jpp ben - auburn universitymajdalani.eng.auburn.edu/publications/pdf/1997 - entezam...pulse...

AIAA 97-2718Modeling of a Rijke-Tube Pulse CombustorUsing Computational Fluid Dynamics B. Entezam and W. K. Van MoorhemUniversity of UtahSalt Lake City, UT 84112

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Conference and Exhibit 6–9 July 1997 Seattle, WA

1American Institute of Aeronautics and Astronautics

MODELING OF A RIJKE-TUBE PULSE COMBUSTORUSING COMPUTATIONAL FLUID DYNAMICS

B. Entezam* and W. K. Van Moorhem†University of Utah, Salt Lake City, UT 84112

andJ. Majdalani‡

Marquette University, Milwaukee, WI 53233

Abstract!

A computational fluid dynamics (CFD) technique isused to investigate the unsteady flowfield inside a Rijketube. This investigation is carried out in an attempt toexplain the coupling that exists in such an environmentbetween heat addition, pressure and velocityoscillations. Similar coupling may exist in variouspulse combustion engines and unstable rocket motors.Due to inherent model complexities, there appears tohave been no direct attempts in the past to approach thisparticular problem numerically. The present modelingemploys unsteady, compressible, two dimensionalflowfields that incorporate heat addition and acousticinteractions. The results show a relationship betweenpressure, velocity and heat transfer oscillations. As onemight expect, heat transfer oscillations are found to bedirectly dependent upon the product of acoustic velocityand pressure. A possible explanation of the heattransfer mechanisms causing heat driven oscillations isoffered in view of the computational findings. Due tothe absence of complete analytical theories regardingthis particular subject, the current numerical solution ishoped to serve as a base upon which a well-formulatedtheory could rely.

Nomenclaturea0 = mean speed of sound inside the Rijke tube

A = oscillatory pressure amplitudeAobs = surface area of obstacle or heater element

Cp = constant pressure specific heat

h = heat transfer coefficientl = internal tube lengthm = longitudinal oscillation mode, m " 1,2,3, ...

p ( )1 = oscillatory pressure component

Q = heat

!*Group Scientist, Andrulis Corporation, Member AIAA.! †Professor, Department of Mechanical Engineering. SeniorMember AIAA.!‡Assistant Professor, Department of Mechanical andIndustrial Engineering. Member AIAA.Copyright © 1997 by B. Entezam, J. Majdalani and W.K.Van Moorhem. Published by the American Institute ofAeronautics and Astronautics, Inc., with permission.

q = heat transfer rate, dQ dt/

q (1) = oscillatory heat transfer rate

T = temperaturet = time

u(1) = oscillatory velocity componentx = axial distance from the bottom-end# = mean ratio of specific heats

$ = acoustic spatial wavelength% = air density

& = circular frequency, m a l' 0 /

Subscriptsin = refers to an incoming quantityobs = refers to the obstacle or heat sourceout = refers to an outgoing quantity( = surrounding mean flow condition

Superscripts(0) = denotes a steady or mean component(1) = denotes an unsteady/oscillatory component

IntroductionThe Rijke tube has been investigated both

experimentally and theoretically1-9; however, none ofthe previous investigations offered a detailedexplanation of the heat transfer mechanisms causing theheat driven oscillations. Due to the model complexity,there has been no pervious attempt to approach theproblem numerically. Due to the lack of sufficientexperimental data that can be used to verify or validatethe semi-analytical formulations, and due to the absenceof numerical results, most theories have remainedlimited and incapable of explaining the observedphenomena. For those reasons, the focus of thisinvestigation will be to model the Rijke tubenumerically, hoping to provide additional informationand explain some of the observed phenomena. We alsohope that the results of this study could be used to pavethe way to more accurate theoretical formulations.

The present analysis begins with a brief classificationof three types of pulse combustors, followed by adescription of the self-excited oscillator that defines thecharacter of the Rijke tube. Next, the computationalmodel is presented, followed by a discussion of the key


results. The numerical results will be shown to be inaccord with existing experimental and theoreticalpredictions to the point of providing useful tools anddetailed flowfield descriptions that tend to clarifyexisting speculations and furnish new means to improveexisting analytical models. In relation to solid andhybrid rocket motors, they will predict a strongcoupling between heat transfer fluctuations andfluctuations in the pressure and velocity fields that isparticularly significant in the forward half of the motor.

