closed twodimensional rectangular chamber is considered for this case. ... gelombang akustik ini digunakan bagi menjana perubahan suhu apabila ... pada satu dinding. ... 25. 4.7. Evolutions of axial velocity (u) for nonadiabatic cases. 26. 4.8 ... c.  Isobaric specific heat. H  Height of resonator k  Thermal conductivity. L ...
NUMERICAL SIMULATION OF ADIABATIC ACOUSTIC WAVES IN A CLOSED RECTANGULAR CHAMBER
AZLI BIN ABD. RAZAK
A project report submitted in partial fulfilment of the requirements for the award of the degree of Master of Engineering (Mechanical)
Faculty of Mechanical Engineering Universiti Teknologi Malaysia
MAY, 2004
To my wife Wan Rozaida Mazlina and my daughter Nur Qaisara Batrisyia
iii
ACKNOWLEDGMENT
I would like especially to thank to advisor Dr. Normah Mohd. Ghazali, for her guidance, encouragement, and patience throughout the research and writing of this work. I also would like to thank Universiti Teknologi MARA for the financial supporting during my study. Many thank to all postgraduate member who help to make this possible. Finally thank to my wife and my daughter for its supports and love all the time.
iv
ABSTRACT
This study of acoustic wave was motivated by the thermoacoustic refrigerator. Normally acoustic wave is used to create a temperature gradient across the stack in a resonant tube. Numerical simulation of adiabatic acoustic waves in a closed twodimensional rectangular chamber is considered for this case. The waves are generated by a membrane boundary condition on one wall. The NavierStoke system is solved here by assuming constant thermophysical properties of the compressible Newtonian fluid and perfect gas. The results for adiabatic and nonadiabatic cases for velocity profile, temperature distribution and pressure distribution are discussed.
v
ABSTRAK
Kajian gelombang akustik ini didorong oleh sistem penyejukan termoakustik. Kebiasaanya gelombang akustik ini digunakan bagi menjana perubahan suhu apabila melalui susunan plat di dalam tiub resonan. Simulasi berangka bagi gelombang akustik yang adiabatik di dalam kebuk segiempat tepat tertutup dua dimensi dipertimbangkan untuk kes ini. Gelombang diuja oleh keadaan sempadan membran pada satu dinding. Penyelesian sistem NavierStoke telah dibuat dengan menganggap bahawa sifat termofizik bendalir Newtonian boleh mampat adalah malar dan gas adalah unggul. Keputusan untuk kes adiabatik dan bukan adiabatik bagi profil halaju, taburan suhu dan tekanan dibincangkan.
vi
TABLE OF CONTENTS
CHAPTER
1
2
3
4
TITLE
PAGE
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
LIST OF TABLE
xi
LIST OF FIGURES
x
LIST OF SYMBOLS
xii
INTRODUCTION
1
1.1
1
History
THEORY
4
2.1
Governing Equations
4
2.2
Normalization
6
2.3
Controlled Equations
7
2.4
Boundary Conditions
10
NUMERICAL FORMULATION
13
3.1
13
Finite Difference Formulation
RESULT AND DISCUSSION 4.1
4.2
17
Comparison of the Velocity Field of Adiabatic and Nonadiabatic Waves
18
Vortex Motion on Adiabatic Case
19
vii
5
4.3
Analysis of the Temperature
19
4.4
Analysis of the Pressure
20
CONCLUSION
36
REFERENCES
37
viii
LIST OF TABLE
TABLE NO. 4.1
TITLE Parameter for test case
PAGE 17
ix
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
Schematic illustration of twodimensional rectangular chamber
11
2.2
Illustration of membrane wave maker
12
2.3
Illustration of piston wave maker
12
3.1
Grid point for a five point formula
14
3.2
The sequence of operation
16
4.1
Schematic diagram of showing the wave length and length of the chamber
18
Comparison of velocity in the centreline of the chamber at t = 2.44
21
Comparison of velocity in the centreline of the chamber for the half wavelength
22
4.4
Velocity profile for the half wave length or at t = 3.142
23
4.5
Evolutions of axial velocity (u) in adiabatic case for a sequence of time.
24
Evolutions of transverse velocity (v) in adiabatic case for a sequence of time.
25
4.7
Evolutions of axial velocity (u) for nonadiabatic cases
26
4.8
Evolutions of transverse velocity (v) in non–adiabatic case for a sequence of time
27
Velocity vector and vortex flow near the membrane at t = 2.702
28
2.1
4.2
4.3
4.6
4.9
x
4.10
Velocity vector and vortex flow near the membrane at t = 3.002
29
Velocity vector and vortex flow near the membrane at t = 3.142
30
The average temperature in the chamber for the five cycle
31
4.13
Temperature at t = 3.002
32
4.14
Temperature at t = 3.142
33
4.15
Pressure at t = 3.002
34
4.16
Pressure at t = 3.142
35
4.11
4.12
xi
LIST OF SYMBOLS
Roman
cp

