pp12 nozzle cd

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Paper: ASAT-13-PP-12

13th

International Conference on

AEROSPACE SCIENCES & AVIATION TECHNOLOGY,

ASAT- 13, May 26 28, 2009, E-Mail: [email protected]

Military Technical College, Kobry Elkobbah, Cairo, Egypt

Tel : +(202) 24025292 24036138, Fax: +(202) 22621908

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An Investigation on the Internal Flow in Simulated Solid RocketMotor Chamber/Nozzle Configuration

M. Nasr*, A. M. Hegab

**, W. A. El-Askary

*, K. A. Yousif

*

Abstract: This paper describes numerical, analytical and experimental investigation of

acoustic wave propagation in a simulated Solid Rocket Motor (SRM) chamber. The

experimental study is carried out on a square-cylinder cross-sectional channel with two

equally permeable sidewalls. An endwall disturbance is imparted using a moving piston

located at the head end while the exit end of the channel is opened to the atmosphere.

Moreover a convergent and convergent-divergent nozzle is changeable fixed at the exit end of

the channel to study the behavior of the complex wave interactions mechanism at different

nozzle areas. The unsteady, compressible, two dimensional Navier-Stokes equations in a

laminar regime are numerically solved by predictor-corrector MacCormack scheme. Axial

acoustic velocity field generated by the end wall disturbances interacts with steady sidewall

injection to generate rotational flow field through the channel. As a result, a steady vorticity is

generated at the sidewall and then is convected toward the centerline by the transverse

component of the total velocity. Furthermore the time-independent, compressible Navier-

Stokes equations with laminar effects are solved. An analytical solution for pure acoustic flowis derived from the reduced form of the full Navier-Stokes equations. The numerical and

analytical solutions are compared with the experimental data. The comparisons show

reasonable agreement between these three approaches. Moreover, the results show that, the

geometry of the variable area parts has significant effect on the generated complex wave

pattern inside the chamber.

Keywords: Solid Rocket Motor chamber, Permeable sidewalls injection, Internal cavity,

Solid rocket nozzle.

Nomenclaturebn wave number

C'o speed of sound, m/s

Cp specific heat at constant pressure

Cv specific heat at constant volume

Et total energy

H dimensionless channel half-heightH' channel half-height, cm

K thermal conductivity

L dimensionless channel length

*Mechanical Power Engineering Department, Faculty of Engineering, Menoufiya University,

Shebin El-Kom, EGYPT**Mechanical Power Engineering Department, Faculty of Engineering, Menoufiya

University, Shebin El-Kom, EGYPT, Corresponding author: [email protected], Tel.

012-7858517


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L' channel length, cm

M Mach number

n wave number index

P static pressure

Patm atmospheric pressure

P'o stagnation pressure, paP% dimensionless pressure perturbation

Pr Prandtl number

Re Reynolds Number

T temperature

T'o stagnation temperature, K

t time

t'a acoustic time, L'/ C'o

u axial speed

U'o reference axial speed, m/s

v transverse speed

V'inj reference injection speed, m/s

x axial coordinateX' dimension axial coordinate

y transverse coordinate

Greek symbols

ratio of specific heat

aspect ratio

transient axial velocity amplitude

' dynamic viscosity, pa.s

density

'o stagnation density, Kg/m3

frequency

Subscripts

' dimension quantities

o stagnation value

inj injection

1. IntroductionThe current work is devoted to examine the time-dependent flow field in a porous channel

with endwall disturbance to describe the effect of adding convergent and convergent-

divergent area parts at the open end on the complex wave pattern inside the chamber. The

presence of sidewall injection with traveling acoustic waves inside long slender square cross-

sectional channel can lead to a rotational flowfield that are decreed by the system geometry.

These waves can, in turn, interact with the solid boundaries to generate acoustic and vortical

wave resulting in complex flow patterns.

In the hope of elucidating the nature of the resulting flowfield, an experimental investigation

was conducted by Brown et al. [1]. They used nitrogen gas injection through uniformly

sintered bronze plates inside a cylinder chamber. In their facility, acoustic waves are

generated from an external rotary valve that controlled the flow exiting the chamber. Their

results verified the accuracy of the analytical model suggested by Culick [2] for the mean

flowfield and also provided substantial data for the resulting acoustic field. Independently of


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the work given in [1], a novel investigation facility was built by Ma [3] to simulate similar

flow conditions in a rectangular chamber. That experiment employed the sublimation of

carbon dioxide, a process that resembled the combustion of propellant, in generating the CO2

gas to mimic the chamber's transpiring wall. Unfortunately, the work by Ma [3] had

experimental difficulties in measuring acoustic pressure and velocity. The results indicated

that the wave generator produced many non-harmonics waves with many higher fundamentalsthat make the interpretation of the complex mechanism is difficult. Moreover, in that

experiment, Ma [3] didnt verify the occurrence of generating turbulence. Barron et al. [4]

introduced an improvement to Ma's experiment by utilizing a Scotch-yoke mechanism to

replace the slider-crank mechanism. The new mechanism led to higher pressure amplitudes,

pure sinusoidal motion and resulted in acceptable validation between numerical and

experimental results.

Dunlap et al. [5] presented an experimental verification for the cold flow simulation of rocket

chamber flow field based on Culick's analysis [2] of steady state flow. The results of Dunlap

et al. [5] revealed that the inviscid flow field solution gives accurate results as long as the

Reynolds number is sufficiently large to ensure that viscous effects are small compared with

pressure gradients.

Flandro [6] provided an early assessment of the importance of vorticity in acoustic boundary

layer. He studied the impact of a small axial pressure gradient, varying harmonically in time,

on the viscous process occurring adjacent to a surface from which a steady spatially uniform

injection occurs. A linear equation for axial velocity contains a balance of convection,

pressure gradient forces and viscous diffusion. The solution described a shear wave

convecting away from the wall, with amplitude that is damped by viscous effects. Flandro [6]

observed intense, transient vorticity in the boundary layer compared to the weaker steady

vorticity associated with the inviscid, rotational Culick's solution [2].

