Study on Methods for Plug Nozzle Design in Two-Phase Flow

8/11/2019 Study on Methods for Plug Nozzle Design in Two-Phase Flow

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STUDY ON METHODS FOR PLUG NOZZLE DESIGN IN TWO-PHASE FLOW

Kan Xie *, Yu Liu , Xiaodong Chen *, Junxue Ren , Yunfei Liao ** School of Astronautics, Beijing University of Aeronautics and Astronautics,

Beijing, 100191, PRC

Abstract '

The previous design methods are not proper todesign plug nozzle contours of solid rocket motorusing the compound propellant, for the existence ofmetallized particle phase. The flow characters of

plug are changed in the two-phase flow condition,therefore the particle-phase influence must beconsidered in the design. The ideal plug contour isdetermined by the expansion waves at the designaltitude, therefore it is a key to understand thegas-particle interactions in the expansion fan of

plug. The method of characteristic (MOC) wasapplied to study the two-phase flow field of plug, inwhich the two-phase flow model was Lagrangemodel. Based on the computational results of MOC,the Angelino method and curve method wereimproved and extended to be able to design the

plugs contour in two-phase flow conditions. Finallythe performances of the plugs created by the two

improved methods were further examined by CFD,which can consider viscosity and turbulent withrespect to the MOC. The examined cases indicatethat the plug created by the improved methods can

be shortened in length while their thrust performances increase at all altitudes, comparing tothe one created by the unimproved methods in thesame two-phase flow parameters.

Nomenclature

a

= stagnation sonic speed

a = local sonic speed A = area

At = throat area of inner nozzle

AA = radius angle of the arc expansion section of

inner nozzle

AE = expansion angle of inner nozzle

C D = drag force coefficient of two-phase flow

m&

* PH.D. candidate, School of Astronautics, BUAA**

PH.D., School of Astronautics, BUAA Professor, School of Astronautics, BUAA Senior Scientist, School of Astronautics, BUAA

C g = specific heat of gas

C p = specific heat of particle

d p = diameter of particles

eT = total expansion ratio of plug nozzle

ein = expansion ratio of inner nozzle

F = thrust

f = mass flow rate ratio

h = geometry height of plug nozzle

K = velocity drag coefficient

L = heat drag coefficient

l i = length of single expansion wave

= mass flow rate

M = gas Mach number

= equivalent Mach number

P = pressure

Pr = Prandtl number

Rg = gas constant

t = static temperatureT c = chamber temperature

v( M ) = Prandtl-Meyer function

v ( M ) = improved Prandtl-Meyer function of

two-phase flowV g = gas velocity

V p = particle velocity

= equivalent specific heat ratio = oblique angle of plug nozzle

= Mach angle = flow deflection angle

= polar angle to x axis for a Mach wave

= gas specific heat ratio

= gas density

= viscositySubscripts

b = background

c = chamber parameters

e =exit of inner nozzle

end = end tip of plug nozzle

America Institute of Aeronautics and Astronautics1

45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit2 - 5 August 2009, Denver, Colorado

AIAA 2009-5329

Copyright 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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g = gas

i = Mach line i

in = inner nozzle

p = particle

Stokes = Stokes flowT = totalSuperscripts

* = stagnation properties

IntroductionAerospike nozzle can enhance rocket motors

performance in the whole flight, owing to itsoutstanding advantages: the automaticallycompensation ability of altitude performance 1-3 .

Although there have been fully developed designmethods for plug nozzles used in the liquid rocketmotors 4-6, it is not proper to use previous methods todesign plug nozzles for the solid rocket motors(SRM) using the compound propellant 7, 8 . For theexistence of metallized particle phase, the flowcharacters and performance of plug nozzles arechanged 7. The existing researches for bell nozzleshave found that the position of sonic line is changedobviously in two-phase flow conditions; the dragforce between particles and gas leads to two-phaselost of performance. Improper design probablydecreases the performance of plug nozzle, increasesthe collision of particles on plug wall anddeteriorates the ablation. Therefore, the

particle-phase influence and the geometryrestriction of decreasing the particle collision must

be considered in a two-phase flow condition 8.For the plug nozzle design in single phase flow,

Angelino put forward an ideal and approximatecontour method early in 1964 4. The Angelinomethod was widely applied in a series ofresearches 9-11 and projects 12. Ref.13 brought upanother curve method using parabola plus cubiccurves, based on the Angelino method. Plugs canget quite high performance designed by both ofthem, validated by a series of experiments 9-12 . Butthey are only fit to apply in single phase flowconditions.

