Beihang University, P.R. China,100083
lichx@buaa.edu.cn
Abstract
Design and optimization of three-dimensional MHD controlled inlets by means of numerical
simulation were devoted in the present study. An airframe/propulsion integrated module was first
established for a waverider-based hypersonic vehicle at design Mach value of 6. Utilizing this
generic configuration, a MHD-based technique for large-scale inlet flow control was studied at the
off-designed Mach value of 8.5. Using mass flow rate as an optimization criterion, an MHD
generator possessing reasonable design parameters would restore the flow field and shock system
as if the scramjet was operated under the design condition, and made the flow characteristics at the
combustor entrance more favorable when flying at higher Mach numbers than the design value. In
order to bring shocks back to the cowl lip, extracting more electric power from the external MHD
generator, and improving the performances of the combustor entrance, the load factor should be
elevated.
1. Introduction
R
ESEARCH on magnetohydrodynamics (MHD) and the potential applications in the area of hypersonic
technology have become increasingly active in many countries. In the early nineties of the last century, Russian
had proposed a new concept on combined propulsion system consisted of a turbine and an MHD enhanced
scramjet for hypersonic cruised aircraft, and established the AJAX program 1,2. The most innovative and principal
issue about this program is how to utilize the MHD system to develop an MHD bypass scramjet technology and to
control the inlet of the propulsion system. According to the AJAX program, the inlet control can be furnished by an
MHD device through the modification of the external flow upstream of the scramjet inlet to provide improvement on
the engine performance at off-design conditions 3~5.
By virtual of unfavorable interferences between the airframe and propulsion module, the overall, fully
integrated performance of the design will be paramount to the success of any hypersonic vehicle 6. Multiple inlet
ramps may generate lift and minimize drag as the flow being pre-compressed upstream of the engine inlet, while the
cowl lip further compresses the flow for the combustor. As a direct consequence, the thermoflow characteristics of
the inlet, such as the total pressure recovery, pressure rise, air capturing factor, etc., can be improved. Nevertheless,
at fight Mach number higher than the design value, the shocks emanated from the nose of the vehicle and corners of
the ramps would make contact with the surface of inlets downstream the cowl lip, creating multiple shock
reflections inside the inlet. Consequently, the flow field inside the inlet would become nonuniform and cause losses
on total pressure, possibly the onset of boundary layer separation, and the engine unstarted, etc.
In this work, an airframe/propulsion integrated module is constructed for a generic waverider-based hypersonic
vehicle at the design Mach number 6. Three inlet ramps are taken to compress the air flow while shocks emanated
from these ramps converge to the cowl lip, and the reflecting shocks are annihilated at the end point of the third
ramp. The inlet further compresses the flow along the spanwise direction downstream the inlet ramps, and thus
increases the static pressure as an isolator. Taking the above generic configuration as the baseline design, a
feasibility of utilizing MHD principle for a large-scale inlet flow control is studied at the off-designed Mach value of
8.5 by means of numerical simulation. The electrical conductivity in the air is assumed to be generated by electron
beams injected along the magnetic lines, and a mathematical model is proposed accordingly. Euler equations with
∗
Graduate Student
Professor
***
Associate Professor
**
1
ponderomotive force and energy interaction terms are employed for the numerical simulation. A second order
entropy conditioned scheme proposed by Dong et al 7,8 will be employed to solve the model equations. The scheme
requires no adjustable parameters to guarantee the satisfaction of the entropy condition.
2. Airframe/Propulsion Integrated Inlet
Due to severe interferences between the airframe and propulsion module, the airframe/propulsion integrated
inlet design becomes one of the crucial parts for the overall, fully integrated performance of the vehicle design (cf.,
Fig.1). Hypersonic vehicle inlets can be categorized into two types, namely, the external and the side-wall
compression inlets. Comparative study on those two types of inlets was performed by Chen, et al 9, where the
computational and experimental results indicated that the external compression inlet would provide higher mass
flow capturing ratio and pressure rise, while the side-wall compression inlet might produce higher total pressure
recovery.
Forebody
Inlet
Combustor
Nozzle
Shock System
Fig.1 Schematic sketch of airframe/propulsion integrated hypersonic vehicle.
A practical inlet should be realized as some compromises of both. In this paper, we present an external/sidewall
mixed compression inlet as a part of the forebody whose lower surface may induce a multi-stage compression.
Under this situation, shocks emanated from the nose of the vehicle and corners of the ramps will make contact with
the inlet surface downstream the cowl lip, and the reflecting shocks will be annihilated at the end point of the third
ramp. The inlet further compresses the flow field along the spanwise direction downstream the inlet ramps, with
which the intercrossed shocks will disappear at the corners of spanwise sidewalls, as shown in Fig. 2.