A mathematical model formulatedby Zhao et al. [7] is used to describe the initiation and

evolution of intense unsteady vorticity in a low Mach number, weakly viscous internal flowsustained by mass addition through the side wall of a long narrow cylinder. The intense

vorticity is formed at walls and is convected into the entire chamber by the steady radial

velocity. The amplitude and the distribution of the vorticity are impacted by weak viscous and

nonlinear effects. It was also demonstrated that there are parameter ranges of Mach number,

Reynolds number and driving frequency for which vorticity is really confined to weakly

viscous acoustic boundary layer, thin compared to the radius of the cylinder but larger than

that obtained by Flandro [6].

Erickson et al. [8] presented analytical study concerning forced gas-dynamic oscillations in

closed, constant diameter cylindrical ducts and ducts whose cross-section area varies axially.

The objective of that work was to determine the effect of duct shape on resulting oscillations

amplitude, wave form, harmonic content, and identify duct shapes that produce largeamplitude oscillations for a given energy content. The results showed that the higher

amplitude pressure oscillations can be forced in horn like shaped ducts as opposed to

cylindrical ducts. Shocks like waveforms existed in constant diameter ducts are caused by the

generation of higher harmonics through efficient non-linear coupling with the fundamental

mode. In contrast, the non-linear coupling between the fundamental mode and its harmonic

was weak in ducts whose cross sectional area varies axially (horn-like ducts). This induces

much lower relative energy content in higher harmonics in that duct and hence decreases the

excitation of harmonics.


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The structure of turbulence in a channel flow with a fully transpired wall has been

investigated experimentally by Deng et al. [9]. The aim of that experiment was to study the

effect of the porous surface boundary conditions on the core flow development and flow

structure in the channel. Air is sucked into the channel through the top honeycomb by high

pressure direct drive blower. It was found that the boundary conditions on the porous surface

are very important to the internal core flow evolution and flow pattern. For a course poroussurface (1/4'' honeycomb), the mean flow differs significantly from classical Culick's solution

[2] and computational results. However, with small pore size (1/8'' honeycomb), the mean

velocity profiles are very close to laminar solution for a considerable downstream length and

that profiles agree well with Culick [2], even though large turbulence intensity was observed.

Hegab [10]presented numerical study that describes the transient flow dynamics generated in

a SRM's chamber model with time-dependent mass injection. The main goal was to

understand the heat transfer and temperature dynamics that accompany the co-existing

acoustic and rotational velocity disturbances. Also the effect of adding variable duct to the

open end on the internal flow-field was considered. The results showed that surprisingly large

transient temperature gradient is presented at the sidewalls and the interior of the channel.

Large gradients at the sidewall imply that there is an unexpectedly heat transfer which mayinfluence the combustion zone above the burning propellant even though the fluid injection is

isothermal. It is observed that the unsteady vorticity across the chamber is sensitive to small

changes in flow dynamics and the maximum amplitude of the vorticity increases as the throat

height is decreased. Also, the time history of pressure amplitude increases as throat height

decreases.

The influence of adding variable area portions to the open end of a circular constant-area tube

on finite amplitude wave deformation and radiation, under resonant conditions was studied

experimentally by Sileem and Nasr [11]. They concluded that, the radiated part of energy

delivered by the piston to the atmosphere depends on the tube end configuration. The noticed

difference between results of the open-end tube and that when variable area portions are

added presumably attributed to the deviation of the natural frequency of the variable areacases from that of the open end tube case.

The internal flow-field forming in the combustion chamber of SRM was analytically studied

by Majdalani et al. [12]. A combined geometric configuration is considered in which a

straight cylinder is connected to a tapered cone and the gases are injected perpendicularly to

the surface. The selection of the injection velocity was followed from the experimental work

of Brown et al. [1]. They concluded that the taper effect is more pronounced as the gases

move away from the head end due to the increasing cross-sectional area. The mean flow

approaches its asymptotic limit in sufficiently long cylinders. It may be worth mentioning that

accurate matching of both numerical and analytical solutions requires that the motor

parameters be chosen within specified limits and the corresponding criteria was shown to be

practical. Similar experimental investigation to that of[12]used for studying the influence ofsolid propellant inclination angle through small-scale cold flow simulation was presented by

Nguyen et al. [13]. The experiments were conducted on a cold gas experimental setup. They

concluded that the sidewall injection angle has significant effect on the internal flow field and

vortex shedding in SRM. For small inclination angles the whole interactions between shear

layer and vortex shedding were decreased and hence decreased the wall vorticity, weaken

pressure wave intensity and oscillation levels. On the contrary, the larger angles enhanced the

influence of wall vortex shedding at the rear end of the chamber, which lead to the increase of

pressure fluctuation levels.


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Hegab and Nasr [14] studied experimentally and numerically the propagation of acoustic

waves (which is generated by oscillating piston) in a long, narrow chamber with endwall

disturbance. Their study includes also, an analytical solution to the two following cases. The

first case is in which straight duct with endwall disturbance at the head end and without

sidewall mass injection. The second case, duct contributes steady sidewall mass injection

from permeable walls and endwall disturbance. They illustrated a reasonable comparisonbetween the experimental, analytical, and the computational results using the two-four

explicit predictor-corrector MaCcormack scheme.

From the mentioned review it is noticed that, the effect of exit geometry on the acoustic

flowfield generated in SRM cavity with/without sidewall mass injection did not take the

sufficient attention. Therefore, the present work focuses on experimental and theoretical

studies for the unsteady flow in a simulated SRM interior cavity. The cavity is represented by

square-cylinder chamber with permeable sidewalls. Moreover, convergent and convergent-

divergent nozzles are changeable installed at the exit end of the channel. The numerical

solution of the laminar, two dimensional, compressible and steady/unsteady Navier-Stokes

equations for the same geometry are considered to validate the experimental work. Also the

analytical solution of the reduced form of Navier-Stokes equations is considered.