For complexity of two-phase flow, it is hard tofind an academic method for the two-phase plugdesign as the conventional maximum trust bell

nozzle14

. The MOC and CFD were both appliedhere to investigate the two-phase flow details and

characters in plugs. Based on the fast calculationresults by MOC, the gas-particle interactions in theflow field of plug, expressed and summarized as avelocity drag coefficient, were analyzed. And thenafter considering the influence of particles on

expansion wave behavior of plug, the Angelinomethod and curve method were improved fortwo-phase plug design. Finally the CFD methodwas applied to validate performance of plugsdesigned by the two improved methods.

Two-phase flow model for MOC andanalysis

NASAs standard program SPP 15, 16 applies such

a method to predict the two-phase nozzle flow of

SRM that CFD is applied in the transonic flow fieldand MOC applied in the rest supersonic flow field.

The calculation results by the SPP program can

agree well with the experimental data. Similar

procedure was applied in the MOC program

developed in this paper.

A. Physical model

(a) Discretization of the expansion fan in plug

nozzle

(b) Geometry of inner nozzle

Fig. 1 Presentation of axial plug nozzle



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Table 1. Two-phase flow parameters

Rg

/J/(kg.K)

T c/K

P c /Pa

C g/J/(kg.K)

C p

/J/(kg.K)

1.2 320 3000 5.0e6 1391.2 1430

Fig.1 shows the typical axisymmetric Aerospike

nozzle of whole length studied. For the reason of

symmetry, only one-sided geometry was computed.

Parameter is the oblique angle between the axial

lines of inner nozzle and the plug. P c is the chamber

pressure and P b is the environmental pressure. P c is

set at 5.0MPa and the total temperature T c is 3000K

in this study. The particle phase is Al 2O3. The

two-phase flow parameters are listed in Tab.1. In

the program, the diameter of particle has only one

size. In the following cases for MOC program, the

flow fields of particle diameters d p=1m andd p=10m were studied individually and compared.

B. The MOC procedure

The particle phase is treated by solving

Lagrangian equations of motion 17, 18 . The model is

based on a dilute particle phase assumption and

particles interaction is neglected. Here NND

scheme is applied to discrete the Euler equations to

compute the transonic flow field of the inner nozzle.

More details about NND scheme is shown in Ref.19.

The supersonic flow field is computed by MOC.

Three main assumptions are adopted in the MOC

program:

1. The gas phase is treated as perfect gas, and has

fixed physical properties. Viscosity of gas is ignored

except when it acts with the particle phase;

2. The particles have only one size. Pressure andvolume of the particle phase is ignored;

3. Only drag force and heat convection are

considered between the two phases. Mass transfer

between the two phases isnt considered. Mass

transfer and energy exchange isnt either considered

at the wall boundary.

The characteristic equations of two-phase flow

and corresponding consistent equations are listed in

Ref.18. 6 kinds of main basic unit-procedures are

applied in the MOC program 20. In the two-phase

flow region (Region between the up and down

limited particle tracks in Fig.2), the gas-particle

interaction procedures are performed and particles

are tracked in the characteristic units. Single

characteristic procedures are performed for the up

and down limited particle tracks. The single gas

procedures are performed out of the two-phase flow

region. Details of those basic procedures were

presented in Ref.20 and 21. When particle tracks

reach the solid wall or axisymmetric line, it is

assumed that the particles do not reflect, but

continue to move along the boundaries; when

particles reach the free streamline boundary,

particles run out of it, never back to the flow field.

The gas boundary conditions are already embedded

in the unit procedures.

C. Analysis and Discussion

(a) Mesh of inner nozzle for NND

(b)Mesh of plug created by MOC

Fig.2 Mesh in the MOC program

The mesh of transonic flow field in the inner

nozzle is shown in Fig.2 (a), and mesh formed by

interlaced characteristics along space in the plug

shown in Fig.2 (b). The gas-particle interaction can

be expressed as the parameter of velocity drag

coefficient K , defined as:

2/ g g p V V V K rrr =



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When K gets close to the value one, it means the

particle can follow the motion of gas; when K gets

close to the value zero, it means the particle lags

behind the gas motion and will pull down the

velocity of gas, which leads to large two-phase lost

in the nozzle. The Mass flow rate ratio is defined

as:

/ p g f m m= & &

Fig.3 shows the variation of K in the flow field

of plug, along the mid particle track (see Fig.2 (b))

at the design altitude. The zero of horizontal

ordinate represents the throat of inner nozzle. The

curves indicate that the size of particles fully

determine the variation law. K suddenly falls downat the point 1 position of plug, where the particles

start to go into the expansion waves generated from

the lip E (see Fig.1 (a)). In the expansion-wave

region, gas velocity increases fast and particle can

not follow it at once, which results in the drop of K

at this position. And then K increases again when

the particles are accelerated by the drag of gas.