θ1
θ2
θ3
M∞ٛ
α
θ4
θ5
Fig.2 Schematic sketch of an external-sidewall mixed compression inlet.
Parameters relevant to the optimal design of the inlet consist of the five compression angles θ1 ~ θ5 as depicted
in Fig. 2 on the upper surface and the sidewalls of the inlet, and the length of the forebody Lf (i.e., the axial distance
2
from the nose to lip). As long as the angle of attack, α , and the reflecting angles θ1 ~ θ5 are given, the inlet
geometry can be determined provided that the surface of the first ramp and the downstream surface of the inlet are
kept parallel.
In order to achieve a high air capturing factor, total pressure recovery, and the pressure rise at the inlet exit, we
select the relevant parameters as Lf = 25m, α = 5o , θ1 = 2.24 o , θ 2 = 2.34o , θ 3 = 2.45o , θ 4 = 4.07 o , θ 5 = 4.33o ,
respectively, according to the computational results obtained by the theory of oblique shock.
3. MHD Controlled Inlet at Off-Design Condition
In order to make up the performance penalties under the off-design condition, inlets can be designed as
variable-geometry capable of providing adequate performances for the Mach number under consideration and the
operating range of the angle of attack. However, the mechanical system for preserving the variable-geometry would
increase substantially the weight and complicate the design tasks of the hypersonic vehicle. It was proven that an
alternative method based on MHD control could improve the inlet performances 3~5.
By creating near-surface magnetic field though magnetic coil and non-equilibrium electrical conductivity
utilizing electron beams along the magnetic lines, MHD generator mounted on the first ramp near the nose of the
forebody may extract energy from the high temperature gas. By virtue of the work done by Lorentz force, j×B, the
flow downstream the MHD interaction region decelerates, altering the structures of the flow field, in particularly the
configuration of the shock system. Through optimizing the relevant parameters of the MHD generator and the
electron beams, MHD generator can bring shocks back to the cowl lip and the reflecting shocks are annihilated on
the inlet surface, as shown in Fig.3.
e-beams
external MHD generator
external flow
B
z
y
combustor
x
Anode Cathode
jy
z
y
B
flow
Fig.3 Schematic sketch of an MHD controlled inlet.
Realization of the MHD controlled inlets leads to some critical issues remained to be verified, such as the
position and length of the MHD interaction region, the distribution of the magnetic flux density in the interactive
region, etc. As discussed in Refs. 5 and 10, and especially, the arguments presented in some details in Ref. 11, two
criteria can be allocated for the parameters, namely,
a) the MHD interaction region should be placed as far upstream as possible;
b) the MHD interaction region should be close to the magnetic coil centerline.
3
In what follows, we shall investigate further the rules played by those parameters to the MHD controlled inlet at offdesign conditions through numerical simulation.
4. Model Equations for MHD Controlled Inlets
4.1 Euler Equations with MHD Interaction
It suffices to consider invicid MHD flows for the present study. The three-dimensional Euler equations with
ponderomotive force and energy interaction terms are nondimensionalized by the following dimensionless
parameters:
t* =
tu∞ *
x
y
ρ * u * v
w *
p
,x =
, y* =
, ρ* =
,u =
,v =
, w* =
,p =
,
L∞
L∞
L∞
u∞
u∞
u∞
ρ∞
ρ∞ u∞2
T* =
Ey0
B
T
σ
, BZ* = Z , σ * =
,k =−
.
T∞
Bmax
u∞ Bmax
σ max
and can be written in Cartesian coordinate system ( x, y, z ) as
0
⎡ρ ⎤
⎡ ρu ⎤ ⎡
⎤
∂ ⎢ ⎥
⎢
⎥
⎢
u + ∇ ⋅ ρ uu + pI =
ρ
Qj × B ⎥
⎢
⎥ ⎢
⎥
∂t ⎢ ⎥
⎢⎣ ρ e ⎥⎦
⎢⎣ ( ρ e + p )u ⎥⎦ ⎢⎣Qj ⋅ E + ψ I ⎥⎦
(1)
where, ρ , u , p , and T are, respectively, the density, velocity, pressure, and temperature of the flow; j, E, and B
are, respectively, the current density, magnetic and electric fields; E y 0 is the y-component of the induced electric
field, ψ denotes the recombination factor of the ions, σ the electric conductivity, and k the loading factor.