2. Experimental SetupThe cavity of SRM is represented by square-cylinder chamber of axial length 44 cm and cross

section height of 2.5 cm. The general arrangement of the experimental set-up layout is shown

in Fig.1. It consists of the following parts: an oscillating piston [2.5 cm diameter and 1.9 cm

stroke] driven by an electric AC motor of YC90S-2 type, 2900 rpm, 1 HP, single phase

electrical input and suitable for variable speed through a pulley-belt system. The number of

revolution of AC motor can be changed (from 3610 rpm to 4774 rpm) by voltage regulator.

The variable speed piston oscillates to generate the acoustic disturbance at one end of the

channel and the other end is open to atmosphere. The square cross-sectional area channel has

permeable sidewalls to inject a guided-steady state flow into the channel. The injected airflows through two similar hoses which connected to two similar injection ducted guided-blade

ducts to obtain uniform injection steady velocity at both sides of the channel. Convergent and

convergent-divergent nozzles are changeably added to the open end of the square cross-

sectional area channel. The detailed dimensions of those parts are given in Table (1). Wall

pressure time histories are recorded at two locations along the channel, where the lower one is

5.5 cm apart from the Top Dead Center (TDC) and the upper one is at 19.5 cm apart from

square cross-sectional area channel end. These pressure histories are measured using

Capacitance Pressure Transducer [model SA, Data Instrument, Action MA01720, USA 0-100

psi, accuracy 1%] connected to Data Acquisition System (DAS) and desktop computer

through a Labview software. The data are recorded in files and plotted as will be shown later.

The uncertainties of the measured pressure shown here are estimated according to the

procedures given by Coleman et al. [15] found to be within 2.5% for a confidence interval of

95.45 %. The reading error of U-tube manometer which is used to measure the injected flow

rate by an orifice meter is 0.25 mm and the total uncertainty of the measured flow rate is

equal to 61.412 10 .


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(16)

(17)

(15)

(13)

(8)

cL

L

2 H

c dL

2 exH 2 thH

10-Distribution board

11- Pressure measuring holes connected

by capacitance pressure transducer

12 To DAS

13- Desktop computer

14- Oscillating piston

15- Pulley-belt system16- AC motor

17- Voltage regulator

18- U-tube manometer

1- Air from atmosphere

2- Reciprocating compressor

3- Air reservoir (volume 500 lit.)

4- Orifice meter

5-Two perforated plates

6- Injection air flow through duct with

guide blades7- Square channel (test duct)

8- Electrical power supply

9- Controlled voltage circuit and

constant voltage power supply device

Fig.1 The experimental test-rig components

Table (1) Detailed dimensions of straight, convergent, and convergent-divergent area

channels.

Part type

Length

(cm)

Exit height,

2H'ex (cm)

Throat height,

2H'th (cm)

Square Duct Constant area 44 2.5 -----------

Convergent Variable area 5 0.65,0.95,1.27 -----------

Convergent- Variable area 14 3,5.78,8.725 0.65,0.95,1.27


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3. Mathematical ModelThe mathematical model is based on solving the unsteady, two dimensional parabolized

Navier-Stokes equations describing both dynamics and acoustic in a perfect gas within a

square-cylinder channel. The gas is initially stationary with a reference-state defined byoP

(the total pressure),o

(the total stagnation density) ando

T (the total temperature). The

associated sound speed is found to be ( / )o o o

C P = , where prime quantities refer to

dimensional values. The characteristic length scales for the axial and transverse variables are

chosen to be the axial length of the channel L and the half-height of the cross section H , respectively. Time is nondimensionalized with respect to the axial acoustic time

a ot L C /= .

The aspect ratio is defined as the ratio of the channel length to the half-height of the

channel, /L H = .The characteristic axial velocity oU is related to the characteristic side-

wall injection velocity of the fluidinj

V through the global mass conservation. The

characteristic axial flow Mach number, Prandtl number and flow Reynolds number are

defined respectively, as follows:

Pro o poM U C C k = / , / = , Re o oU L /=

The governing flow equation can be written in its dimensionless form as follows;

Q F

t x y

+ + = 0 (1)

Where Q, E and F are column vectors given by [16]

t

uQ

v

=

( 1)t

u

pu

uv

u up

+

=

+

, ,

/ Re

= /( )

[ ( 1) ]

Re.Pr

y

yt

M v

uv u

F v p

v p

+

+

The dimensionless form of the equation of state for a perfect gas is;

p T= (2)

The non dimensional form of the total energy is represented by

[ ( ) ] ( 1)2

t v

v

uC + = + (3)

Where, Cv and are the specific heat at constant volume and the specific heat ratio,

respectively.

The velocity and temperature gradients read;

yu

uy

=

, yT

y

=

(4)


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In general >>1, M>1 and Pr=O(1)

In the present work, two solution techniques including numerical as well as analytical

methods are used. They will be discussed in the following subsection.

3.1. Numerical Method

An accurate flow-field time history in SRM chamber can be obtained using finite-differencescheme which shows the evolution of the flow variables in the axial and transverse directions

after many acoustic wave cycles. The present study employs higher-order accuracy difference

equations to minimize the impact of numerical diffusion which was found to affect the results

obtained from the second order explicit MacCormack code, see Hegab [17]. Near the

boundaries, the second order explicit predictor-corrector scheme, developed by MacCormack

[18] is used to descretize the two-dimensional, unsteady, compressible, Navier-Stokes

equations. At the interior points, the Navier-Stokes equations are descretized using the Two-

Four explicit, predictor corrector scheme, developed by Gottlieb and Turkel [19], which is a

fourth-order variant of the fully explicit MacCormack scheme. This method, applied by

Kirkkopru et al. [20], is highly-phase accurate and therefore suitable for describing many

wave cycles and wave interaction problems. Hegab [21] used the same numerical method in

similar problems with good numerical predictions.