Fig.3 (a) shows that 1m particles can quickly

adjust their parameters to follow the gas motion

when they run across the expansion fan of plug, and

K s value fast stabilizes at about 0.97 at last. Fig.3

(b) shows that 10m particles adjust their

parameters less quickly and K goes up slowly in the

expansion-wave region. The mass weighted

assembly average of K for 10m particles in the

expansion-wave region is about 0.55.

(a) d p=1m, f =30%

(b) d p=10m, f =30%Fig.3 K s variation along the axial line of inner

nozzle

The ideal plug contour is determined by the

expansion waves at the designed altitude. In

two-phase flow conditions the expansion waves are

disturbed by the particles, therefore the two-phase

plug contour can not be design using the previous

methods without considering the disturbance.

Especially at the position near point 1, the

disturbance can not be neglected. The disturbance

on the expansion waves of particles can beconsidered using the mass weighted assembly

average of K , which can represent the gas-particle

interaction in a simplified way. And the MOC

program provides a fast way to predict the values of

K in the flow field of plug nozzle.

Improved methods for two-phase plugdesign

A. The improved Angelino methodThe Angelino method assumes that at the outlet

of the inner nozzle, gas has uniform flow

parameters and velocity vector is parallel to the

axial line of the inner nozzle. At the design altitude,

gas generates a branch of linear, constant properties

expansion waves from the lip E shown in Fig.1.

And after going through the expansion waves,

velocity vector of gas turns to parallel to the plugs

centerline at the tip of plug (point F in Fig.1).

Therefore the solid wall of plug can be treated as a



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gas streamline boundary of the linear expansion

waves. The Mach angle i for a Mach line i in the

expansion region (See Fig.1 (a)) can be calculated

using the formula (1). Flow deflection angle imeasured by the Prandtl-Meyer angle of M e (See

Fig.1 (a)) is: i=v( M i)-v( M e). v( M i) is the

Prandtl-Meyer function listed in formula (2). Then

the polar angle i of Mach line i is determined by i

and i. According to mass conservation, the Mach

waves length l i is able to be calculated by formula

(3). When all the i and l i of each Mach line in the

expansion region are calculated, the ideal contour

for the plug can be drawn.

1sin (1/ )i i

M = (1)

( ) ( )1 2 11 1 11 1i i

v M tg M tg M

+ = +

2 1i

)

(2)

( 1 2/ ( ) ( )i g i i i il m M f M f M W = & (3)

The specific expressions of 1( M i), 2( M i) and

W i in formula (3) has relation with gas velocity V g ,

gas density and flow area Ai of each Mach line i.

their expressions are listed in Ref.4.

Here we use an assembly average of K in the

flow field of plug to represent the disturbance of

particles on gas motion. And it is assumed that K is

a constant. Several attached assumptions on

two-phase flow of plug are applied as follows:

1. It is assumed that particles keep the state of

Stokes flow, therefore C D+ = Nu+ =1.0, whose

definitions are as follows:

,

D D

D Stokes

C C

C

+ (4)

,

24 24 D Stokes

e g p

C R d V V

= =

(5)

2 stokes

Nu N Nu

Nu+ = = u (6)

C D is called drag force coefficient20, 22 .

2. Particles stay liquid, no phase change is

considered for particles;

3. The specific heats of gas and particles are

constant.

On those assumptions, velocity drag coefficient

K and heat drag coefficient L have one-to-one

correspondence. L is defined as follows:

c p

c g

T t L

T t

=

(7)

K and L have a relational expression 20, 22 :

1

11 3Pr pl

p

c K L

c K

= +

(8)

For K is a constant, one-dimensional control

equations for such two-phase flow can be simplified

as the same form with the ones for steady isentropic

flow of perfect gas (Note that in one-dimensional

two-phase flow, K s expression can be simplified as

K =V p /V g). The only difference is that gas parameters

and M in gas control equations are replaced by

equivalent parameters of two-phase flow and .Their definitions are as follows 20, 22 :

1 ( 1) BC

= + (9)

CM = (10)

, and

21

1 pl

p

fK Bc

f Lc

+=+

( ) ( )21 1 1 pl p

cC f K K K BL

c

= + + +

Then the similar aerodynamic functions as

single gas phase can be used to resolve one

dimensional two-phase flow with constant velocity

drag coefficient K . The following aerodynamic

functions exist in such two-phase flow:

12 1

22 2

1 2 11

1 2t

A M

A M

+ = +

+

(11)

2112 g

T M

T = + (12)

1121

1 2 M

= + (13)



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12112

p M

p

= +

(14)

To calculate the linear expansion wave length in

the two-phase flow with a constant K , a new

Prandtl-Meyer function need to be derived

considering the particle disturbance. It is different

with the formula (2) of single gas phase.