Isobaric specific heat
H

Height of resonator
k

Thermal conductivity
L

Length of the resonator
L/H

Aspect Ratio
M

Mach number
p

pressure
Pe

Peclet number
Pr

Prandtl number
R

Specific gas constant
Re

Reynolds number
t

Time
T

Temperature
u,v

Velocity component
Uo

Forcing amplitude
V

Volume
α

⎛ k ⎞⎟ Thermal diffusivity ⎜ ≡ ⎜ ρc ⎟ p ⎠ ⎝
φ

Nonlinear terms
Φ

Viscous Dissipation
γ

Specific heat ratio
λ

Wave length
Greek
µ  Dynamic viscosity
xii
ν

Kinematic viscous
ρ  Density τ

Viscous stress tensor
ω

Forcing frequency
∇

Vector differential operator
Subscript i,j

Grid location in x, y direction
m

Mean
Superscript *

Denote a nondimensional quantity
‘

Fluctuating part
xiii
CHAPTER 1
INTRODUCTION
A brief overview of major theoretical, experimental and numerical study advancements in thermoacoustic and acoustic wave is presented in this section. This will serve as a general motivation for this research. A review of the theory will be given first, followed by a summary of experimental investigation and ended with numerical and simulation study. Currently few researchers have developed numerical interpretation of the thermoacoustic effects (Aranha, Yue and Mei 1982; Wolikar and Knio, 1996; Normah, 2001). The numerical simulation of acoustic wave in a closed rectangular chamber is very important in order to ensure the phenomena of acoustic effects. The acoustic wave is widely used in thermoacoustic refrigerators and thermoacoustic engines.
1.1
History
The use of thermoacoustic effects started over a century ago. The fundamentals on thermoacoustics can be referred to in Rott’s review article and Swift’s review article. Here we expand upon the representative results of those works. Rott is generally considered the initiator of the field, being the first to write down a full theoretical description of the thermoacoustic effect (N. Rott, 1980). However, he attributes the beginnings of the theoretical work to Kirchhoff who modified the result of the HelmholtzRayleigh theory of sound attenuation in ducts to
1
consider the heat transfer between the gas and the isothermal duct wall. In his review, Rott summarized the results obtained over more than a decade ago by him and his colleagues which originally intended to explain theoretically how Taconis vibration occurs. The momentum, continuity and the energy equations, all in the smalloscillating limit, were considered. Ideal gasses were assumed. He showed how a temperature difference can arise in the walls of a narrow tube due to the timeaveraged entropy flow in a gas forced into oscillation and found the expression of second order heat flux associated with these oscillations. In his review, Swift (1988) took a decisive step toward implementing Rott’s theory of thermoacoustic phenomena into creating practical thermal engines. He considered and extended the fundamental theory of Rott’s to calculate the heat and work flux in a stack as a collection of individual narrow thermoacoustic elements, placed in a standingwave resonator. The efficiency of a thermoacoustic device, either as a primemover or as a refrigerator, was calculated. The derivations were done in increasing steps of complexity. The limiting cases, like that of zero viscosity, illustrate the basic concepts in a more clear way and make the theory more accessible. Also, the pictorial representation of the moving parcel of gas in Lagrange frame of reference has helped many beginners in the field to gain a more intuitive understanding of the phenomenon. He also derived the equation for absence of plate wherein the acoustic wave is adiabatic. Another important theoretical approach in thermoacoustic was presented in