3.1.a Boundary ConditionsThere are three kinds of boundary conditions used in this study. The first, case (1) in which

the air is injected uniformly across the sidewalls of the channel and the head end is closed

while the other free end is open to atmosphere. Also, variable area nozzles are changeably

added at the open end. At the head end, x=0 the no slip boundary conditions are applied, u=

v=0. While, at the duct exit, the pressure is taken as an atmospheric pressure and the other

parameters are extrapolated using zero gradient at exit. At the sidewalls, the no slip condition

is applied for the streamwise component of velocity, but the crosswise velocity is taken to be

the injection velocity V'inj. The second, case (2) of the boundary conditions is concerned with

studying the internal flow field when only acoustic waves are generated at the closed end

while the other free end is open to atmosphere. Also variable area nozzles are added to thechannel free end. The same boundary conditions for the case (1) are applied except that; the

no slip condition is applied also to the velocity components at sidewalls, i.e. u=v=0. At the

head end, x=0, the streamwise velocity is considered in the form u= sin (t). In the third,

case (3) of boundary conditions the air is injected uniformly across the sidewall of the

chamber and acoustic waves are generated at the head end while the other free end is open to

atmosphere. The variable area nozzles are also included at the free end. The same boundary

conditions for the case (1) are applied except that; at the head end (x=0), u= sin (t). At the

sidewalls, v'(X', 0) =V'inj and v'(X', 2H') =-V'inj, (v=v'/V'inj, V'inj=0.1273 m/s)

3.2. Analytical Solution

3.2.a Analytical Solution for Case (1)

In this case a steady uniform injection from the sidewalls of square-cylinder channel is

considered. Culick [2] and Hegab [10] showed that the axial velocity distribution is

independent of viscosity when Reynolds number is sufficiently large. Culick [2] derived the

following equations for axial, transverse velocities and static pressure:

o

u ycos[( ) (1 )]

U 2 H

= (5-a)


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inj

v ysin[( ) (1 )]

V 2 H

= (5-b)

2( / )o

o atm

P PL

P P

=

(5-c)

Where Uo is the center line dimensionless velocity at the channel exit, and Po is thedimensionless stagnation pressure at the channel head end.

3.2.b Analytical Solution for Case (2)

In the case (2), endwall disturbance is fitted at the duct head end while the other end is open

to atmosphere. The analytical approach is based on the reduced form of the Navier-Stokes

equations using asymptotic techniques [9, 14 and 20]. The final solution for pressure and

velocity is as follows:

The non resonance acoustic solution for axial velocity is:

2 2 20

( , ) sin 2 ( sin sin )sinn nn

n n

u x t t t b t b xb b

== + %

forbn* (6)

Where bn is the wave number, ( 1/ 2)nb n = + with n as the wave number index, is theforced frequency and bn* is the wave number at resonance conditions. The perturbation

pressure ( , )p x t% can be obtained using the following equation [20]:

( , ) 1 ( , )u x t p x t

t x

=

%%(7)

The boundary condition at the duct head end, x=1 reads p=1. Integrating equation (7) yieldsthe pressure perturbation in the form [20]:

20

2( , ) ( 1)cos (cos cos )cos

n nn

n

p t x x t t b t b xb

== % (8)

The solution for resonance case can be also found in [20] as:

* * *

*

1( , ) sin { sin( ) cos( )}sin( )n n n

n

u x t t b t t b t b xb

= +%

( )** * *( , ) cos( ) ( 1) 2 cos sin cosnn n np x t t x b t b t b t b x = % (9)

Where *nb = and represents the wave amplitude

4. Results and DiscussionsThe present results are classified into three categories. The first category of the results (case

(1)) deals with the steady state internal flow generated by a steady mass addition from the two

sidewalls of the square-cylinder channel (nozzless duct). Moreover, the effect of adding

convergent and convergent-divergent area nozzles to the end of the duct is considered. The

second category of the results (case (2)) considers the same channel with endwall disturbance


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at the head end, while variable area nozzles are added changeably at the channel exit. The last

category (case (3)) deals with the channel as in the second category with the presence of

uniform injected air from permeable sidewalls.

The axial pressure distribution along the chamber wall is shown Fig. 2 for experimental,

numerical and analytical solutions as in case (1). In such case, air is injected uniformly acrossthe sidewalls of the square chamber with 0.1273 /inj

V m s = and the head end is closed while

the other free end is open to atmosphere (nozzless duct). It is noticed that, the static pressure

reaches its maximum value at the closed duct head end at ( / 0X L = ) and graduallydecreases toward the channel exit. The comparison between the numerical and analytical

approaches with the experimental data shows reasonable agreement. In Fig. 3, the axial

velocity distribution across the channel height from the present numerical code, the analytical

solution and the results obtained using the commercial code Fluent [22] at different axial

locations is shown. The development of the velocity profiles is attributed to the mass addition

from the walls as axial distance increases. It is noted that the velocity profiles are found to be

very close to the laminar velocity profiles and symmetrical about the centerline. Consistent

results for Fig. 3 have been shown also in [10]. Figure 4 shows transverse velocity at different

axial locations for the same conditions given in Fig. 3. The numerical and analytical resultsare almost coincides in the all domain, while small deviation occurs near the duct inlet and

exit. This deviation is attributed to the increase in the axial velocity due to the mass

accumulation at aft end and in turn makes the flow to be convected towards downstream as

soon as it is injected from the walls. However, the analytical solution is proved to be the best

choice to simulate such flowfield case when the air is injected uniformly along the sidewall,

like the gasification of propellant in SRMs.