Noting that the asterisk denoting the dimensionless quantities has been dropped in Eq. (1), and in what follows,
for simplicity without causing any confusion. The resulting dimensionless parameters are defined as follows:
Magnetic interaction parameter: Q =
2
σ max Bmax
L∞
ρ ∞ u∞
Ionization energy consumption parameter: I =
qion L∞
ρ ∞ u∞
3
Here, qion is the energy injected by electron beams which is given by
qion =
jbε b
Δz ( x )
where jb denotes the electron beam current, ε b the electron beam energy in electron volt, and Δz ( x ) is the depth
in flow field from the surface of forebody.
According to Refs.11 and 12, the electrical power provided by electron beams for ionization is substantially
lower than the power generated by the external MHD generator. Hence, we may neglect the ionization power in our
calculations.
4.2 Definition of Electric Field
The electric current density and electric field must be specified prior to solve the Euler equations 13. Assume
that the velocity component in x-direction, the magnetic flux density on y-axis, and the electrical conductivity are
given, respectively, as u = u ( x, y , z ) , Bz = Bz ( x, y , z ) , and σ = σ ( x , y , z ) . The electromotive force between the
electrodes on the MHD generator is given as E y 0 H , where, H the distance between the anode and cathode. Then,
the total current produced by the external loads gives
4
I = ∫∫∫ − j y ( x, y , z )dxdydz
x, y,z
= ∫∫∫ σ uBz dxdydz − E y 0 ∫∫∫ σ dxdydz
x, y,z
(2)
x, y ,z
Applying Ohm law for the whole circuit, we obtained
⎛
⎞
E y 0 H = IR = R ⎜ ∫∫∫ σ uBz dxdydz − E y 0 ∫∫∫ σ dxdydz ⎟
⎜
⎟
x ,y ,z
⎝ x ,y ,z
⎠
(3)
where R is the electrical resistance of the external loads. The internal electric resistance is readily given as
r=
H
∫∫∫ σ dxdydz
(4)
x ,y ,z
Thus, by solving Eqs. (2) through (4), we may obtain the electric field as
Ey0 =
k ∫∫∫ σ uBz dxdydz
x ,y ,z
∫∫∫ σ dxdydz
(5)
x ,y ,z
Here, the loading factor k =
R
.
R+r
As for the present interests, the Faraday’s electric current density can be determined with the aid of Eq. (5) as
j y ( x, y , z ) = σ ( x, y , z ) ⎡⎣u( x, y , z ) Bz ( x, y , z ) − E y 0 ⎤⎦
(6)
4.3 Electrical Conductivity Model
There are many possible approaches to create nonequilibrium conductivity. However, in all of these cases, the
electron beams have demonstrated to be an effective one for cold gas, the energy consumed for creating an ionized
molecule or atom is Wi=35eV for air 14, which is only a few times greater than the ionization energy of an air
molecule. Kuranov and Sheikin 15 argued that, when the recombination of electrons and positive ions becomes the
dominant part in the reaction process, the electron density can be written as
ne ≈
qion
Wi ken
where ken denotes the recombination coefficient. Thus, the conductivity of the plasma can be expressed as
2
σ=
2
e ne
qion
e
=
mnk en mnk en Wi k dr
where, m is the electron mass, n the concentration of the neutral molecule, and ne the electron concentration.
The energy deposited by electron beams, qion , depends on the spatial distribution of electric conductivity. By
utilizing the “forward-back” method in conjunction with Monte Carlo calculations (MCC), Macheret 4 found that the
energy deposition was very close to the Gaussian distribution:
qion (ξ ) = a +
⎡ ⎛ ξ − zm ⎞ 2 ⎤
b
exp ⎢ −2 ⎜
⎟ ⎥
w
⎣⎢ ⎝ w ⎠ ⎦⎥
where, ξ = Δz ( x ) , zm ≈ LR / 3.21 , w ≈ 1.64 zm . Here, LR is the beam effective length of ionization. There consist
two constraints for the energy deposition, namely,
5
LR
∫ q (ξ ) d ξ =
qion ( LR ) = 0 , and
ion
0
jb ε b
e
where the electron beam energy, ε b , and effective length of ionization, LR , can be as follows:
εb
1.7
LR = 1.1 × 10
21
n
Consequently, we may optimize the effective length of ionization, LR , via adjusting the electron beam energy, ε b ,
to make the MHD generator more operationally effective.
According to the analysis given above, we may establish the electric conductivity for solving 3-D Euler
equations (1). Here, we are simply taking the normalized conductivity into account which is given as
σ = C 1+
⎛ ⎛ ξ − zm ⎞ 2 ⎞
A
exp ⎜ −2 ⎜
⎟
⎜ ⎝ w ⎟⎠ ⎟
w
⎝
⎠
(7)
Here, the quantities A and C can be determined by the following two constraints:
σ ( LR ) = 0 , σ max (ξ ) = 1
5. Numerical Formulation
The second order entropy conditioned scheme is employed to solve the 3-D Euler equations due to the fact that
the scheme requires no adjustable parameters to guarantee the satisfaction of the entropy condition7, 8.