the paper by Keller and Millman (1979). They studied on compressible wave travelling in a rigid cylindrical waveguide. This paper created the governing equation by ignoring the effect of viscosity and heat conduction. Gogate and Munjal (1992) did the analytical solution of the laminar mean flow wave equation in a line or unlined twodimensional rectangular duct. This analysis used the NewtonRaphson technique to solve the boundary conditions. The analysis was limited to the laminar flow only. Experimental study on thermoacoustic was done by Merkli and Thomann (1975). They described an experiment that tests quantitatively the predictions of thermoacoustic theory. They measured the thermoacoustic time average heat flux along the walls of a tube driven to resonance by a piston. Heating at the close end (high pressure compression) and cooling at the velocity antinodes (low pressureexpansion), were observed. The results of thermal measurement in air (standard 2
temperature and pressure in equilibrium) were compared with calculations made on the same line with the earlier results of Rott. The study was limited to the calculation of the heat flux entering the tube wall because it only solved firstorder quantities of differential equation. Another experimental work on thermoacoustic was done for a flow through a thermoacoustic refrigerator (R.S. Reid and G.W. Swift, 2000). The focus on this experiment was done on a standingwave thermoacoustic refrigerator with parallel superimposed steady flow. The flow was moved by a piston driver. Numerical simulation was used to overcome the problem created by experiments in thermoacoustic. By using a numerical method we can predict the result and it is also inexpensive to develop the model compared to an experimental rig. The Worlikar and Knio review article (1996), the first numerical work on thermoacoustic refrigerator concentrated on unsteady adiabatic flow around the stack. They used central finite difference methodology on rectangular grids. Their numerical study however covered only the region enclosed by two plates and without oscillating flow anywhere in their computational domain. They also neglected the thermal diffusion. Ahmad Zakaria et al. (2000) used the central finite difference scheme to model propagation in time domain of acoustic wave in shallow water. Their numerical method was fourth order and second order accurate in time. They showed that the finite difference scheme on this acoustic wave problem is stable. However, their scope is on incompressible fluid.
Another numerical study on nonlinear
acoustic wave was done by Aranha et al. and S. Lichter and J. Chen (1982). They showed that in their study the nonlinear initial boundary value problems were solved using a semiimplicit finite difference scheme of the CrankNicolson. The numerical study on acoustic wave was performed by Mohd Ghazali, Normah (2001) on a rectangular chamber. The effects of chamber with a plate and without a plate were simulated. The finite difference spatial discretization and semiimplicit time marching procedures was used in the numerical study. However the study considered the acoustic wave as nonadiabatic. This study is an extension of the work of Mohd. Ghazali, Normah but an adiabatic case is now being considered. The purpose is to see if there are any differences in the two cases which may or may not justify the extra terms added for the nonadiabatic case to the controlled equation modelling the physical domain. 3
CHAPTER 2
THEORY