Experimental data

0 0.2 0.4 0.6 0.8 10.1 0.3 0.5 0.7 0.9X' / L'

1.00000

1.00010

1.00020

1.00030

1.00040

1.00050

1.00005

1.00015

1.00025

1.00035

1.00045

P'/P'atm

Numerical 1

Analytical

Fig. 2 Static Pressure distribution along the duct wall for nozzless duct

without acoustic (case (1)).[1-persent code]


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Numerical 2Analytical

0.0

0.2

0.4

0.6

0.8

1.0

0.1

0.3

0.5

0.7

0.9

Y

'/2H'

Numerical 1

0 2 4 61 3 5 7

U' (m/s)

Fig. 3 Transverse axial velocity distribution at different axial locations for

nozzless duct without acoustic (case (1)).[1-persent code, 2-Fluent code]

X=0.25

X=0.5

X=0.75

0.0

0.2

0.4

0.6

0.8

1.0

0.1

0.3

0.5

0.7

0.9

Y'/2H

'

0.0

Analytical

-0.1 0.1-0.15 -0.05 0.05 0.15

v'(m/s)

X=0.95

Fig. 4 Transverse velocity distribution for nozzless channel without acoustic

(case (1)).

The effect of adding convergent area nozzle on the static pressure distribution along the wall

of the channel for case (1) is shown in Fig. 5. The convergent area nozzle has total height at

exit of 2 0.65exH cm = and its converging length about 5 cm. It is appeared that the static

pressure decreases slowly through the channel and then it decreases with strong gradient

through the nozzle due to the flow acceleration in the nozzle. The over-prediction of the

experimental results with numerical may be related to uncertainty of the experimental data.


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L' L'c

H'ex

H'

0.0 0.4 0.80.2 0.6 1.0

X'/L'

1.000

1.001

1.002

1.003

1.004

1.005

P'/P'atm

Experimental

Numerical 1

1.1136

Fig. 5 Static Pressure distribution along the wall of duct ended by convergent

nozzle, 2H'in=2.5 cm, 2H'ex=0.65 cm, L'c=5 cm and V'inj=0.1273 m/s,

(case (1)).

The second set of the results for case (2) discusses the internal flowfield due to endwall

disturbance using reciprocating piston at the head end of the channel along with and without a

convergent nozzle added at the end of the channel. The main purpose of this set is to describe

the effects of forced frequency, axial location and the nozzle geometry on the generated

acoustic field inside the interior cavity. In Figs. 6 and 7, a comparison between the

experimental and numerical results for nozzless duct at the same dimensional flow

characteristics as in Fig. 5 is presented. These Figures demonstrate the pressure time history at

two dimensionless distances from the head end or to be more precise from the top dead center

of the piston ( / 0.125X L = and 0.557 ) and constant dimensionless forced frequency=0.65. It is observed that, there is small deviation in the static pressure mean value and the

phase angle between the experimental and numerical results. Moreover, higher harmonic

oscillations are seen in the experimental data which represents the eigenfunction mode

contributions. The reasons behind these deficiencies may be related to the sudden change of

cross-sectional area from circular duct (in which the piton moves) at the TDC to the squarecross-sectional channel. Moreover, using slider-crank shaft mechanism cant produce a pure

sinusoidal piston motion. Since the piston movement is not purely sinusoidal, many

harmonics are introduced into the acoustic velocity and pressure fields as discussed by

Barron et. al [4]. Furthermore, it is found that, the amplitude of the measured static pressure

oscillation decreases as the axial distance increases downstream. That may be attributed to the

wave attenuation due to the boundary layer development on the duct sidewalls and acoustic

streaming.


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L

2HcL

cR

sinu t =

Numerical 1

0 20 40 60 80 100

t

0.992

0.996

1.000

1.004

1.008

P

'/P

'atm

Experimental

Fig. 6 Pressure-time history for nozzless channel fitted by forced

oscillation, =0.65, =0.045 and X'/L'=0.125 (case (2)).

0 20 40 60 80 100

t

0.992

0.996

1.000

1.004

1.008

P

'/P

'atm

Experimental

Numerical 1


oscillation, = 0.65, =0.045 and X'/L'=0.557 (case (2)).


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Figure 8 represents a comparison between analytical and numerical solutions for the pressure

time history at =1.1, =0.1, / 0.125X L = and =20. There are small deviations in bothphase and amplitude between the numerical and analytical results. This deviation may be

related to the assumption with the analytical solution that considers an incompressible,

inviscid flow and linear system. Similar trend was found by Hegab [10]. During one complete

cycle of the piston, the acoustic wave completes two round trips across the channel length asdiscussed by Sileem and Nasr [11]. A comparison of the analytical and numerical results near

resonance frequency for pressure time history is presented in Fig. 9 at =1.55, =0.1,

/ 0.557X L = and =20. For t10, the results show good agreement in amplitude and phase

Analytical

0 20 40 60 80 100

t

0.960

1.000

1.040

0.940

0.980

1.020

1.060

P'/P

'atm

Numerical 1


oscillation, =1.1, X'/L'=0.125, M=0.1 and =0.1 (case (2)).

0 10 20 30 40 50

t

0.6

0.8

1.0

1.2

1.4

0.5

0.7

0.9

1.1

1.3

1.5

P'/P'atm

AnalyticalNumerical 1


oscillation, =1.55, X'/L'=0.557, M=0.1 and =0.1 ((case (2)).


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between numerical and analytical approaches. While for t>10 the analytical solution shows

linear growth with time rather than beats with the numerical solution. Similar results were

previously noticed by Hegab [21], who discussed this phenomenon and gave interpretation to

this trend as the nonlinearity effect with the numerical solution.

The effect of forced frequency on the axial velocity time history is introduced in Figs. 10 and11. Data are recorded at / 0.125X L = and =0.52 and 0.65, respectively. It is observedthat, increasing the forced frequency leads to an increase of the axial velocity amplitude with

the same phase angle.