The numerical flux along i-direction can be expressed as
δ Fi = F
i+
F
i−
F
i+
F ( Qi − 2 , Qi −1 , Qi , Qi +1 , ξ x , ξ y , ξ z ) = F
1
2
1
2
E
1
2
−F
i−
1
2
= F ( Q i −2 , Q i −1 , Q i , Q i +1 , ξ x , ξ y , ξ z )
= F ( Q i −1 , Q i , Q i +1 , Q i + 2 , ξ x , ξ y , ξ z )
( Qi −1 , Qi ) + R i + 1
2
⎧λl
⎪⎪ i − 12
⎨
⎪ 2
⎪⎩
⎛ dL −dR1
i−
⎜ i − 12
2
⎜
2
⎜
⎝
and likewise for the fluxes along j- and k-direction. Here,
d
d
L
1
i−
2
R
1
i−
2
⎛
⎞
⎛ l
⎞
l
l
= ⎜ 1 − λ 1 Δt ⎟ superbee ⎜ L 3 , L 1 ⎟
i−
2
⎝
⎠
⎝ i− 2 i− 2 ⎠
⎛
⎞
⎛ l
⎞
l
l
= ⎜ 1 + λ 1 Δt ⎟ superbee ⎜ L 1 , L 1 ⎟
i−
⎝
⎠
⎝ i− 2 i+ 2 ⎠
2
L
R
Λ
i−
1
2
⎧ l ⎫
E
= ⎨L 1 ⎬ = L ( Qi −1 , Qi )
i−
⎩ 2⎭
i−
1
2
⎧ l ⎫
E
= ⎨R 1 ⎬ = R ( Qi −1 , Qi )
i−
⎩ 2⎭
i−
1
2
⎧ l ⎫
E
= ⎨λ 1 ⎬ = Λ ( Qi −1 , Qi )
i−
⎩ 2⎭
6
⎞ λl 1
i−
⎟
2
⎟−
2
⎟
⎠
⎫
R
L
⎡
d 1 + d 1 ⎤⎪
i−
i− ⎥ ⎪
⎢ l
2
2
⎢ Li − 1 −
⎥⎬
2
⎢ 2
⎥⎪
⎦⎪
⎣
⎭
where, F E , LE , R E and Λ E represent, respectively, the flux, left eigenvector, right eigenvector, and the matrix for
eigenvalues.
The entropy conditions are readily given as
λ ( Q− ) ≥ λ ( Q+ )
⎧ El
l
⎪ λ ( Q − , Q+ ) = λRoe ( Q− , Q+ )
⎪⎪ E l
l
⎨ L ( Q− , Q+ ) = L Roe ( Q− , Q+ )
⎪ E
F Q + F ( Q+ )
⎪ F ( Q− , Q+ ) = ( − )
⎪⎩
2
λ ( Q− ) < λ ( Q+ )
l ⎛ Q− + Q+ ⎞
⎧ El
⎟
⎪λ ( Q − , Q + ) = λ ⎜
2
⎝
⎠
⎪
⎪ El
l ⎛ Q− + Q+ ⎞
⎨ L ( Q− , Q+ ) = L ⎜
⎟
2
⎝
⎠
⎪
⎪ E
⎛ Q− + Q+ ⎞
⎪ F ( Q− , Q+ ) = F ⎜
⎟
2
⎝
⎠
⎩
l
l
l
l
6. Numerical Results
In order to investigate the MHD characteristics over the forebody, numerical computations of MHD flows over
the designed airframe/ propulsion integrated inlet (cf. Sec.2) at design and off-design conditions were performed.
The results are presented in the followings.
6.1 Numerical Results at Design Condition without MHD Control
Recall that the designed Mach number M ∞ = 6 , relevant parameters Lf = 25m, α = 5o , θ1 = 2.24 o , θ 2 = 2.34o ,
θ 3 = 2.45o , θ 4 = 4.07 o , θ 5 = 4.33o , and the freestream conditions at designed Mach number are listed in Table 1.
The grid is divided into three blocks, as shown in Fig.4, where the block 1 (red) consists of 77× 49×59 grid points,
block 2 (green) consists of 100× 49×40, and block 3 (blue) consists of 50×49×20, respectively.