The governing equations for two dimensional compressible flows are derived in this chapter. These are the equations of continuity, momentum and energy. The governing equations are derived by following the procedure describes in Mohd. Ghazali(2001) which is summarized below. First, the NavierStokes equations are made dimensionless using appropriate choice of normalizing parameters.
2.1
Governing Equations
The formulation is based on the assumption that: (a) thermophysical properties of the compressible Newtonian fluid are constant: (b) the flow is twodimensional. The twodimensional equations can be expressed as (Dale A.Anderson et al, 1984)
Continuity equation: ∂ρ ∂ ( ρu ) ∂ ( ρv ) + + =0 ∂t ∂x ∂y
(2.1)
Where, ρ is the density of the fluid and u and v is the velocity component in horizontal and vertical drive respectively.
4
Momentum equation:
∂τ xy ⎛ ∂u ∂u ∂u ⎞ ∂p ∂τ ρ⎜⎜ + u + v ⎟⎟ = − + xx + ∂y ∂x ∂y ⎠ ∂x ∂x ⎝ ∂t ∂τ xy ⎛ ∂v ∂v ∂v ⎞ ∂p ∂τ ρ⎜⎜ + u + v ⎟⎟ = − + xx + ∂x ∂y ⎠ ∂y ∂x ∂y ⎝ ∂t
(2.2a)
(2.2b)
τ xx , τ yy and τ xy represent the viscous stress tensor given by, τ xx =
2 ⎛ ∂u ∂v ⎞ µ⎜ 2 − ⎟ 3 ⎜⎝ ∂x ∂y ⎟⎠
τ yy =
2 ⎛ ∂v ∂u ⎞ µ⎜ 2 − ⎟ 3 ⎜⎝ ∂y ∂x ⎟⎠
⎛ ∂u ∂v ⎞ τ xy = µ⎜⎜ + ⎟⎟ ⎝ ∂y ∂x ⎠
Energy equation: ⎡ ∂ 2T ∂ 2T ⎤ ⎡ ∂p ⎡ ∂T ∂p ∂p ⎤ ∂T ∂T ⎤ +v ⎥+Φ +u + v ⎥ = k⎢ 2 + 2 ⎥ + ⎢ + u ρc p ⎢ ∂x ∂y ⎦ ∂x ∂y ⎦ ∂y ⎦ ⎣ ∂t ⎣ ∂t ⎣ ∂x
(2.3) where Φ represent the viscous dissipation given by, 2 2 2 ⎡ ⎛ ∂u ⎞ 2 ⎛ ∂v ⎞ ⎛ ∂v ∂u ⎞ 2 ⎛ ∂u ∂v ⎞ ⎤ Φ = µ ⎢2⎜ ⎟ + 2⎜⎜ ⎟⎟ + ⎜⎜ + ⎟⎟ − ⎜⎜ + ⎟⎟ ⎥ 3 ⎝ ∂x ∂y ⎠ ⎥ ⎢⎣ ⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂x ∂y ⎠ ⎦
(2.4)
Here c p is the isobaric specific heat, µ is the dynamic viscosity, and k is thermal conductivity. For this case the bulk viscosity is negligible and body forces were not included.
5
Gas has been assumed as ideal, so the equation of state is given by
p = ρRT
(2.5)
The equations above contain five unknowns ρ, u, v, p and T. This been
solved by introducing (2.5) into (2.1) and eliminating the density. The equation is then put in dimensionless form.
2.2
Normalization
The dimensionless parameters is defined as follows x* =
x , H
t * = tω,
y , H ρ ρo* = o , ρm y* =
u , ωH T T* = , Tm u* =
v , ωH p , p* = ρ m RTm
v* =
where, H = width of resonator
ω = forcing frequency ρm = mean density Tm = mean temperature
The dimensionless parameters are then applied to the NavierStokes equation. By using a suitable combination of continuity, momentum, and energy equations, the normalized NavierStoke equations become, D ⎛ p ⎞ ⎛ p ⎞⎛ ∂u ∂v ⎞ ⎜ ⎟ = ⎜ − ⎟⎜ + ⎟ Dt ⎝ T ⎠ ⎝ T ⎠⎜⎝ ∂x ∂y ⎟⎠
(2.6)
ρ
1 ∂p 1 2 1 ∂ ⎛ ∂u ∂v ⎞ Du ⎜ + ⎟ =− 2 + ∇ ⋅u + 3 Re ∂x ⎜⎝ ∂x ∂y ⎟⎠ Dt M γ ∂x Re
(2.7)
ρ
1 ∂p 1 2 1 ∂ ⎛ ∂u ∂v ⎞ Dv ⎜ + ⎟ =− 2 + ∇ ⋅v + 3 Re ∂y ⎜⎝ ∂x ∂y ⎟⎠ Dt M γ ∂y Re
(2.8)
6
ρ
⎛ γ − 1 ⎞ Dp M 2 1 2 DT ⎟⎟ (γ − 1)Φ ′ = ∇ T + ⎜⎜ + Dt Pe ⎝ γ ⎠ Dt Re
Here, γ is the specific heat ratio, M =
ωH γRTm
(2.9)
is the Mach number, Re =
H 2ω ρH 2ω H 2ω = is the Reynolds number, v is the kinematic viscosity, Pe = is µ ν α
Peclet number, α =
K where K is thermal conductivity, ρ is the density, and ρC p
C p is the specific heat.
2.3
Controlled Equations
The pressure, temperature, and density in equation (2.6 – 2.9) now will be separated into the mean plus the fluctuating parts. This is given by p = p m + p'
(2.10)
T = Tm + T '
(2.11)
ρ = ρm + ρ'
(2.12)
The subscript m indicates mean quantities and the superscript (’) refers to the fluctuating parts quantities. Equations (2.10 – 2.12) are then substituted into equations (2.6 – 2.9). After expanding the equation and dropping the circumflex, the equation of continuity becomes
(1 + T ) Dp = (1 + p ) DT − (1 + p )(1 + T )(∇.u ) Dt
Dt
(2.13)
Further, momentum and energy equations become
7
∂u 1 ∂p 1 2 1 ∂ ⎛ ∂u ∂v ⎞ ⎜ + ⎟ − φx =− 2 + ∇ u+ ∂t 3 Re ∂x ⎜⎝ ∂x ∂y ⎟⎠ M γ ∂x Re
(2.14)
∂v 1 ∂p 1 2 1 ∂ ⎛ ∂u ∂v ⎞ ⎜ + ⎟ − φy =− 2 + ∇ v+ ∂t 3 Re ∂y ⎜⎝ ∂x ∂y ⎟⎠ M γ ∂y Re