0 20 40 60 80 100

t

-0.2

0

0.2

-0.3

-0.1

0.1

0.3

u'/U

'o


Fig. 10 Axial velocity time-history for nozzless channel fitted by forced

oscillation, =0.52, X'/L'=0.125 and =0.1 (case (2)).

0 20 40 60 80 100

t

-0.2

0

0.2

-0.3

-0.1

0.1

0.3

u'/U

'o


Fig. 11 Axial velocity time-history for nozzless channel fitted by forced

oscillation, =0.65, X'/L'=0.125 and =0.1 (case (2)).


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Figures 12 and 13, show the acoustic axial velocity across the channel at x=0.25 and 0.5 for

=1.5, M=0.1 and =0.1 at four times t=10, 20, 40 and 78. One can observe the presence of

the acoustic boundary layer development at the same location with different boundary layer

thickness. Rotational flow is considered to be confined in that region. The explanation is

related to the interaction between the duct sidewalls and the acoustic wave motion. Baum [23]

noticed an overshoot of the axial velocity at the edge of acoustic boundary layer and showedthat, acoustic boundary layer thickness and axial velocity overshoot at its edge are strongly

influenced by mean flow Reynolds number and frequency of oscillations.

0.50

0.60

0.70

0.80

0.90

1.00

Y'/

2H

'

t=78

t=40

t=20

-1 0 1 2-0.5 0.5 1.5

u' / U'o

t=10

Fig. 12 Transverse axial velocity distribution for nozzless channel fitted by

forced oscillation, =1.5, X'/L'=0.25 and =0.1 and (case (2)).

0.50

0.60

0.70

0.80

0.90

1.00

Y

'/2H

'

t=78

t=40

t=20

-1 0 1 2-0.5 0.5 1.5u' / U'o

t=10

Fig. 13 Transverse axial velocity distribution for nozzless channel fitted by

forced oscillation, =1.5, X'/L'=0.5 and =0.1 (case (2)).


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The effect of installed convergent and convergent-divergent nozzles at the end of the channel

on the complex wave interaction mechanism inside the chamber is illustrated experimentally

in Figs. 14 and 15, respectively. The angular frequency is taken to be =0.52, and the

convergent nozzle length and total exit height are 5cL cm = , 2 0.65exH cm = , while the

convergent-divergent nozzle length, total throat and exit heights are

14cdL cm = , 2 0.65thH cm = and 2 3exH cm = , respectively. The exit height for the convergentarea nozzle is equal to the throat height of the convergent-divergent area nozzle with an equal

converging length. It is shown that, the static pressure amplitude for channel ended with

nozzles (at x=0.125) has higher amplitude with deviation in the mean value compared to

nozzless duct. It may be attributed to the decreasing of the flow area at the nozzle exit which

leads to a reduction in transmitted energy waves from nozzle exit and hence increases the

interactions between incident and reflected waves from boundaries which in turn increases the

pressure amplitude. The nozzle and nozzless results show higher harmonic oscillations as

discussed by Barron et. al [4].

0 20 40 60 80 100

t

0.992

0.996

1.000

1.004

1.008

P'/P'atm

Nozzless channel

Channel ended by convergent nozzle

Fig. 14 Pressure-time history for channel fitted by forced oscillation ended

by convergent nozzle,(2H'ex=0.65 cm) and nozzless channel

at =0.045, =0.52 and X'/L'=0.125 (case (2)).


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0 20 40 60 80 100

t

0.992

0.996

1.000

1.004

1.008

P'/P'atm

Nozzless channel

Channel ended by convergent-divergent nozzle

Fig. 15 Pressure-time history for channel fitted by forced oscillation endedby convergent-divergent, (2H'th=0.65 cm, 2H'ex=3 cm)and nozzles

channel at X'/L'=0.125, =0.52 and =0.045 (case (2)).

Moreover, the significant difference between the convergent and convergent-divergent nozzle

results is presumably related to the deviation of the natural frequency producing by adding

variable area portions at the end of the duct . Also, it is noticed that, the effect of adding

convergent-divergent nozzle results in higher amplitude in static pressure but still lower

amplitude compared with that of convergent nozzle. Sileem and Nasr [11] and Hegab [10]

have noticed similar trend in their results.

The effect of adding convergent and convergent-divergent nozzle at the open end of thechannel with sidewall mass addition is shown in Fig. 16 for case (3). The sidewall injection

velocity is 0.1273 /inj

V m s = and endwall disturbance at the head end with =0.65. It is

shown that, the steady sidewall injection reduces the pressure amplitude significantly.

(Compare Figs. 15 and 16). It may be explained that, the flow is injected through porous

surface on the duct sidewalls across two parallel perforated plates which act as jet velocity

through the generated flow field and hence attenuate the acoustic energy especially when that

energy is nearly small and the effect of exit from the nozzle doesnt play an explicit role in the

pressure amplitude.


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0 20 40 60 80 100

t

0.992

0.996

1.000

1.004

1.008

P'/P'atm

Channel ended by convergent nozzle

Nozzless channel

Channel ended by convergent-divergent nozzle

Fig. 16 Pressure-time history for nozzless channel, channel ended by convergent

nozzle and convergent-divergent at forced frequency

=0.65, =0.045 and X'/L'=0.125 (case (3)).

The effect of the exit height on the acoustic flowfield generated in the duct ended by

convergent nozzle (without air flow injection) is shown in Fig. 17 for case (2) at

/ ' 0.125X L = , = 0.52 and different exit heights; 2 1.27,0.95exH = and 0.65cm . It is

shown from this figure that, with deceasing the exit height of the convergent nozzle, the

pressure amplitude increases. It is clearly seen by decreasing the exit height causes an

increase in the amount of reflected acoustic energy which in turn interacts with the incident

waves from the piston and hence enhances the pressure amplitude. Figure 18 shows the effect

of throat height on the pressure time history for duct ended with convergent-divergent nozzle.