Table 1 Freestream conditions at designed Mach number
H/km
ρ/kg/m3)
p/pa
M
35
8.463×10-3
574.6
6
Fig.4 Computational grids
7
The shock system and flow profile in the integrated inlet are depicted in Fig.5 and 6. Computational results
reveal the features as expected. Shocks emanated from the forebody make a contact with cowl lip, and afterwards
reflect to the end point of the third ramp where the shocks are annihilated as illustrated in Fig.5. The inlet further
compresses the flow in spanwise direction. Intercrossed shocks downstream the inlet ramps disappear at the corners
of spanwise surfaces. The nature of the flow field can also be seen from the distribution of stream line, Mach
number, pressure and temperature on the x-z mid-plane as depicted in Figs.6-9, respectively, and the pressure
distribution on the x-y mid-plane shown in Fig.10. The flow field inside the inlet turns uniform due to the
disappearance of the shock system.
Fig.5 Shock system in Mach 6 design case
Fig.6 Stream lines on x-z midplane in Mach 6 design case
Fig.8 Pressure contours on x-z mid-plane in Mach 6 design
case
Fig.7 Mach number contours on x-z mid-plane in Mach 6
design case
Fig.10 Pressure contours inside inlet on x-z mid-plane
in Mach 6 design case
Fig.9 Temperature contours on x-z mid-plane in Mach 6
design case
8
6.2 Numerical Results at Off-Design Condition without MHD Control
As for the same geometry designed for Mach 6, an inlet flow with neutral air is first studied at the off-designed
Mach value of 8.5 for comparative reason. To keep the dynamic pressure at the inlet about the same as the designed
Mach number 6, the flight altitude should be raised to 37 km to reach Mach 8.5. We choose this flight environment
as the freestream conditions for the test case, and are listed in Table 2. The numerical grids used in the computations
are the same as in the previous test case for the designed condition.
Table 2 Freestream conditions at off-design Mach number 8.5
H/km
ρ/kg/m3)
p/pa
M
37
6.235×10-3
433.2
8.5
The shock system and flow field in the integrated inlet are illustrated in Figs.11-16. It is seen that the shocks
emanated from the nose of the vehicle and corners of the ramps would interact with the inlet surface downstream the
cowl lip, creating multiple shock reflections inside the inlet. Comparing with the results computed for the designed
case given in the Sec.6.1, it is seen that the flow field inside the inlet becomes highly disturbed. One may anticipate
that the non-uniformities would deteriorate the performances of the inlet noticeably.
Fig.11 Shock system at Mach 8.5 off-design case with no
MHD interaction
Fig.12 Stream lines on x-z mid-plane at Mach 8.5 offdesign case with no MHD interaction
Fig.13 Mach number contours on x-z mid-plane at Mach
8.5 off-design case with no MHD interaction
Fig.14 Pressure contours on x-z mid-plane at Mach 8.5 offdesign case with no MHD interaction
9
Fig.16 Pressure contours inside inlet on x-z mid-plane in
Mach 8.5 off-design case with no MHD interaction
Fig.15 Temperature contours on x-z mid-plane at Mach 8.5
off-design case with no MHD interaction
6.3 Numerical Results for MHD Control Inlet Flows at Off-Design Condition
A parametric study on the MHD-based large-scale inlet flow control at off-designed Mach value of 8.5 is
performed over the forebody/inlet system with the same geometry. Following the two criteria stated in Sec.3
concerning the location and length of the MHD interaction region, the MHD generator is located upstream the lip of
the inlet, 10m away from the nose of the forebody (cf. Fig.3). While taking the spanwise length of the forebody into
account, the MHD interaction is in the range of 10m ≤ x ≤ 10.5m . The magnetic profile along z direction is assumed
to be Gaussian, and can be expressed in a normalized form having a maximum value of one on the wall as
⎡ ⎛ Δz ( x ) ⎞ 2 ⎤
B = exp ⎢ − ⎜
⎟ ⎥
⎢⎣ ⎝ 16 ⎠ ⎥⎦
According to the analysis given above, the performances of the MHD generator on inlet flow control are
determined by two dimensionless parameters, namely, the magnetic interaction parameter, Q, and the load factor, k.
The magnetic interaction parameter is contributed by the maximum electrical conductivity and magnetic flux density.
If magnetic interaction parameter and incident flow condition are given, we have
2
σ max Bmax
=
Q ρ ∞ u∞
L∞
As far as the MHD inlet performances are concerned, parametric study will be performed primarily on three
parameters, including the effective length of ionization, LR, load factor, k, and magnetic interaction parameter, Q.