(a) Two-phase flow over a budge

(b) Gas deflection at Mach line i Fig.4 Physical process for the two-phase flow over

a budge

Performing a similar progress of deducting the

basic differential equations of flowing over a bulge

for single gas phase, the basic differential equations

for two-phase flow with constant K (See Fig.4) are

attained and given as follows:

( )2 2

1 1

gi i i

gi i i

dV d Cd d

V

(15)

And from the expression , the

following equation is attained:

a M V i gi =

gi i i

gi i i

dV dM dM da daV M a M

= + = +a

(16)

The relational expression (17) for the stagnation

sonic velocity and the local sonic velocity exists in

the two-phase flow with a constant K :

( )( )2 2 21 / 2 ia a a M = + 2 (17)

It is an isentropic process when gas flows across

the expansion contour, therefore a *=constant.

Making expression (17)s differential form

substitute in equation (16), the following expression

is attained:2

2 212 12

gi i

gii i

dV dM V M M

= +

(18)

Substitute expression (18) into (15); we get:2

2112

ii i

i i

M CdM d Cd

M M

= = +

i , and its

integral form is

( )1 2 1 21 1 1 11 1i itg M tg M D + i= + + + (19)

D is an integral constant, and make

( ) ( )1 2 11 1 1 11 1i i

v M tg M tg M

+ 2i= +

(20)

2

i

i M M C C M

= = =

Formula (20) is the new Prandtl-Meyer function

for two-phase flow with a constant K . The

difference between formula (20) and (2) is that the

specific heat ratio of gas is taken place by

equivalent parameter of two-phase flow, but theMach number in function (20) is still the gas Mach

number M i. The formula (20) indicates that the main

influence of particle phase on gas defection in the

expansion region is changing the specific heat ratio

of mixture. The equivalent specific heat

ratio contains two parts of disturbance of particle phase: drag force disturbance and heat disturbance

(Expressed as parameters K and L individually informula (9)). Then the linear expansion waves in



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two-phase flow can be fixed as following: for a

given M i, relation expressions (9) and (10) are

applied first to gain equivalent Mach

number i and equivalent specific heat ratio . By

using the new functions (19) and (20) and formula(1), polar angle i for the inclined linear expansion

wave i of two-phase flow can be calculated.

Formulas (11)-(14) are used to resolve the

parameters on expansion wave i. At last, the length

l i of expansion wave i after considering the

disturbance of particles can be calculated using the

following mass convection function:

( ) ( )( )1 2/i g i i i il m M f M f M W = & (21)When i and l i are fixed, expansion waves and

ideal plug contour are fixed. Formulas (11)-(14), (1),

(19) and (20) are the final design formulas of the

improved Angelino method. The new design

formulas contain the new parameters K , C g /C p, f , t p,

and i which can represent the disturbance of particles on the linear expansion waves in some

degree. The new formulas also show different

behaviors of expansion waves in the two-phase flow

conditions. Note that when f =0, formulas (20) and

(21) turn the same as the ones of single gas phase,

and i i M = , = .

B. The improved Parabola plus Cubic Curve

curve Method

Fig. 5 Presentation of curve method

Ref.13 uses curves to approximate the ideal plug

contour generated by the Angelino method. Two

curves are used: the front one is parabola; the back

one is cubic curve (see Fig. 5). At the outlet of the

inner nozzle, the static pressure and Mach number

of gas are noted as P e and M

e. At the design altitude,

gas fully expands until the pressure equals to the

environment pressure P b. The Mach number at the

end of plug is noted as M end , and the gas flow

deflection angle (see Fig.5) is: = ( M end )- ( M e).

Only two expansion wave lines, EF and EG

(See Fig.5), are resolved in the curve method. EF is

the last wave, through which the gas flow direction

turns parallel to the plugs centerline and the static

pressure turns the same as the environment.

Formula (3) is used to calculate the length of EF .

The position of point F is then determined by EF s

length and the angle . The gas flow deflection

angle is /2 when gas flows through the wave EG .