ClassificationThe so-called pulse combustors10 that have been

developed to date can be divided into three categoriesaccording to their operational system geometry. Theseare: 1) the Schmidt tube (closed-open, quarter-wavesystem), 2) the Helmholtz resonator (closed-open oropen-open system), and 3) the Rijke tube (open-open,half-wave system).10

The Schmidt tube that was developed in Germany11

can be generally divided into three sections: 1) theinlet, 2) the combustion chamber, and 3) the tail orexhaust pipe (see Fig.1). The inlet section consists ofone-way valves which open or close depending onwhether the combustor pressure is lower or higher thanthe pressure upstream of the valves.

The Helmholtz resonator12 consists of a rigid-wallcavity which has at least one short and narrow neckthrough which the enclosed fluid can communicate withthe external medium (see Fig. 2). Due to the neckeffect, the size of the device is less than a quarter of awavelength.

Fig.1 Schematics of a Schmidt tube pulse jet engine.

Fig. 2 Simple Helmholtz resonators.

The third type of pulse combustors is the Rijke tubewhere a heat source in the lower half of a vertical tuberesults in acoustic mode excitation.

Rijke TubeAmong the first accounts of thermoacoustical

oscillations is that of Rijke1 in 1859. In his work, Rijkediscovered that strong oscillations occurred when aheated wire screen was placed in the lower half of anopen-ended vertical pipe shown schematically in Fig. 3.Reported acoustic oscillations were found to stopaltogether when the top end of the pipe was sealed,indicating that upward convective air currents in thepipe were essential for thermoacoustically drivenoscillations to take place. As could be inferred fromFig. 3, the Rijke tube is a half wave pulse combustorsince the acoustic wavelength is actually twice thelength of the tube. From experimental reports,oscillations reached maximum amplification when theheater was located at the middle of the bottom half. Atthat location, both acoustic pressure and velocity arenon-zero. For heater positions in the upper half of thepipe, damping instead of driving occurred. Rijkebelieved that rising convection currents expanded in theregion of the heated screen and compressed downstream

L

Sound Generated

Outflow

(Hot Flow)

L/4x<L/2

Inflow

(Convection Flow)

$)*Half WaveLength

P' AcousticPressureStructure

u' AcousticVelocityStructure

Heater Element

Fig. 3 The Rijke tube and associated fundamentalacoustic wave structure.

Tail PipeCombustion ChamberDiffuserAir

Fuel SprayInlet Section

Spark Plug

(a) (b) (c)

x

OutflowHot Air

Inflow Cold Air

l

HeatSource l/4

u(1)

p(1)

Half WaveLength

SoundGenerated

$/2


from the heater due to cooling at the pipe walls.Accordingly, production of sound was attributed tosuccessive expansions and contractions. Thisexplanation remained limited since it could not addressthe details of heat exchange mechanisms causing theactual oscillations.

Since Rijke’s discovery, there has been a number oftheoretical and experimental attempts to explain thisphenomenon1-9. Lord Rayleigh1 proposed a criterion,later named after him, in which he addresses therelationship between heat addition and sound waves.According to Rayleigh’s criterion, energy is fed into anacoustic disturbance when heat is added to a soundwave at the high temperature phase of its cycle;conversely, energy is lost from a sound wave when heatis added at the low temperature phase (Fig. 4). Sincecompression in a sound wave is adiabatic, pressure andtemperature fluctuations are in phase. As a result, heataddition during a positive pressure disturbanceincreases the amplitude of the sound waves. Clearly, anopposite effect is seen when the pressure disturbance isnegative. This is similar to the effect of heat addition inthermodynamic cycles. When heat is added at the highpressure phase of a cycle, the system gains energy. Inthe Rijke tube, the grid (or heat source) heats the airaround it causing it to rise. Acoustically inducedparticle displacements are therefore superimposed onthe naturally convected flow. When acoustic particledisplacements are positive upwards, fresh cold aircrosses the heated grid, but when negative downwards,hot air passes through the grid. During the upwardpass, maximum heat transfer occurs between the heatsource and the air. Since the timing in the acousticcycle is such that maximum heat transfer corresponds toa positive particle displacement with favorable pressure,an ideal situation for acoustic wave growth isestablished.

Heat addition increasesacoustic amplitude -

+

Time

T p(1) (1),

Heat addition decreasesacoustic amplitude

Fig. 4 Rayleigh’s criterion.