There are five test cases with different parameters being presented here as listed in Table 1. The numerical grids
Table 3 Computing Cases
Case1
Case2
Case3
Case4
Case5
k
0.8
0.8
0.8
0.8
0.5
LR
2
3
4
5
3
used in the computations are the same as in the previous test cases. For all cases, the flow field will be restored to
the one at design condition, i.e. the one satisfying the ‘shock-on-lip’ (SOL) condition, by adjusting the magnetic
interaction parameter. As an illustration, the electrical conductivity distribution for the case of LR = 3m is depicted in
Figs.17 and 18.
The flow filed in the integrated inlet for k = 0.8, Q = 1.3, and LR = 3m is given in Figs.19-23. It is seen that the
presence of MHD generator causes the shocks move back to the cowl lip, and the reflecting shocks interact with the
10
end point of the third ramp, where shocks spread downstream weakly. The flow field downstream the third ramp is
very similar to the designed Mach 6 case. It is seen that the flow field inside the inlet returns to uniform under the
effects of MHD interaction.
Fig.17 Electrical conductivity on x-z mid-plane at Mach
8.5 off-design case with MHD
Fig.18 Electrical conductivity along magnetic line at Mach
8.5 off-design case with MHD
Fig.19 Stream lines on x-z mid-plane at Mach 8.5 offdesign case with MHD interaction, k=0.8
Fig.20 Mach number contours on x-z mid-plane at Mach 8.5
off-design case with MHD, k=0.8
Fig.21 Pressure contours on x-z mid-plane at Mach 8.5 offdesign case with MHD interaction, k=0.8
Fig.22 Temperature contours on x-z mid-plane at Mach 8.5
off-design case with MHD interaction, k=0.8
11
Fig.23 Pressure contours on x-y mid-plane at Mach 8.5 offdesign case with MHD interaction (k=0.8)
6.4 Parametric Analysis
Key parameters for maintaining the performances of the propulsion system include total pressure recovery
ratio η , mass flow rate m , flow mean parameter ε , pressure rise pr , and the flow and state variables at the inlet exit,
such as the Mach number, the static and total temperatures, as well as the total pressure, etc. All the variables at the
inlet exit are averaged with respect to the mass flow rate over the cross section of the exit, except the total pressure.
On the other hand, the enthalpy extraction ratio and electric efficiency are the most important parameters
representing the performances of the MHD generator. The enthalpy extraction ratio is defined as the ratio of the
difference of total enthalpy at the entrance and exit of the external MHD generator and the total enthalpy of the incoming flow, while the electric efficiency represents the percentage of extracted electric power from the work done
by the Lorentz force. The definitions of the relevant parameters are summarized in Table 4.
Table 4 Key Parameters
Parameter
Mass flow
rate
Definition
Parameter
m = ∫ ρVdA
Total
pressure
recovery
ratio
Definition
η=
γ
T3 =
∫ ρVTdA
∫ ρVdA
Pressure
rise
Average
total
temperature
at inlet exit
T0,3 =
∫ ρVT dA
∫ ρVdA
Flow
mean
parameter
∫ P dA
∫ dA
Enthalpy
extraction
ratio
ηN =
Electric
efficiency
ηe =
Mach
number at
inlet exit
P0,∞
⎛ γ − 1 2 ⎞ γ −1
⎜ 1+ 2 M∞ ⎟
Pr = η ⎜
γ −1 2 ⎟
⎜ 1+
M3 ⎟
2
⎝
⎠
Average
static
temperature
at inlet exit
Average
total
pressure at
inlet exit
P0,3
P0,3 =
M3 =
0
0
2 T0,3
− 1)
(
γ − 1 T3
12
ε =
∫ ( M − M )dA
M ∫ dA
3
3
Δh0
h0 ∞
PjE
PL
In addition, the maximum allowable static temperature for the combustor inlet ranges from 1400~1700K. Due
to the design constraints being put forth by the tolerable temperature of the engine material and the type of the
cooling methodology, and to avoid the so-called “chocking” effects that may put additional energy into the
combustor, the inlet stagnation temperature should not exceed 3000K. Consequently, the static temperature and
stagnation temperature at the inlet exit are very critical parameters to optimize the inlet.
6.4.1 Influences of effective length of ionization
The variations of mass flow rate with respect to the magnetic interaction parameter of the external MHD
generator under different effective lengths of ionization are demonstrated in Fig.24. As seen from the sketches that
there is an inflexion point cuts every curve into two parts, before which the mass flow rate is almost constant, and
decreases dramatically after that point. The phenomenon can be explained based on the reflecting position of the
shocks emanated from the nose and corners of the ramps as follows: Before the inflexions, the shocks would contact
with the surface of the inlet downstream the cowl lip, while, after the inflexions, the flow spillages. Consequently,
the inflexions depicted in Fig. 24 are the positions, under various LR , where the shocks would make contact with the
cowl lip. These values can be used to determine the magnetic interaction parameters to satisfy the SOL condition.