In the same way, EG s length then can be resolved

and the position of point G is determined. In

addition, the position of point D and the tangentialangles at points D, G and F are known. Therefore

the mathematical expressions of the parabolic curve

and cubic curve can be attained.

In the improved curve method, the same curves

are still applied to fit the ideal contour points

generated by the improved Angelino method. The

differences with the unimproved curve method are

that expressions (20) and (21) are used to replace

the expressions (2) and (3), and the aerodynamic

functions are taken place of by functions (11)-(14).

The solution process is the same as the unimproved

method mentioned above.

C. Design results and analysis

Table 2. Basic design conditions for the referenced

two-phase plug

eT ein At /m2 AA AE K d p/m

30 4.5 5.71e-4 16 o 0o 0.3 0.55 10

The design conditions for two-phase axial plug

and geometry variables are listed in Tab. 2. In the

design process, it was found that the convergent

angle and radius of the arc convergence section of

inner nozzle, chamber pressure P c, total temperature

T c, and Prandtl number Pr (Here is 0.72) , have little

influence on physical dimension of the plug. The

main factors of influencing the outline of the plug

obviously are parameters AA, AE , ein, eT , and K. The



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total pressure ratio eT (Correspondent to a design

M end ) are usually set before design. Therefore the

examined design variables are AA, AE (See Fig.1

(b)), ein, and K , of which the parameter K

determines the plugs outline chiefly.

When using the improved methods to design

two-phase plugs, K should be known beforehand as

one of design conditions. It can be estimated by the

MOC program developed in this paper. The specific

resolving process is as follows: 1) under the same

design conditions, the unimproved methods is used

first to get an initial plug contour; 2) using the MOC

program the two-phase flow field of initial plug

nozzle can be fast simulated, and an assembly

average value of K is attained; 3) using the initialvalue of K , the two-phase plug can be designed

using the improved methods; 4) a new value of K

will be attained by MOC program for the improved

plug. After 2-3 times of iterations, the value of K

can stabilize at a fixed value. And the plug contour

is fixed at the same time. As discussed before, K is

determined chiefly by particle size. The influence of

K on plug outline implies the influence of particle

size. For the examples studied here, the results of

MOC show that the corresponding assembly

average of K for 10 m particles with f =30% isabout 0.55.

Fig.6 (a) compares the design results by the

unimproved and improved methods with the same

design parameters listed in Tab.2. The length of

two-phase plugs designed by the improved

Angelino method (Called two-phase ideal face) and

the improved curve method (Called two-phase

curve face) are shortened by 33%, comparing to the plug contours by unimproved methods (Called pure

gas curve face and pure gas ideal face). The plug of

two-phase idea face in Fig.6 (a) is noted as the

referenced plug nozzle for the following discussion.

Because of heat drag between the gas phase and

particle phase, the temperature of particles is always

higher than the gas in the same position of plug

flow field. The heat convection between the two

phases makes the streamlines of gas divergent

outward. If the plug in two-phase flow condition is

designed using the previous methods of single

phase, no heat influence of particles can be

considered. Then at the design altitude, the free

streamline boundary is no longer parallel to the

centerline of plug nozzle but divergent outward.

The design M end (At the design altitude) can either

not be reached for the drag of particles.

Fig.6 (b) gives different plug contours by the

improved Angelino method with different values of

K . As K increases to the value one (Namely the

particle size is smaller), the two-phase flow

approaches a two-phase equilibrium flow and the

two-phase plug turns to have a larger oblique angle

(See the enlarged view of inner nozzle in Fig.6 (b))

and a shorter length. As K changes toward theopposite end of value zero, two-phase plug turns the

same as the plug of single gas phase (See the view

of whole plug nozzles in Fig.6 (b)). The design

results indicate that the parameter increases in a

two-phase flow condition with considering the heat

disturbance of particles. And Note that the heat

disturbance of particles is related to the velocity

drag coefficient K through the formula (8).

(a) Plug contours created by three methods

View of whole plug nozzles



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The enlarged view of inner nozzle

(b)Plug contours vs. K

(c) Plug contours with different inner nozzles Fig.6 Results for two-phase plug design

Fig.6 (c) shows the plug contours by the

improved Angelino method as the inner nozzles

variables change comparing to the referenced

parameters listed in Tab.2. As AA decreases, the

inner nozzle tends to have a longer length, and its

appearance is more like an elongated cone nozzle;

the plug tends to have a larger and a larger height

h. As AE increases, the inner nozzle tends to have a

shorter length, a smaller , and a smaller h, but

there is little change on the shape of the whole plug.