If the grid is now placed in the upper half of the pipe,acoustic velocity and pressure have an unfavorablephase between them. Pressure acts adversely, in adirection that opposes particle motion. Under suchunfavorable environment, acoustic amplification is notpossible. A similar mechanism is used to explain theso-called reheat buzz, a jet engine instability of thereheat system (after-burner) which can be triggered in asimilar fashion. There, acoustic velocity in a pipedisturbs the reheat afterburner flame and alters the rateof combustion. If the phase relationship betweenpressure and heat release rate is suitable, disturbancesgain energy and grow in magnitude. Resulting pressureperturbations become large and lead to undesirablestructural damage.

The Self-Excited OscillatorThe wave equation with the heat addition term acting

as a driving function can be written for acousticpressure and velocity as follows.14

2 (1) (1)2 (1)

2 2 (0)0

1 1

p

p qp

tt C Ta

+ +,- "

++(1)

2 (1) (1)2 2 (1)

02 (0)

1

u qa u

xt

#%

+ , +, - " ,

++(2)

Equations (1) and (2) are characteristic of self-excitedor “feedback” oscillations owing to the nature of theright-hand-side terms. Self-excited oscillations aredifferent from the majority of oscillation-type problemsin nature which are either of the free or of the forcedtype.

In a free oscillator, the amplitude decreases with timeas a result of friction. In a forced oscillator, the energyassociated with the motion is typically supplied by anoscillating force, an electric current or fluid flow orheat, which is modulated by the oscillator. This is thecase in various mechanical clocks, electronicoscillators, musical wind instruments, bowed stringinstruments, the human voice, pulse combustions, andnumerous types of whistles, control valves and pipes.In a forced oscillation, the sustaining alternating forceexists independently of the motion and persists evenwhen the oscillatory motion is stopped.

In a self-excited oscillation, the alternating forcesustaining the motion is induced or controlled by themotion itself; when the motion stops, the alternatingforce disappears. Under certain conditions, the motionof an oscillator generates disturbances which can be fedback to the energy associated with the existingoscillations, acting effectively as a driving force. Suchfeedback can cause the amplitude to grow rather than


decay. Since the unsteady driving force q ( )1 that

appears in Eqs. (1-2) is not externally controlled, butrather induced by fluctuations in other properties withinthe system, the Rijke tube is of the self-excited type.

As could be inferred from Eqs. (1-2), in order to seekanalytical solutions to the problem and determine theoscillatory pressure and velocity distributions, the

dependence of q ( )1 on the acoustic pressure and/or

velocity must be known. Conditions controlling theflowfield on both sides of the heat source must also beknown. For such reasons, one of the primary goals of

this study will be to determine the form of q ( )1 , relating

it to p(1) , u(1) , or both.

A literature search has revealed that no exact closed

form expression relating p(1) , u(1) , and q ( )1 has ever

been derived. Computationally, no numerical techniquehas ever been attempted to extract information that canbe exploited to achieve a better understanding andcharacterization of the peculiar coupling mechanismassociated with the primary variables. In the remainderof this work, a theoretical explanation of the couplingmechanism will be sought via numerical analysis in anattempt to unveil the true character of the couplingmechanism and, particularly, to establish a functional

dependence relating p(1) , u(1) , and q ( )1 .

The Computational ModelThe two-dimensional unsteady flowfield inside the

Rijke tube has been computed with FLOW-3Ddeveloped by Flow Science Corporation.15

The pipe is 90 centimeters in length and has aninternal diameter of 5 centimeters. Only one crosssection of the pipe is modeled, thus taking advantage ofprevalent geometric symmetry (since the pipe is a bodyof revolution and material properties, boundaryconditions and other effects are symmetric with respectto the centerline). This problem consists of modeling ahollow cylinder inside a box. The hollow cylinderrepresents the Rijke tube and the box itself representsthe room. Figure 5 gives a three-dimensional renderingof the Rijke tube inside the room. The computationalmesh for the domain representing half of the Rijke tubeand room in the x-z directions is defined in Fig. 6. Theambient atmospheric conditions, including pressure,temperature and density, are used to define the initialconditions at pipe inlets and outlets and inside the box.Air properties at standard sea level are used to calculatethe density, thermal conductivity, specific heat, gasconstant, and dynamic viscosity.

A solid porous obstacle with a diameter of 3.75 cm isadded inside the tube at a location of 22.5 cm from thebottom-end to model the heat source. An obstacle

porosity value of 0.9 was used which represents a 90percent open area. Thermal conductivity, density, andheat capacity of steel were used to specify the obstacleproperties. Heat had to be released inside the obstacle(at the source) with a time step shown in Fig. 7, sincereleasing the heat suddenly (say, at t = 0 sec) causes theprogram to crash.