(2.15)
⎛ γ − 1 ⎞ ∂p ∂T 1 2 ⎟⎟ − φT = ∇ T + ⎜⎜ ∂t Pe ⎝ γ ⎠ ∂t
(2.16)
where φ is the nonlinear terms which is given as, φx = ρ
Du ∂u ∂u +u +v Dt ∂y ∂x
(2.17)
φy = ρ
Dv ∂v ∂v +u +v Dt ∂y ∂x
(2.18)
φT = ρ
∂p ⎞ DT ∂T ∂T ⎛ γ − 1 ⎞⎛ ∂p ⎟⎟⎜⎜ u + v ⎟⎟ +u +v − ⎜⎜ ∂y ⎠ Dt ∂x ∂y ⎝ γ ⎠⎝ ∂x
(2.19)
The viscous dissipation has been dropped from the equation above. The viscous dissipation was important only for laminar flow condition or very low Reynolds number. The density term, ρ in the equation (2.17 – 2.19), are neglected and it can be justified with a Boussinesq approximation argument. To solve the nonlinear and linear terms in equation (2.13) they have been separated and written as (Mohd. Ghazali, Normah, 2001) ∂p ∂T = − (∇.u ) − φ p ∂t ∂t
(2.20)
Here φ p is define as,
⎛ ∂p ∂p ∂p ⎞ φ p = T ⎜⎜ + u + v ⎟⎟ − ∂x ∂y ⎠ ⎝ ∂t
⎛ ∂T ∂T ∂T ⎞ ⎟ p⎜⎜ +u +v ∂x ∂y ⎟⎠ ⎝ ∂t
⎛ ∂p ∂p ⎞ + ⎜⎜ u + v ⎟⎟ + ( p + T + pT )∇ ⋅ u ∂y ⎠ ⎝ ∂x
(2.21)
8
Then equation (2.20) were use to eliminate p in equation (2.14 – 2.16) to obtain [12],
1 ⎛ ∂ 2 u ∂ 2 v ∂ 2T ⎞ 1 ∂ 2 ∂ 2u ⎜ ⎟+ − ∇ u = + ∂t 2 M 2 γ ⎜⎝ ∂x 2 ∂x∂y ∂x∂t ⎟⎠ Re ∂t
(
+
)
1 ∂ ⎛ ∂ 2u ∂ 2 v ⎞ 1 ∂ ⎜⎜ 2 + ⎟⎟ + 2 (φ p ) − ∂ (φ x ) ∂x∂y ⎠ M γ ∂x 3 Re ∂t ⎝ ∂x ∂t
(2.22)
∂ 2v 1 ⎛ ∂ 2 u ∂ 2 v ∂ 2T ⎞ 1 ∂ 2 ⎟+ ⎜ = + − ∇ v ∂t 2 M 2 γ ⎜⎝ ∂x∂y ∂y 2 ∂y∂t ⎟⎠ Re ∂t
( )
+
1 ∂ ⎛ ∂ 2u ∂ 2v ⎞ 1 ∂ ⎜⎜ (φ p ) − ∂ (φ y ) + 2 ⎟⎟ + 2 ∂t 3 Re ∂t ⎝ ∂x∂y ∂y ⎠ M γ ∂y
∂T γ 2 = ∇ T + (1 − γ )(∇ ⋅ u ) + (1 − γ )φ p − γφT ∂t Pe
(2.23)
(2.24)
The equations (2.22 – 2.24) only solved for velocity in x and y direction (u and v) and temperature. For the pressure it would be determined using expression given below.
∂p γ ⎛ ∂T 1 2 ⎞ = − ∇ T + φT ⎟ ⎜ ∂t γ − 1 ⎝ ∂t Pe ⎠
(2.25)
For the adiabatic case Peclet number is large, therefore diffusion term in (2.24) is dropped. Then equation (2.24) was used to remove the temperature term in (2.22) and (2.23). This will give the expression below. 1 ⎛ ∂ 2u ∂ 2 v ⎞ 1 ∂ 2 1 ∂ ⎛ ∂ 2u ∂ 2 v ⎞ ∂ 2u ⎟ ⎟ ⎜ ⎜ + ∇ + + = + u 3 Re ∂t ⎜⎝ ∂x 2 ∂x∂y ⎟⎠ ∂t 2 M 2 ⎜⎝ ∂x 2 ∂x∂y ⎟⎠ Re ∂t
(
)
9
+
1 ∂ (φ p + φT ) − ∂ (φ x ) 2 ∂t M ∂x
1 ∂ 2v = 2 2 M ∂t
+
(2.26)
⎛ ∂ 2u ∂ 2 v ⎞ 1 ∂ 2 1 ∂ ⎛ ∂ 2u ∂ 2v ⎞ ⎟ ⎜ ⎜⎜ ∇ v + + + 2 ⎟⎟ + 3 Re ∂t ⎜⎝ ∂x∂y ∂y 2 ⎟⎠ ⎝ ∂x∂y ∂y ⎠ Re ∂t
( )
1 ∂ (φ p + φT ) − ∂ (φ y ) 2 ∂t M ∂y
(2.27)
For this adiabatic case, the temperature and pressure will be determined using, ∂T = (1 − γ )(∇ ⋅ u ) + (1 − γ )φ p − γφT ∂t
(2.28)
∂p γ − 1 ⎛ ∂T ⎞ = + φT ⎟ ⎜ ∂t γ ⎝ ∂t ⎠
(2.29)
In this study, the equations 2.22 through 2.29 have been converted to a numerical scheme by Mohd. Ghazali, Normah (2001).
2.4
Boundary Condition
The derivation of the governing equation is completed by specifying the boundary conditions. The boundary conditions are based on Figure 2.1 and is to be assumed as noslip, nopenetration, and adiabatic at the inner wall of chamber (no heat flux). given as below: u (0, y, t ) = f ( y, t )
⎞ ⎛L ⎞ ⎛L u ⎜ , y , t ⎟ = v⎜ , y , t ⎟ = 0 ⎝H ⎠ ⎝H ⎠ u ( x,0, t ) = v( x,0, t ) = 0 u ( x,1, t ) = v( x,1, t ) = 0
10
L is the length of the resonator and H is the height of the resonator. Therefore
L is H
the aspect ratio. The function u (0, y, t ) has been chosen to drive the acoustic waves.
L wall U Simulation domain V
H
close end
wall
y x
Figure 2.1: Schematic illustration of twodimensional rectangular chamber
The input to the system is the velocity. For this case the waves are generated by imposing the normal velocity component in the open end. This initial condition is, the velocity at x = 0 , therefore u (0, y ) = U o h( y )sin t . Here, U o is defines as the forcing amplitude of velocity and h( y ) is a shape function. The shape function depends on the wave generated either by a flat vibrating piston or membrane. For the flat vibrating piston the value of h =1 and for membrane the value of h = sin y. Figures 2.2 and 2.3 illustrate the schematic diagram of generating wave using membrane and flat vibrating piston respectively.