As for Cases 1 through 4, the magnetic interaction parameters for SOL condition are obtained as Q = 1.9, 1.3, 1 and
0.9, respectively. It is clear that, the longer the effective length of ionization for the electron beam be, the less the
magnetic interaction parameter is needed to meet the SOL condition.
Fig.24 Mass flux vs. magnetic interaction parameter for
various LR
Figs.25-28 depicted the variations of total temperature, average Mach number, total pressure recovery ratio,
and pressure rise at the inlet exit with respect to the magnetic interaction parameter under different the effective
lengths of ionization, LR. It is seen that, due to energy extraction by the external MHD generator, the flow
decelerates. This leads to the drops of the total temperature and the average Mach number at the inlet exit, and the
total pressure recovery ratio, as the magnetic interaction parameter increases. On the other hand, as the magnetic
parameter increases, the pressure rise would increases sharply before the value of the parameter riches the point
where the SOL condition is met, and then varies slowly thereafter.
For the flow mean parameter at the inlet exit, however, the variation tendencies are different for different
effective lengths of ionization. As shown in Fig.29, for LR = 2, as the value of magnetic interaction parameter
increases, the value of flow mean parameter also increases; for LR = 3 and 4, however, the flow mean parameters
increase at first to certain peaks, and then turns decreasing as the value of magnetic interaction parameter increases;
while the flow mean parameter for LR = 5 shows slowly varying tendency in the lower side of the magnetic
interaction parameter, and then turns to increase with further increasing the value of the magnetic interaction
parameter.
13
Fig.25 Total temperature at the inlet exit vs. magnetic
interaction parameter for various LR
Fig.26 Average Mach number at the inlet exit vs. magnetic
interaction parameter for various LR
Fig.27 Total pressure recovery ratio vs. magnetic
interaction parameter for various LR
Fig.28 Pressure rise vs. magnetic interaction parameter for
various LR
Fig.29 Flow mean parameter vs. magnetic interaction
parameter for various LR
14
6.4.2 Performance comparisons for different effective lengths of ionization at SOL condition
Variations of the optimal magnetic interaction parameter, mass flow rate, total temperature, average Mach
number, and the total pressure recovery ratio at the inlet exit vs. the effective length of ionization at SOL condition
are shown, respectively, in Figs. 30 through 34. It can be seen that all the parameters decrease as the effective length
of ionization, LR, increases. For the pressure rise and the flow mean parameter, on the other hand, each possesses a
peak at the effective length of ionization LR = 3m as depicted in Figs.35 and 36, respectively.
Noting that, the deceleration zone in the external MHD generator would be enlarged as the effective length of
ionization increases, bringing the mass flow rate decreases as the effective length of ionization increases at the SOL
condition (cf. Fig.31), and causing more spillage near the cowl lip, which can be comprehended in Figs.37 through
40. The sketches also reveal that the Mach number would drops noticeably inside the inlet for all cases, although the
flow field still maintains supersonic.
Fig.30 Magnetic interaction parameter vs. effective
length of ionization at SOL condition
Fig.31 Mass flow rate vs. effective length of ionization at SOL
condition
Fig.33 Average Mach number at inlet exit vs. effective
length of ionization at SOL condition
Fig.32 Total temperature at inlet exit vs. effective
length of ionization at SOL condition
15
Fig.34 Total pressure recovery ratio vs. effective length
of ionization at SOL condition
Fig.35 Pressure rise vs. effective length of ionization at SOL
condition
Fig.36 Flow mean parameter vs. effective length of ionization
at SOL condition
Fig.38 Mach number contours on x-z mid-plane at Mach 8.5
off-design case with k = 0.8, LR = 3m
Fig.37 Mach number contours on x-z mid-plane at Mach
8.5 off-design case with k = 0.8, LR = 2m
16
Fig.40 Mach number contours on x-z mid-plane in Mach 8.5
off-design case with k = 0.8, LR = 5m
Fig.39 Mach number contours on x-z mid-plane in Mach
8.5 off-design case with MHD, k = 0.8, LR = 4m
6.4.3 Parametric comparisons for different load factors
Numerical results indicate that, for neutral flows at off-design case, the total pressure decreases due to the
entropy-generated dissipation induced by the multiple shock reflections inside the inlet. Although the shock system
is recovered to the state close to the design condition under the influences of the ionized flows, the total pressure
decreases even more. Actually, it is the work done by the Lorentz force, u ⋅ ( j × B) , being responsible to the total
pressure loss, which consists of an electric power, j·E, and a Joule heating,
j
2
σ
. The electric energy is converted
from the total enthalpy by the external MHD generator, while the process would inevitably induce a Joule heating in
the flow field.