Numerical StrategyThe numerical strategy consists of two stages

reminiscent of transient and steady-state stages. Thefirst stage carries the problem from an initial state ofrest to a time of 20 seconds. After the first 20 secondsthe problem reaches a terminal condition characterizedby the presence of constant amplitude oscillations.

The second stage carries the problem from 20seconds to 20.025 with a much smaller time interval inorder to track more precisely the progressive acousticwave growth. Virtual probes are located inside theRijke tube model at numerous axial locations in order tomonitor pressures, temperatures, densities andvelocities. Primary input variables into thepreprocessor are summarized in Table 1 below.

Fig. 5 Three-dimensional representation of theRijke tube inside a room.

x

y

z

50 cm50 cm

Room

RijkeTube

90 cm150 cm HeatSource

5.0 cm


Fig. 6 Computational mesh.

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

250

300

350

400

450

Hea

t Pow

er I

nput

(W

)

Time (s)

Fig. 7 Heat input at the source as a function of time.

Results and DiscussionThe symmetric flowfield inside the Rijke tube has

been computed with FLOW-3D using the fullcompressible flow option. The results will be presentedin several separate sections. Discussion of each topicwill be addressed as pertinent results are presented.

Standard RunFor the standard conditions described in Table 1

(corresponding to standard pipe properties, standardheat source, and standard gas properties), 430 watts ofheat are released inside the tube at a location of l / 4from the bottom.

As the source temperature starts to rise, it produces atemperature difference between the source and thesurrounding air. As a result, the local air density isreduced, causing air to be displaced in a convective cur-

Table 1 Standard input properties____________________________________________

Unit InputStandard Pipe PropertiesMaterial - SteelDiameter cm 5Length cm 90Wall thickness cm 0.5Thermal conductivity W/m/K 36Density-heat capacity J/m3/K 3.77x106

productHeat transfer coefficient W/m2/K CalculatedInitial temperature K 293.0____________________________________________Standard Obstacle PropertiesMaterial - SteelDiameter cm 3.75Thickness cm 1.0Location from bottom cm 22.5Porosity - 0.9Power input W 430Thermal conductivity W/m/K 36Density-heat capacity J/m3/K 3.77x106

productHeat transfer coefficient W/m2/K CalculatedInitial temperature K 293.0____________________________________________Standard Gas PropertiesGas medium - AirGas constant J/Kg/K 287Dynamic viscosity kg/m/s 1.824x10-5

Specific heat J/kg/K 718Thermal conductivity W/m/K 0.0251Initial temperature K 293.0____________________________________________

h

h

RigidWalls

h

HeatTransfer

TestCells

x

zCenter Line

-30.0

-15.0

0.0

15.0

22.5

30.0

45.0

60.0

75.0

90.0

105.0

120.0 cm


rent. The heat transferred from the source to thesurrounding air is a function of source temperature,ambient air temperature, source area, and averageconvection heat transfer coefficient, as given byNewton’s cooling law:

. /q hA T Tout obs obs" , ( (3)

Figure 8 is a graphical representation of the pressure,radial velocity, axial velocity, temperature, density andsource heat transfer versus time for the first 20 seconds.It shows that, after about 8.5 seconds, heat, pressure,and velocity oscillations begin. This coincides with thetime when the air temperature around the sourceapproaches its terminal condition characterized by aleveling out of the temperature curve (Fig. 8d).

Figure 9 shows the pressure, radial velocity, axialvelocity, temperature, density and source to fluid heattransfer versus time for the second stage (20 to 20.025sec) using a smaller time step to track the acoustic wavegrowth more accurately. This figure indicates thatperiodic oscillations are present in all the variables witha frequency of about 200 Hz which matches veryclosely the predicted natural frequency of the pipe givenby f ma l" 0 2/ ( ) .

Figure 10 shows the pressure and axial velocityoscillations at various locations along the pipe. Anexamination of Fig. 10a shows that pressure oscillationsreach their maxima at the center of the pipe and minimaat both ends. It also shows that pressure oscillations arein phase at any axial location, as one would expect fromacoustic wave theory. Figure 10b, on the other hand,shows that velocity oscillations reach their maxima atboth ends and are at a minimum at the center of thepipe. The velocity oscillations are 180° out of phase inthe lower and upper half-domains.