11
Closed rectangular chamber
Membrane wave maker H
u = U o sin y sin t
x x=0
x=L
Figure 2.2: Illustration of membrane wave maker
Figure 2.3: Illustration of piston wave maker
12
CHAPTER 3
NUMERICAL FORMULATION
In this chapter the numerical scheme is developed for the simulation of the system of governing equations. The numerical simulation of the governing equations is based on a finite difference methodology. Spatial derivatives are approximated using secondorder central difference. The construction of the scheme was composed into two parts. First, the linear terms are treated as implicit containing first order temporal derivative dealt with CrankNicholson method. Second, the nonlinear terms treated as explicit dealt with AdamsBashforth method (Mohd. Ghazali, Normah, 2001).
3.1
Finite Difference Formulation
The central difference method use internal grid point as shown in Figure 3.1. This five point formula is the most commonly use to describe the finite difference formulation. Example of typical explicit equations in a two dimensional system was taken. The explicit equation is approximated as, ∂u ∂u u i +1, j − u i −1, j u i , j +1 − u i , j −1 ≅ + + 2∆x 2∆y ∂x ∂y
(3.1)
∂ 2 u ∂ 2 u u i +1, j − 2u i , j + u i −1, j u i , j +1 − 2u i , j + u i , j −1 + ≅ + ∂x 2 ∂x 2 (∆x) 2 (∆y ) 2
(3.2)
13
and then onesided derivative on the boundaries is written as,
∂u − u i + 2, j + 4u i +1, j − 3u i , j ≅ 2∆x ∂x
(3.3)
∂ 2 u 2u i , j − 5u i +1, j + 4u i + 2, j − u i +3, j ≅ ∂x 2 ∆x 2
(3.4)
Figure 3.1: Grid point for a five point formula
In the equations, second order linear temporal derivative should be handled carefully. This is to prevent the possibility of artificial acoustic waves damping. The approximation to second order temporal term can be written as, u n +1 − 2u n + u n −1 f ≅ 2 ∆t
n +1
+ f 2
n −1
(3.5)
14
Then, nonlinear terms are approximated for the first order as,
∂φ x φ nx − φ nx −1 ≅ ∂t ∆t
(3.6)
The equations 2.21 through 2.29 have been continued by Mohd. Ghazali, Normah. In the current study, the terms signifying the nonadiabatic cases have been dropped. The expression
1
γ
no longer exist in equations 2.26 and 2.27. A new subroutine has
been created to handle the nonlinear term of the energy equations, φ T . The flow chart (Figure 3.2) shows the sequence of this operation.
15
Figure 3.2: The sequence of operation
16
CHAPTER 4
RESULT AND DISCUSSION
The adiabatic and nonadiabatic acoustic waves are analyzed in this chapter. The comparison is only for acoustic waves in the absence of a stack which is normally present in a thermoacoustic resonant tube. The working fluid tested is helium gas one of the media generally used by past researchers. Details of the parameters considers in this case are summarized in Table 4.1 (Mohd.Ghazali, Normah, 2001)
Pressure, p
101 kPa
Temperature, T
293 K
Mach number, M
0.7854
Reynolds number, Re
800000
Peclet number, Pe
600000
Prandtl number, Pr
0.667
Aspect ratio, L/H
4/1
Uo
0.001
∆x
0.01
∆t
0.001 Table 4.1: Parameter for test case
The length of the chamber chosen is half of the wave length and described as, L =
λ 2
as shown in Figure 4.1.
17
λ 2
u = U o sin t
H
L
Figure 4.1: Schematic diagram of showing the wave length and length of the
Chamber
4.1
Comparison of the Velocity Field of Adiabatic and Nonadiabatic Waves
A membrane wave maker has been modelled. The results of the flow pattern for adiabatic and nonadiabatic conditions at t = 2.442 and the half wave length at t = 3.142 s as display in Figure 4.2 and 4.3 respectively which is taken at the centreline of the chamber (at