As ein increases, the plug has a smaller , while h is

almost the same with the referenced plug.In general, the most sensitive variables for

two-phase plug design are AA, ein, and K . It is noted

that, for the referenced plug, the particles will be

unavoidable to collide on the plug wall. That is

because the outlet of inner nozzle points to the front

part of plug wall. A basic and direct design rule is

that: the farmer the down limited particle track gets

away from the plug (see Fig.2 (b)), the smaller

probability there is for particles to collide on the

wall. Therefore to decrease the probability and

frequency of particle collision, the design geometry

parameters AA, AE and ein, should be chosen

carefully. And a larger but pertinent ein, a smaller

AA, and a larger AE comparing to the reference

geometry parameters are recommended in practical

design. This is called geometry restriction with

considering the probability of knocking on the plug

wall for the down limited particle track.

Performance validation and discussion

A. CFD model and mesh

Fig. 7 Two-phase flow phenomena in the plug

nozzle

Fig.8 Mesh of plugs flow field for CFD

Fig.7 is a sketch of the flow phenomenon in a

two-phase plugs flow field. Although the MOC

program developed by this paper can calculate the

performances of two-phase plugs, the control

equations of gas for MOC are Euler equations.

Therefore the MOC model cant calculate the lost

caused by viscosity and consider the turbulent effect.

To review the combined effects of all factors, CFD

method is applied to resolve the two-phase flow.

The model of particle phase is the Lagrange model.The turbulence model applied is S-A model with a



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logarithmic law of the wall 23. All the calculation is

performed in FLUENT (V6.3). In the following

examples, the particles have only one size and

d p= 10m is chosen to review. Fig.8 is the region ofcalculation and the mesh for FLUENT.

B. Flow field analysis

The flow fields and performances of four

two-phase plugs were calculated to compare, which

are two two-phase ideal plugs 1 and 2 (By the

improved Angelino method), a two-phase curve

plug 3 (By the improved curve method), and a pure

gas ideal plug 4 (By the unimproved Angelino

method). Their design total expansion ratio eT is 30.

The geometry variables of expansion section for the

four inner nozzles are listed in Tab. 3. The other

design conditions are the same with the ones listed

in Tab.1 and 2. The oblique angle was calculated

after designing the plugs. The two-phase ideal plug

2 was designed after considering the geometry

limitation of decreasing the possibility of particles

collision on the plug based on the flow field

analysis of plug 1.

Table 3. The geometry parameters of four plugs AA AE e in

Two-phase ideal plug 1 16 o 0o 4.5 33.89 o

Two-phase ideal plug 2 10 o 6o 8 16.15 o

Two-phase curve plug 3 16 o 0o 4.5 33.89 o

Pure gas ideal plug 4 16 0 0o 4.5 32.73 o

Fig.9 compares the results of two-phase plugs 1

and 2 predicted by CFD at the design altitude.

Influenced by the heat transfer from particles, the

gas stream lines of plug 1 are divergent away from

the axial line of the plug, which leads to divergence

loss (see Fig.9 (a)). The gas free streamline

boundary in plug 2s flow field is approximately

parallel to the center line of the plug, which has less

divergent loss than the plug 1(See Fig.9 (b)). Fig.9

(c) and (d) show the initial knocking point of the

down limited particle track on the plug wall. The

initial knocking point of plug 2 moves backward

comparing to the initial one of plug 1 and thenumbers of particles knocking tracks in plug 2 are

smaller than those in plug 1. It implies that the

probability of particles knocking on the wall of

plug 2 is smaller than that of plug 1. Therefore it is

truly effective for lightening the scouring of

particles on plug wall by considering the geometry

restriction of the down limited particles track. And

the particles in the plug 2s flow field have a

uniform distribution which is favorable for energy

exchange of two phases and full expansion of gas.

The flow field analysis above indicates the

importance of choosing geometry parameters of AA,

AE and ein and necessity of considering the

geometry limitation of decreasing the possibility of

particles collision in two-phase plug design.

(a) (b)

(c) (d)

Fig.9 Parameters contours of plug 1 and 2: (a)

Mach number and total pressure contour of plug 1;

Mach number and total pressure contour of plug 2;

(c) Particles tracks of plug 1; (d) particles tracks of

plug 2.Fig.10 gives the results of two-phase curve plug

3 at the design altitude. The oval region noted in

Fig.10 (a) is corresponding to the one in Fig.10 (b).