These results exhibit the expected pattern predictedby acoustic theory, eliminating the possibility of beingthe mere outcome of computational error. They alsoagree with experimental observations in Rijke tubes.1-5

Conditions at the Upper and Lower Sides of the SourceFigures 11-13 show the pressure, temperature and

density distributions along the length of the tube. Thenumerical data provides, for the first time, completeinformation regarding the flow properties, especially, atboth ends of the heat source. Such information isnecessary for setting up the boundary conditions that areneeded to formulate an analytical model that allows

determining p(1) , u(1) , or q ( )1 . As one would expect,

the steady pressure (in Fig. 11) is almost constant alongthe pipe. The temperature (in Fig. 12) remains constantin the lower section of the tube; it experiences a sudden

jump crossing the source location; it then decaysexponentially above the source. The density variation,plotted in Fig. 13, shows that the air expands afterpassing through the heat source. This similar butinverted dependence, by comparison to the temperaturevariation, is expected from the ideal gas equation.

Effect of Source LocationThe location of the source is a key factor in

producing oscillations inside the Rijke tube. When thesource is placed in the lower half of the pipe, largeamplitude oscillations are seen to occur. The resultingoscillations are found to have the largest amplitudeswhen the source is located at ( / )l 4 from the bottomend. This is in accord with experimental findings donepreviously.1-5 During separate runs, the source wasrelocated to the middle ( / )l 2 , and to the upper section( / )2 3l of the pipe in order to observe whether or notthe oscillations would occur.

Figures 14-15 are graphical representations of thepressure, radial velocity, axial velocity, temperature,density and source heat transfer versus time at the( / )l 2 and ( / )2 3l source locations. The resultsindicate that when the source is positioned at ( / )l 2 ,where the amplitude of the pressure oscillation ismaximum (corresponding to zero velocity), nooscillations are seen. The same can be said when thesource is at ( / )2 3l , where the amplitude of velocityoscillations is a maximum (corresponding to zeropressure). It is observed that oscillations occur onlywhen the product of the velocity and pressure ispositive, and maximum when this product is amaximum as well. Figure 16 shows the temperaturedistribution of the source versus time at variouslocations. It is encouraging to see that the heat sourcehas a lower temperature when placed at ( / )l 4 than at( / )l 2 and ( / )2 3l . This can be attributed to the largertransfer of energy to the acoustic waves when the sourceis located at ( / )l 4 where heat conversion into acousticenergy is optimized.

Effect of Heat Input to the SourceFor the standard conditions several different heat

inputs (125.5, 376.6, 408.0, 426.9, 430.0, 433.1, and439.4 watts) were released at the source at ( / )l 4 inorder to examine the effect of varying the heat input onthe induced acoustic motion.

Figure 17 is a graphical representation of thepressure, radial velocity, axial velocity, temperature,density and source heat transfer versus time for the first20 seconds. Figure 17a (125.5 watts of heat input) doesnot show any sign of oscillations in the tube. This is anindication that the heat input level, which feeds the


0 4 8 12 16 20

101.4

101.6

101.8

Pre

ssur

e (k

Pa)

Time (s)

(a)

0 4 8 12 16 20

-0.8

-0.4

0.0

Rad

ial V

eloc

ity (

cm./s

)

Time (s)

(b)

0 4 8 12 16 20

0

100

200

300

Axi

al V

eloc

ity (

cm./s

)

Time (s)

(c)

0 4 8 12 16 20

300

400

500

600

700

Tem

pera

ture

(K

)

Time (s)

(d)

0 4 8 12 16 20

0.60

0.80

1.00

1.20

Den

sity

(kg

/m3 )

Time (s)

(e)

0 4 8 12 16 20

0.00

0.40

0.80

1.20

1.60

Hea

t Tra

nsfe

r (W

)

Time (s)

(f)

Fig. 8 Time-dependent (a) pressure, (b) radial velocity, (c) axial velocity, (d) temperature, (e) density,and (f) heat transfer versus time during the first 20.0 seconds (Stage I). This a standard run with 430watts at the heater location which coincides with the virtual probe at l / 4 .


20.000 20.005 20.010 20.015 20.020 20.025

101.5

101.6

101.7

101.8

101.9Pr

essu

re (

kPa)

Time (s)

(a)

20.000 20.005 20.010 20.015 20.020 20.025

-0.8

-0.4

0.0

Rad

ial V

eloc

ity (

cm/s

)

Time (s)

(b)

20.000 20.005 20.010 20.015 20.020 20.025

160

200

240

280

Axi

al V

eloc

ity (

cm/s

)

Time (s)

(c)

20.000 20.005 20.010 20.015 20.020 20.025651.0

651.5

652.0

652.5

653.0

Tem

pera

ture

(K

)

Time (s)

(d)

20.000 20.005 20.010 20.015 20.020 20.025

0.5425

0.5432

0.5439

0.5446

Den

sity

(kg

/m3 )

Time (s)

(e)

20.000 20.005 20.010 20.015 20.020 20.025

1.25

1.30

1.35

1.40

1.45

Hea

t Tra

nsfe

r (W

)

Time (s)

(f)

Fig. 9 Time-dependent (a) pressure, (b) radial velocity, (c) axial velocity, (d) temperature, (e) density,and (f) heat transfer versus time at a time step of 0.0025 seconds (Stage II). This a standard run with430 watts at the heater location which coincides with the virtual probe at l / 4 .