Some parametric comparisons among typical test cases for various loading factors are listed in Table 5,
, pressure rise Pr , enthalpy extraction ratio η N ,
including the total pressure recovery ratio η , mass flow rate m
electric efficiency ηe , the static temperature and stagnation temperature at the inlet exit, and the optimal magnetic
interaction parameter. It is seen that, there is 30% increase in pressure rise when the MHD flow control is applied. It
is also found that, under the same effective length of ionization, the flow mean parameter for k = 0.8 is less than the
case for k = 0.5. This implies that, at SOL condition, we may maintain more uniform flow at the inlet exit for larger
load factors.
Table 5 Parametric comparisons
Neutral air flows
M=6
M = 8.5
Qopitmized
η
m （kg/s）
Pr
ε
0.855
293.01
44.36
0.0058
0.469
307.31
57.83
0.0897
ηN
ηe
T3（K）
T0,3（K）
731.2
1940.4
958.6
3742.1
MHD flows（M = 8.5, LR = 3）
k = 0.5
k = 0.8
0.25
1.3
0.239
0.277
305.08
311.2
79.91
75.78
0.121
0.112
0.0323
0.0833
0.503
0.802
1233.3
1112.5
3621.4
3458.2
Comparative analysis on typical test cases at off-design Mach 8.5 reveals that the magnetic interaction
2
parameter Q for optimal flow field at load factor k = 0.5, and thus σ max Bmax
is less than the case at k = 0.8, which
means that, at small load factor, lower energy electron beams and magnetic flux density can bring the flow field
17
back to the state at design condition. However, the external MHD generator at load factor k = 0.8 converts 8.33% of
the kinetic energy of the flow field into electric power. For converting this amount of kinetic energy into the electric
power, there is only 20% of the work done by the Lorentz force being dissipated through the Joule heating. For the
case of k = 0.5, the external MHD generator converts only 3.23% of the flow kinetic energy into the electric power,
but with 50% of the work done by the Lorentz force turning into the Joule heating.
It is equally important to notice that, due to the lowering of Joule heating, more reductions on both static
temperature and stagnation temperature at the entrance of combustor can be attained at larger value of load factor.
In summary, in order to bring the shocks back to the cowl lip, extracting more electric power from the external
MHD generator, and improving the performances of the combustor entrance, the load factor associated with strong
electron beam power and magnetic flux density should be elevated.
7. Conclusion and Discussions
In this work, an airframe/propulsion integrated module was established for waverider-based hypersonic vehicle
at a design Mach value of 6. Three inlet ramps were taken to pre-compress the flow field, while shocks emanated
from these ramps converged on the cowl lip and the reflecting shocks were annihilated at the end point of the third
ramp. The inlet further compressed the flow in spanwise direction downstream the inlet ramps to increase the static
pressure as an isolator. As for the same geometry designed for Mach 6, a large-scale inlet flow control applying
MHD concept was studied at off-designed Mach value of 8.5. The electrical conductivity in air was supposed to be
generated by electron beams injected along the magnetic lines. The second order entropy conditioned scheme was
chosen to solve the 3-D Euler equations.
Numerical simulation gave the 3-D flow field over the airframe/propulsion integrated inlet at design condition,
and confirmed the shock system stated above. Under the off-design condition at Mach 8.5, the flow field inside the
inlet became highly disturbed when the MHD control was not applied. The non-uniformities would deteriorate the
performances of the inlet noticeably. Through the energy conversion and large scale flow control process provided
by an MHD generator, however, one could produce favorable parameters to restore the flow field and shock system
as if the scramjet was operated under the design condition, and made the flow characteristics at the combustor
entrance more favorable when flying at mach numbers higher than designed Mach number.
There are advantages to have a large effective length of ionization, since it keeps the key parameters, namely,
the optimal magnetic interaction parameter, total temperature, average Mach number and flow mean parameter at
the inlet exit small. However, it will deteriorate the inlet performances such as the mass flow rate, total pressure
recovery ratio, and pressure rise.
Parametric study for different load factor also indicates that small load factor can make the optimal magnetic
interaction parameter small. This implies that we need only low electron-beam energy and small magnetic flux
density to restore the flow field and shock system back to the design condition. However, a large load factor will
extract more electric power from the external MHD generator, and improve the performances of the inlet.
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19