H ). The adiabatic velocity is 20% faster than the nonadiabatic 2
velocity. This can be clearly seen in Figure 4.2, which is taken at the same time. In both of the cases the acoustic waves not yet reach the wall. Adiabatic acoustic wave at this time propagate 99% of overall length of the wall. While for the nonadiabatic acoustic wave propagate 79% of the overall length. Note that in Figure 4.3 the front of the adiabatic wave has already reached the wall. The acoustic waves get reflected and then merge with the oncoming generated waves. The resulting magnitude of the adiabatic wave makes the amplitude of the axial velocity higher near the wall. This difference in speed phenomenon occurs because in the adiabatic case there is no energy transfers from the gas particle to another particle when it collides with each other. For the nonadiabatic case the collisions of the gas molecules transfers energy to each other. Because of the no energy transfer the gas molecules move faster in adiabatic process compare to the nonadiabatic process. Figure 4.4 18
show the comparison of the profile of the half wave length for the nonadiabatic and adiabatic at the resolution of 400×100. This was captured at t = 3.142. The difference in speed is again obvious here at the right wall of the domain. The comparison of evolution axial velocity and transverse velocity can be seen in Figure 4.5 through 4.8. The numerous peaks and valleys may be attributed to mode competition before a stable limiting cycle is reached. In this study, simulation is focused on the generation and subsequent progress of the acoustic waves.
4.2
Vortex Motion on Adiabatic Case
The sequence of velocity vector for the adiabatic and nonadiabatic acoustic waves is shown in Figure 4.7 through 4.9. The plotting area is taken at the bottom left corner which is adjacent to the wavemaker. The vortex motion effect in the adiabatic case is the same as that for the nonadiabatic case. The vortex motion is in the clock wise direction. It means that when the backward flow reach the membrane the outward flow force the flow to the front. The vortex motion is symmetric with respect to the chamber centreline.
4.3
Analysis of the Temperature
The temperature variations inside the chamber occur due to the collisions of the gas particles to the wall during compression and expansion. The comparison has been done by taking the average temperature for the both cases (adiabatic and nonadiabatic). Figure 4.10 show the average temperature in the domain for the both cases. This average temperature in the domain given by,
T =
1 V
∫∫ TdV
(5.1)
where T is the average temperature and V is the average volume.
19
The average temperature shows that whether it is adiabatic or nonadiabatic, the average temperature inside the chamber looks almost the same. However, the computed values of adiabatic are 5.08% lower than nonadiabatic. Figure 4.11 show the contour plot of temperature to compare the adiabatic and nonadiabatic. At any instant in a cycle the temperature for each case is different. And since it is faster than the other, the mean temperature computed at approximately each cycle is different as mentioned earlier.
4.4
Analysis of the Pressure
The pressure and the temperature is also related to each other. When the temperature increases the pressures also increases or vice versa because they are in phase. At the end wall the pressure growth is very fast. This situation happens because of the same reason given for the temperature increase near the wall. Figure 4.13 (at t = 3.002) shows the contour plots for pressure. The amplitude of the pressure is lower in the adiabatic case compared to the nonadiabatic case. Figure 4.14 show the contour plot for pressure at a latter time (t = 3.142). In general the pressure amplitude at any instant is higher for nonadiabatic compared to adiabatic.