20.000 20.005 20.010 20.015 20.020 20.025-200

-150

-100

-50

0

50

100

150

200

Velocity probe at 15l/16

Velocity probe at l/16

Velocity probe at l/2V

eloc

ity A

mpl

itude

(cm

/s)

Time (s)

(a)

20.000 20.005 20.010 20.015 20.020 20.025-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Pressure probe at 15l/16

Pressure probe at l/16

Pressure probe at l/2

Pres

sure

Am

plitu

de (

kPa)

Time (s)

(b)

Fig. 10 (a) Pressure, and (b) axial velocityoscillations at various locations along the pipe versustime (Stage II).

0 10 20 30 40 50 60 70 80 90101.00

101.25

101.50

101.75

102.00l/4 l0

t = 8.0 st = 6.0 s

t = 4.0 s

t = 2.0 st = 0.0 s

Pres

sure

(kP

a)

Axial Position, x (cm)

Fig. 11 Steady pressure at various times before theonset of oscillations (Stage I).

0 10 20 30 40 50 60 70 80 90250

300

350

400

450

500

550l/4 l0

t = 8.0 st = 6.0 s

t = 4.0 s

t = 2.0 s

t = 0.0 s

Tem

pera

ture

(K

)


Fig. 12 Steady temperature at various times beforethe onset of oscillations (Stage I).

0 10 20 30 40 50 60 70 80 900.6

0.8

1.0

1.2

1.4l0 l/4

t= 6.0 s

t = 8.0 s

t= 4.0 s

t= 2.0 sD

ensi

ty (

kg/m

3 )


Fig. 13 Density at various times before the onset ofoscillations (Stage I).

acoustic oscillations, has a threshold value. Figures17b-17d show the presence of oscillations in the tubefor heat inputs ranging from 376.6 to 426.9 watts.Figures 17e-f demonstrate that exceeding certain criticalvalues of heat input triggers an instability characterizedby growing acoustic wave amplitudes.

Figure 18 provides a trace of maximum pressuremagnitudes at the various heat inputs. Pressureoscillations appear to grow exponentially withincreasing heat input.

Pressure, Velocity and Heat Transfer CouplingFigure 19 shows the pressure and velocity oscillations

along the tube at different time periods during a cycle.Figure 19a shows that the pressure oscillation reachesits maximum amplitude at the center of the pipe and isminimum at both ends. This is in accord with plane-wave acoustic theory. Figure 19b shows that thevelocity oscillation reaches its maxima at both ends and


0 4 8 12 16 20

101.3

101.4

101.5

101.6

101.7P

ress

ure

(kP

a)

Time (s)

(a)

0 4 8 12 16 20

-0.1

0.0

0.1

Rad

ial V

eloc

ity (

cm/s

)

Time (s)

(b)

0 4 8 12 16 20

0

100

200

Axi

al V

eloc

ity (

cm/s

)

Time (s)

(c)

0 4 8 12 16 20

300

400

500

600

700

Tem

pera

ture

(K

)

Time (s)

(d)

0 4 8 12 16 200.4

0.6

0.8

1.0

1.2

Den

sity

(kg

/m3 )

Time (s)

(e)

0 4 8 12 16 20

0.0

0.4

0.8

1.2

1.6

Hea

t Tra

nsfe

r (W

)

Time (s)

(f)

Fig. 14 Time-dependent (a) pressure, (b) radial velocity, (c) axial velocity, (d) temperature, (e)density, and (f) heat transfer versus time for the first 20.0 seconds. This a standard run with 430watts at the heater location which coincides with the virtual probe at l / 2 .