It is concluded that, at the same forced frequency and axial location, for no air flow injection;

decreasing the throat height increases the pressure amplitude but still smaller than the effect

of adding a convergent nozzle with the same throat heights.

The power spectrum density (PSD) is calculated from the experimental data for case (2) and

is shown in Fig. 19 for nozzless channel and without injection. It is shown that, the gas

oscillates at the same frequency of the piston (79.6 HZ). Higher harmonics of very small

energy content could be noticed before and after the main frequency.

The effect of adding convergent nozzle by different exit heights on the energy content in PSD

is shown in Fig. 20. It is concluded that, decreasing the exit height of the convergent nozzleincreases the amount of reflected waves which in turn increases the wave's interactions and

energy content. Moreover, higher harmonics increases as the exit height of the convergent

nozzle decreases. This is an indication of small wave deformation. This confirms the

aforementioned understanding of the mechanism of wave amplitude increasing, which is

mainly due to off-resonant situation caused by the exit area reduction.


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0 20 40 60 80 100

t

0.992

0.996

1.000

1.004

1.008

P'/P'atm

2H'ex=1.27 cm

2H'ex=0.95 cm

2H'ex=0.65 cm

Fig. 17 Pressure-time history for the channel ended by convergent nozzle at

different exit heights at forced frequency =0.52, =0.045 and

X'/L'=0.125 (case (2)).

0 20 40 60 80 100t

0.992

0.996

1.000

1.004

1.008

P'/P'atm

2H'th=1.27 cm

2H'th=0.65 cm

2H'th=0.95 cm

Fig. 18 Pressure-time history for the channel ended by convergent-divergent

nozzle at different throat heights at forced frequency =0.65, =0.045 and

X'/L'=0.125 (case (2)).


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21/24

0 40 80 120 160 200

0

4E-005

8E-005

0.00012

0.00016

0.0002

PSD

Fig. 19 Power spectrum density for nozzless channel at X'/L'=0.125 (case (2)).

0 40 80 120 160 200

0

4E-005

8E-005

0.00012

0.00016

0.0002

P

SD

2H'ex=1.27 cm

2H'ex=0.95 cm

2H'ex=0.65 cm

Fig. 20 Power spectrum density for channel ended by convergent

nozzle at X'/L'=0.125 (case (2)).

PSD for channel ended by convergent-divergent nozzle is shown in Fig. 21. It is obvious that,

installing convergent-divergent nozzle at the channel end increases the energy content of the

wave. Moreover, higher harmonics are clearly seen in this figure. It may be related to energy

dissipation and transfer from the fundamental mode to its higher harmonics. Also the

frequency deviation for the three throat heights may be result from the difference of the

natural frequency between the nozzless channel and channel ended with convergent-divergent

nozzle. Moreover, decreasing the throat height increases the PSD at the same frequency. The

maximum and minimum pressure amplitudes for nozzless channel, channel ended by

convergent nozzles and convergent-divergent nozzles are shown in tables (2), (3) and (4),

respectively.


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22/24

0

4E-005

8E-005

0.00012

0.00016

0.0002

PSD

2H'th=1.27 cm

0 40 80 120 160 200

2H'th=0.95 cm

2H'th=0.65 cm

Fig. 21 Power spectrum density for channel ended by convergent-divergent

nozzle at X'/L'=0.125 (case (2)).

Table (2) The maximum and minimum pressure amplitudes for nozzless channel.

Pmin Pmax

0.52 0.9955 1.0029

0.63 0.99515 1.003

0.65 0.99405 1.0032

Table (3) The maximum and minimum pressure amplitudes for channel ended by

convergent nozzle.

2H'ex=1.27 cm 2H'ex=0.95 cm 2H'ex=0.65 cm

Pmin Pmax Pmin Pmax Pmin Pmax

0.52 0.99732 1.0026 0.99612 1.0038 0.99512 1.00425

0.63 0.9953 1.00275 0.99525 1.0039 0.995 1.0049

0.65 0.99517 1.0028 0.99745 1.0033 0.99472 1.00527

Table (4) The maximum and minimum pressure amplitudes for channel ended by

convergent-divergent nozzle.

2H'th=0.65cm

and

2H'ex=3 cm

2H'th=0.95cm

and

2H'ex=5.78 cm

2H'th=1.27 cm

and

2H'ex=8.725 cm

Pmin Pmax Pmin Pmax Pmin Pmax

0.52 0.99599 1.00388 0.9951 1.00382 0.99737 1.0027

0.63 0.99633 1.00498 0.9952 1.0041 0.9955 1.0038

0.65 0.99521 1.0052 0.9956 1.0042 0.99618 1.00395


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5. ConclusionsThe effect of adding convergent and convergent-divergent nozzles at the end of square-

cylinder channel on the internal flow field is studied experimentally, analytically and

numerically. The flow field in the channel is generated by either steady mass addition from

sidewalls and/or endwall disturbance. The following statements could be drawn from the

current study:1. For the considered injection mass rate, the whole flow is found to be subsonic laminar.

2.The flow is highly rotational near walls and it then weakens until reaching the core region

as demonstrated in case (2).

3. The internal acoustic flowfield is highly affected by some parameters such as forced

frequency, boundary conditions treatment and exit area geometry.

4.Adding a convergent nozzle at the end of the channel leads to partially transmitted acoustic

waves at the duct exit and the remained are reflected from the solid walls. The latter increases

when the exit height decreases and hence increasing the wave amplitude.

Finally, the effect of convergent-divergent nozzle existence at the end of the channel shows

wave interaction mechanism, which is completely different than that with either nozzless or

convergent nozzle existence. Moreover, the flow injected through the porous sidewallattenuates the generated acoustic field throughout the channel. These acoustic waves interact

with the steady sidewall injection to generate vorticity across the chamber. The generated

rotational flow may impact the burning rate in the real solid rocket motor propellant.