There exists catastrophic deposit of particles in the

oval region; therefore there is distortion in the flow

field which results in a wake at the end of plug. The

Mach number contours approximately have a

laminated distribution in the oval region, but no

separation of gas happens there (see Fig.10 (b)). In

practical application, this oval region can be cut off



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and the plug turns a shortened form.

(a) (b)

Fig.10 Parameters contours of plug 3: (a) Mach

number and total pressure contour; (b) particlestracks and gass stream line

Fig.11 Mach number and total pressure contour of

plug 4

For the particles drag, the gas total pressure

falls down. The total pressure contour can illustrate

the distribution of intensive action between gas and

particles. The lower parts in Fig.9 (a), 10 (a) and 11

show the total pressure contours. The results showthat there is a large region of energy exchange

between particles and gas for the two-phase plugs

created by the improved methods. But there exist

apparent energy exchange only in the region near

the solid wall for plug 4 by unimproved Angelino

method (See Fig.11).

C. Performance and further discussion

The thrust performance of examined plugs is

calculated using the following formula:

, x g g x ainlet inlet inlet

x F pdA V V dA p dA = +

x w xwall wall

pdA dA (22)In formula (22), F is total thrust of the plug

nozzle; the first term and the second term in

right-hand side together represent the gass force on

the chamber head (The inlet in Fig.8); the third term

is the force of environmental pressure on the

chamber head. The fourth term is the integration of

gass pressure on the wall. The final term is the

friction force of gas on the wall.

Fig.12 Comparison of thrust performance

Table 4. Work conditions and trusts of the four

plugs

Work conditions

/pa101325 50000 15244 5000 1000

Thrust of Plug 1

(N)4698.6 4943.0 5316.3 5475.5 5553.4

Thrust of Plug 2

(N)4918.6 4943.2 5289.2 5394.8 5776.0

Thrust of Plug 3

(N)4828.4 5096.9 5527.1 5710.3 5774.9

Thrust of Plug 4

(N)4629.7 4853.2 5219.8 5379.1 5444.2

Notes: 15244 Pa is the pressure of the design altitude;

Fig.12 gives trusts variation curves as the

environmental pressure changes for the fourexamined plugs. The symbol of background



12/13

pressures value is negative in the graph. The curves

show that two-phase ideal plugs 1 and 2, and

two-phase curve plug 3, which are all created by the

improved methods of considering the particle-phase

influence, have better thrust performances at all

altitudes than the pure gas plug 4 created by the

unimproved Angelino method, under the same

design conditions. Especially the trust of plug 3 is

increased by 4% comparing to plug 1 (See Tab. 4).

Plug 2 has better performance especially in lower

and in higher altitudes than plug 1, but at around the

designed altitude it has a worse performance.

Therefore the performance of the plug nozzle and

geometry limitation of decreasing the probability of

particle collision should be balanced in the practical plug nozzle design. As mentioned before, the length

of plugs created by the improved methods is

shortened by 33% comparing to the ones created by

the unimproved methods. In general, after

considering the disturbance of particle phase, a

shorter length and better performance can be

attained for plug nozzle design in two-phase flow

conditions. The improved methods can consider the

particle-phase influence on gas motion at some

degree as to increase the performance of plug in

two-phase conditions, comparing to the unimproved

ones.

ConclusionThe computation results by MOC program

indicate that the disturbance of particle phase on theexpansion waves is mainly related with the particlesize in a dilute two-phase flow of the plug. Anapproximate assumption of constant velocity dragcoefficient K in the two-phase flow of plugs was

brought up to simply the analysis. By introducingthe summarized parameter K for gas-particleinteraction, the Angelino method and curve methodwere improved and extended for the plug design intwo-phase flow conditions. The new designformulas can consider the velocity and heatdisturbance of particles on expansion wave behavior.Meanwhile the down limited particle track isconsidered as an additional geometry restriction in

the design. In the two phase flow condition with afixed mass flow rate ratio, the main parameters to

influence the plugs outline are AA, AE, e in , and K. K is the most important parameter to influence the plugs outline obviously. Its value can be fastestimated by the MOC program developed in this

paper. As the particle size increases, the two-phase

plugs tend to have a larger and a shorter length.The examined cases by CFD prediction indicate thatthe plug created by the improved methods can beshortened in length, while their performancesincrease at all altitudes, comparing to the onecreated by the unimproved methods in the sametwo-phase flow parameters.

AcknowledgementThis work has been carried out at the support of

National Natural Science Foundation of China(General Program 50476002). The authors aregrateful to Dr Wuye Dai and Dr Lizi Qin for theirhelp.