0 4 8 12 16 20

101.3

101.4

101.5

101.6

101.7P

ress

ure

(kP

a)

Time (s)

(a)

0 4 8 12 16 20

-0.8

-0.6

-0.4

-0.2

0.0

Rad

ial V

eloc

ity (

cm/s

)

Time (s)

(b)

0 4 8 12 16 20-50

0

50

100

150

200

Axi

al V

eloc

ity (

cm/s

)

Time (s)

(c)

0 4 8 12 16 20

400

600

800

Tem

pera

ture

(K

)

Time (s)

(d)

0 4 8 12 16 20

0.4

0.6

0.8

1.0

1.2

Den

sity

(kg

/m3 )

Time (s)

(e)

0 4 8 12 16 20

0.0

0.4

0.8

1.2

1.6

Hea

t Tra

nsfe

r (W

)

Time (s)(f)

Fig. 15 Time-dependent (a) pressure, (b) radial velocity, (c) axial velocity, (d) temperature, (e)density, and (f) heat transfer versus time for the first 20.0 seconds. This a standard run with 430watts at the heater location which coincides with the virtual probe at 2l / 3 .


0 4 8 12 16 20

300

400

500

600

700

800

900

Heat Input Rate = 430 watts

Heat Source Location = l/4, l/2, and 2l/3

Probe Location = l/4, l/2, and 2l/3

l/4

l/2

2l/3H

eat S

ourc

e T

empe

ratu

re (

K)

Time (s)

Fig. 16 Temperature of the source versus time atvarious locations.

is a minimum at the center of the pipe. This is also inagreement with acoustic theory. An interesting result isalso seen in Fig. 19b which shows a jump in velocityamplitude right at the source location. This can beattributed to the strong local coupling between velocityand heat release at the source.

The Acoustic Phase AnglesFigure 20 shows the relationship between pressure,

velocity and heat transfer oscillations at the sourcelocation. The pressure oscillation is leading thevelocity oscillation by 90° and the heat oscillation isleading the velocity oscillation by 45° and lagging thepressure oscillation by 45°. This numerical phase resultis in agreement with Carvallo’s conclusions5 that thereshould be a time lag between the fluid velocity and theheat transferred to the flow. Carvallo also concludesthat, in general, the phase between the acoustic pressureand heat transfer cannot be 90°. This is in agreementwith our result for a corresponding phase lag of 45°. Inhis study, Carvallo uses the following equation forpressure oscillations:

. /p Am x

lt(1) sin sin" 0

'& 1 (4)

where 1 , being the phase lead between p( )1 and q (1) ,

is left to be determined. From our numerical results, weconcluded that 1 '" / 4 in Eq. (4). This is a key result

in assessing the relationship between pressure, velocityand heat oscillations. Using this newly found result, thetime lag could be now incorporated into the expressionfor pressure.

The Form of CouplingThe current computational work shows that the

magnitude of heat transfer oscillations is of the sameorder as the product of acoustic velocity amplitude,acoustic pressure amplitude, and the area of the source(see Table 2). This is indicative of a coupling betweenthe acoustic heat transfer and the product of acousticpressure and velocity in the standing wave field.Simply stated, it is a further indication of a directconversion of acoustic intensity (mechanical energy)into heat and vice versa. An analytical expression forthe heat transfer that satisfies existing criteria andconditions can be now proposed. This equation is:

q Const p u(1) (1) (1)" 2 (5)

Concluding RemarksThe numerical results described above, which agree

with experimental observations, are supportive of atheory that attributes heat transfer coupling to thecombined effects of pressure and velocity. Bothexperimental observations and present numericalsolutions concur that, unless the product of pressureand velocity is positive, acoustic damping will occur.This leads us to believe that velocity or pressurealone cannot be solely responsible for driving theoscillations, convincing us to be in favor of a theorythat proposes coupling between the unsteadycomponents of pressure, velocity and heat transfer.

AcknowledgmentThe authors would like to thank Dr. Michael

Barkhudarov of Flow Science, Inc., for his help andsupport. Without his helpful contributions, this workwould not have been possible.

References1Rayleigh, J.W.S., The Theory of Sound, Vol. 1 and

2, Reprint of 1894-6 Edition, Dover Publications, NewYork, 1945.

2Zinn, B.T., Miller, N., Carvalho, J.A. Jr., and Daniel,B. R., “Pulsating Combustion of Coal in a Rijke TypeCombustor,” Proceedings, 19th InternationalSymposium on Combustion, 1982.

3Evans, R.E., and Putnam, A.A., “Rijke TubeApparatus,” J. Appl. Phys., No 360, 1966.

4Collyer, A. A., and Ayres, D. J., “The Generation ofSound in a Rijke Tube Using Two Heating Coils,” J.Phys: Appl. Phys., Vol. 5, 1972.