6. AcknowledgmentThis work is supported by the Science and Technology Development Fund (STDF) through

the project ID-108.

References:[1] Brown, R.S., Blackner, A.M., Willoughby, P.G., and Dunlap, R., "Coupling between

Acoustic Velocity Oscillations and Solid Propellant Combustion", J. of Propulsion and

Power, Vol. 2, No.5, 1986, pp. 428-438.

[2] Culick, F.E.C., "Rotational Axisymmetric Mean Flow and Damping of Acoustic Waves

in a Solid Propellant Rocket", AIAA Journal, Vol. 4, No. 8, 1996, pp.1462-1464.

[3] Ma, Y., "A Simulation of the Flow Near a Burning Propellant in a Solid Propellant

Rocket Motor", Ph D. Thesis, University of Utah, Salt Lake City, USA, 1990.

[4] Barron, T., Van Moorhem, W.K, and Majdalani, J., "A Novel Investigation of the

Oscillatory Field over a Transpiring Surface", J. of Sound and Vibration, Vol. 235, No.

2, 2000, pp. 281-297.

[5] Dunlap, R., Willoughby, P.G, and Hermsen, R.W., "Flow-Field in the Combustion

Chamber of a Solid Propellant Rocket Motor", AIAA journal, Vol. 12, No. 10, 1974,pp.1440-1442.

[6] Flandro, G.A., "Solid Propellant Acoustic Admittance Corrections", J. of Sound and

Vibration, Vol. 36, No. 3, 1974, pp. 297-312.

[7] Zhao, Q., Kassoy, D.R, and Kirkkorpru, K., "Acoustically Generated Vorticity in an

Internal Flow", J. of Fluid Mech., Vol. 413, pp. 247-285.

[8] Erickson, R., Markopoulos, N., and Zinn, B.T., "Finite Amplitude Acoustic

Waves in Variable Area Ducts", AIAA paper 2001-1099, 39th

Aerospace Sciences

Meeting and Exhibit, Jan 8-11, Reno, NV., USA.


24/24


[9] Deng, Z., Addrian, R.J, and Tomkinson, C.D.," Structure of Turbulence in Channel

Flow with a Fully Transpired Wall", AIAA paper 2001-1019, 39th

AerospaceSciences Meeting and Exhibit, Jan. 8-11, Reno, NV, USA.

[10] Hegab, A.M., "Study of Acoustic Phenomena in a Solid Rocket Engines", Ph.D. Thesis,

Mech. Power Engineering Dept., Menoufiya Univ., Egypt, carried out at Center for

Combustion Research, University of Colorado at Boulder, USA ,1998.[11] Sileem, A.A., and Nasr, M., "An Experimental Study of Finite Amplitude Oscillations

in Ducts: The Effect of Adding Variable Area Part to the Open End of Constant-Area

Resonant Tube", Alexandria Engineering Journal, Vol.42, No. 4, 2003, pp. 379-409.

[12] Majdalani, J., Sams IV O.C., and Saad, T., "Mean Flow Approximations for Solid

Rocket Motors with Tapered Walls", J. of Propulsion and Power, Vol. 23, No2. , 2007,pp. 445-456.

[13] Nguyen, C.C., Plourd, F., and Doam, S.K., "Analysis of Injecting Wall Inclination on

Segmented Solid Rocket Motor Instability", J. of Propulsion and Power, Vol. 24. No. 2,

2008, pp. 213-223.

[14] Hegab, A.M., and Nasr, M., "Unsteady Vorticity Fields in a Long Narrow Channel with

Endwall Disturbances and Sidewall Injection",EG-220, 9th International Congress ofFluid Dynamics & Propulsion, Dec. 18-21, 2008, Alexandria, Egypt.

[15] Coleman, W.H., and Glenn S.W., "Engineering Application of Experimental

Uncertainty Analysis", AIAA J, Vol. 33, No. 10, 1995, pp. 1888-1896.

[16] Anderson, J.D., "Computational Fluid Dynamics", McGraw-Hill, Inc, New York, NY,

USA, 1995.

[17] Hegab, A.M., " Vorticity Generation and Acoustic Resonance of Simulated Solid

Rocket Motor Chamber with Transient Sidewall Mass Injection", Proceedings of the 9th

International Congress of Fluid Dynamics & Propulsion, ICFDP9-265 December 18-21,

2008, Alexandria, Egypt.

[18] MacCormack, R.W., "The Effect of Viscosity in Hypervelocity Impact Crating", AIAA

Hypervelocity Impact Conference, Cincinnati, Ohio, April, 1969, No.69-354.

[19] Gottlieb, D., and Turkel, E., "Dissipative Two-Four Methods for Time-Dependent

Problems", Int. J. Mathematics and Computation, Vol. 30, No. 136, 1976, pp. 703-

723.[20] Kirkkopru, K., Kassoy, D.R., and Zhao, Q., "Unsteady Vorticity Generation and

Evaluation in a Model of Solid Rocket Motor", J. of Propulsion and Power, Vol. 12,

No.4, 1996, pp. 646-654.

[21] Hegab, A.M., "Vorticity Generation and Acoustic Resonance of Simulated Solid Rocket

Motor Chamber with High Wave Number Wall Injection", J. of Computers & Fluids,

Vol. 38, No. 6, 2009, pp. 1258-1269.[22] Fluent-6.3.26. User's Manual, (2006), Fluent Inc., Lebanon, USA

[23] Baum, J.D., Investigation of Flow Turning Phenomenon Effects of Frequency and

Blowing Rate, AIAA paper 1989-0297, 27th

Aerospace Sciences Meeting and Exhibit,

Jan. 9-12, Reno, NV., USA.

pp12 nozzle cd

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