Reference1Ruf, J. H., McConnaughey, P. K., The Plume

Physics Behind Aerospike Nozzle AltitudeCompensation and Slipstream Effect, AIAA Paper97-3218, July 6-9, 1997.

2Muss, J. A., and Nguyen, T.V., Evaluation ofAltitude Compensating Nozzle Concepts for RLV,AIAA Paper 97-3222, July 1997.

3Angelino, G., Theoretical and ExperimentalInvestigation of the Design and Performance of aPlug-type Nozzle, Training Center ForExperimental Aerodynamics, Technical Note 12,July 1963. Rhode-Saint Genese, Belgium.

4Angelino, G., Approximate Method for Plug Nozzle Design, AIAA Journal , Vol. 2, No. 10, 1964, pp. 1934-1835.

5Humphreys, R. P., Thompson, H. D. andHoffmann, J. D., Design of Maximum Thrust Plug

Nozzles for Fixed Inlet Geometry, AIAA Journal ,Vol.9, No. 8, Aug. 1971, pp. 1581-1587.

6Greer, H., Rapid Method for Plug NozzleDesign, ARS Journal , Vol. 31, No. 4, 1961, pp.560-561.

7Ren, J. X., Numerical Simulation of Two-phaseFlow in Two-dimensional Plug Nozzle, 2005International Autumn Seminar on Propellants,Explosive and Pyrotechnics. Beijing, China, Oct.

2005.8Hoffman, J. D., A General Method for



13/13

America Institute of Aeronautics and Astronautics

13

Determining Optimum Thrust Nozzle Contours forGas-Particle Flows, AIAA Journal , Vol. 5, No. 4,April 1967, pp. 670-676.

9Dai W. Y., Liu Y. and Cheng X. C., SimulatedTests of Aerospike Nozzle and Data Acquisition

System, Journal of Propulsion Technology , Vol. 21, No. 4, April 2000, pp. 85-88.

10Dai, W. Y., Liu, Y., Cheng, X. C., and Ma, B.,Analytical and Experimental Studies ofTile-Shaped Aerospike Nozzles, Journal of

Propulsion and Power , Vol. 19, No. 4, July 2003, pp. 640-645.

11Liu, Y., Zhang G. Z., Dai W. Y., Ma, B., Cheng,X. C., Wang, Y. B., Qin, L. Z., Wang, C. H., and Li,J. W., Experimental Investigation on Aerospike

Nozzle in Different Structures and WorkingConditions, AIAA 2001-3704, July 2001.

12Rocketdyne Inc., Final Report, AdvancedAerodynamic Spike Configurations. Report No.AFPRL-TR-67-246, September 1967.

13Qin, L. Z., Contour Design and Optimizationof Aerospike Nozzles, Ph.D. Thesis, BeijingUniversity of Aeronautics and Astronautics, PRC,May 2002.

14Rao, G. V. R., Exhaust Nozzle Contour forOptimum Thrust, Jet Propulsion , Vol. 28, No. 2,

1958, pp. 377-382.15Coats, D. E., French, J. C. and Dunn, S. S.,Improvement to the Solid Performance Program(SPP), AIAA Paper 2003-4504, July 2003.

16Coats, D. E., Solid Performance Program(SPP), AIAA paper 87-1701, 1987.

17Marc, B., Olivier, S., On the Prediction ofGas-solid Flows with Two-way Coupling UsingLarge Eddy Simulation, Phys. Fluids , Vol. 12, No.8, 2000, pp. 2080-2090.

18Hwang, C. J., Numerical Study of Gas-ParticleFlow in a Solid Rocket Nozzle, AIAA Journal , Vol.26, No. 6, 1987, pp. 682-689.

19Zhang, H. X., Non-Oscillatory and Non-Free-Parameter Dissipation Difference Scheme, ACTA

Aerodynamica Sinica , Vol. 6, No. 2, 1988, pp.143-165.

20Zucrow, M. J., Hoffman, J. D., GasDynamics, Beijing, National Defense Press, 1984,

pp. 83-187.21Dai, W. Y., and Liu, Y., Applications of

Method of Characteristics on Aerospike Nozzle, Journal of Aerospace Power , Vol. 15, No. 4, Oct.

2000, pp. 371-37.22Kliegel, J. R., Gas Particle Nozzle Flows,

Ninth International Symposium on Combustion, New York: Academic Press, 1963, pp. 811-826.

23Spalart, P., and Allmaras, S., A One-equation

Turbulence Model for Aerodynamic Flows, AIAAPaper 92-0439, 1992.

Study on Methods for Plug Nozzle Design in Two-Phase Flow

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