5Carvalho, J.R., Ferreira, C., Bressan, C., andFerreira, G., “Definition of Heater Location to DriveMaximum Amplitude Acoustic Oscillations in a RijkeTube,” Combustion and Flame, Vol. 76, 1989.


0 4 8 12 16 20

101.0

101.5

102.0

102.5

Heat Power Input = 125.5 watts

Pre

ssur

e (k

Pa)

Time (s)

(a)

0 4 8 12 16 20

101.0

101.5

102.0

102.5


Pre

ssur

e (k

Pa)

Time (s)

(b)

0 4 8 12 16 20

101.0

101.5

102.0

102.5


Pre

ssur

e (k

Pa)

Time (s)

(c)

0 4 8 12 16 20

101.0

101.5

102.0

102.5


Pres

sure

(kP

a)

Time (s)

(d)

0 4 8 12 16 20

101.0

101.5

102.0

102.5


Pres

sure

(kP

a)

Time (s)

(e)

0 4 8 12 16 20

101.0

101.5

102.0

102.5Heat Power Input = 439.4 watts

Pres

sure

(kP

a)

Time (s)

(f)

Fig. 17 Pressure versus time for different power inputs for the first 20 seconds (Stage I). This astandard run with a heater location that coincides with the virtual probe at l / 4 .


410 415 420 425 430 435

0.2

0.4

0.6

0.8

Input Heat Power (watts)

Pre

ssur

e A

mpl

itud

e (k

Pa)

Maximum Pressure

130

135

140

145

150

SP

L (

dB)

SPL

Fig. 18 Maximum pressure versus time at variousheat power inputs.

0 10 20 30 40 50 60 70 80 90-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25l/4 l0

Pre

ssur

e (k

Pa)


(a)

0 10 20 30 40 50 60 70 80 90-100

-75

-50

-25

0

25

50

75

100ll/40


Vel

ocity

(cm

/s)

(b)

Fig. 19 (a) Pressure and (b) velocity versus axiallength along the pipe at various time steps duringone period.

20.000 20.005 20.010 20.015 20.020 20.025

20.000 20.005 20.010 20.015 20.020 20.025

u(1)

q(1) p

(1)

Prim

ary

Var

iabl

es

Time (s)

(a) During Stage II

20.002 20.003 20.004 20.005 20.006 20.007

Pri

mar

y V

aria

bles

Time (s)

q(1)

u(1)

p(1)

*'3')*'')*0

(b) During a cycle in Stage II

Fig. 20 Relationship between pressure, axialvelocity and heat transfer oscillations at locationl / 4 versus time (Stage II).

Table 2 Coupling variables vs. heat power input

qin

(W)Aobs

(cm2)pmax

(1)

(Pa)umax

(1)

(m/s)Aobs pmax

(1)umax(1)

(W)q(1)

(W)125.5 11.03 - - - -376.6 11.03 33.65 0.0868 3.22x10-3 18.7x10-3

408.0 11.03 71.45 0.1856 1.46x10-2 4.04x10-2

426.9 11.03 151.25 0.3942 6.58x10-2 8.62x10-2

430.0 11.03 208.80 0.6586 0.152 0.12433.1 11.03 837.20 2.622 2.42a 0.485439.4 11.03 924.40 2.435 2.25a 0.491

a Corresponding to growing acoustic instability.

6Carrier, G.F., “The Mechanics of Rijke Tube,” Q.appl. Math., Vol. 12, 383, 1955.

7Maling, G.C., “Simplified Analysis of the RijkePhenomenon,” Journal of the Acoustical Society ofAmerica, Vol. 35, 1058, 1963.

8Miller, J. Carvalho, J.A.S., “Comments on RijkeTube,” Scient. Am., No 204, 180, 1961.


9Friedlander, M.M., Smith, T.J.B., “Experiments onthe Rijke Tube Phenomenon,” Journal of the AcousticalSociety of America, Vol. 36, 17, 1964.

10Zinn, B.T., “State of the Art and Research Needs ofPulsating Combustion,” American Society ofMechanical Engineers, 84-WA/NAC-19, December1984.

11AGARD, History of German Guided MissilesDevelopment, First Guided Missiles Seminar, Munich,Germany, 1956.

12Kinsler, L.E., Frey, A.R., Coppens, A.B., andSanders, J.V., Fundamentals of Acoustics, John Wileyand Sons, New York, 1982.

13Dowling, A.P., and Efows Williams, J.E., Soundand Source of Sound, Ellis Horwood limited, WestSussex, England, 1983.

14Chu, B.T., NACA RM 56D27, 1956.15Flow Science Incorporated, Los Alamos, New

Mexico.

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