20
0.0008
0.0007
0.0006
0.0005
adiabatic
U 0.0004
nonadiabatic
0.0003
0.0002
0.0001
0 0
0.5
1
1.5
2
2.5
3
3.5
4
Length
Figure 4.2: Comparison of velocity in the centreline of the chamber at t = 2.44
21
0.0007
0.0006
0.0005
0.0004 adiabatc
U
nonadiabatic 0.0003
0.0002
0.0001
0 0
1
2
3
4
Length
Figure 4.3: Comparison of velocity in the centreline of the chamber for the half
wavelength ( t = 3.142)
22
4
x 10 8
6
4
2
0
2 100 80 60 40
H
20 0
0
50
100
150
200
250
300
350
400
L
(a) Nonadiabatic at t = 3.142
4
x 10 8 6 4 2 0 2 4 100
80 60 H
40 20 0
0
50
100
150
200
250
300
350
400
L
(b) Adiabatic at t = 3.142 Figure 4.4: Velocity profile for the half wave length or at t = 3.142
23
3
3
x 10
x 10
1
2
0
0
1 30
2 30
20
10
3
x 10
50
100
0 0 (a) t = 3.142
x 10 5
0
0
20
10
3
x 10
0 0 (c) t =18.852
50
100
5 30
0
0
100 10
0 0 (e) t = 37.740
10
x 10 5
20
20 3
5
5 30
10
3
5
5 30
20
50
5 30
20
50
100
0 0 (b) t = 6.284
0 0 (d) t = 25.136
50
100
100 10
0 0 (f) t = 50.272
50
Figure 4.5: Evolutions of axial velocity (u) in adiabatic case for a sequence of time.
24
4
4
x 10
x 10
2
5
0
0
2 30
5 30
20
100 10
4
x 10
0 0 (a) t = 3.142
50
x 10 5
0
0
5 30
5 30
10
4
x 10
50
100
0 0 (c) t = 18.852
0
0
5 30
5 30
100 0 0 (e) t = 37.704
10
x 10 5
10
20 4
5
20
100 10
4
5
20
20
50
20
0 0 (b) t = 6.284
50
50
100
0 0 (d) t = 25.136
100 10
0 0 (f) t = 50.272
50
Figure 4.6: Evolutions of transverse velocity (v) in adiabatic case for a sequence of
time.
25
3
3
x 10
x 10
1
2
0
0
1 30
2 30
100
20
10
3
x 10
0 0 (a) t = 3.142
50
x 10 2
0
0
5 30
2 30
10
3
x 10
50
100
0 0 (c) t = 18.852
0
0
5 30
5 30
50 0 0 (e) t = 37.740
10
x 10 5
10
20 3
5
20
100 10
3
5
20
20
100
20
10
0 0 (b) t = 6.284
50
50
100
0 0 (d) t = 25.136
50
100
0 0 (f) t = 50.272
Figure 4.7: Evolutions of axial velocity (u) for nonadiabatic cases
26
4
4
x 10
x 10
5
5
0
0
5 30
5 30
20
10
4
x 10
50
100
0 0 (a) t = 3.142
x 10 5
0
0
20
100 10
4
x 10
0 0 (c) t = 18.852
5 30
50
0
0
5 30
5 30
0 0 (e) t = 37.704
50
100
20
50
100
0 0 (b) t = 6.284
100 10
x 10 5
10
20 4
5
20
10
4
5
5 30
20
10
0 0 (d) t = 25.136
0 0 (f) t = 50.272
50
50
100
Figure 4.8: Evolutions of transverse velocity (v) in non adiabatic case for a
sequence of time .
27
20
18
16
14
12
10
8
6
4
2 0
5
10
15
20
25
30
(a) Vector plot at t = 2.702 for adiabatic 20
18
16
14
12
10
8
6
4
2 5
10
15
20
25
30
(b) Vector plot at t = 2.702 for nonadiabatic
(c) Schematic diagram of the chamber to shows the vector plot region
Figure 4.9: Velocity vector and vortex flow near the membrane at t = 2.702
28
20
18
16
14
12
10
8
6
4
2
5
10
15
20
25
30
(a) Vector plot at t = 3.002 for adiabatic 20
18
16
14
12
10
8
6
4
2
0 0
5
10
15
20
25
30
(b) Vector plot at t = 3.002 for nonadiabatic
(c) Schematic diagram of the chamber to shows the vector plot region
Figure 4.10: Velocity vector and vortex flow near the membrane at t = 3.002
29
20
18
16
14
12
10
8
6
4
2
0
5
(a)
10
15
20
25
30
Vector plot at t = 3.142 for adiabatic
20
18
16
14
12
10
8
6
4
2 5
(b)
10
15
20
25
30
Vector plot at t = 3.142 for nonadiabatic
(c) Schematic diagram of the chamber to shows the vector plot region
Figure 4.11: Velocity vector and vortex flow near the membrane at t = 3.142
30
2.50E04
2.00E04
1.50E04 adiabatic
Tmean
nonadiabatic
1.00E04
5.00E05
0.00E+00 0
5
10
15
20
25
30
time
Figure 4.12: The average temperature in the chamber for the five cycle.
31
4
x 10
100
3 80 2
60 40
1
20 0 50
100
150
200 250 (a) adiabatic case
300
350
400
4
x 10
100 80
3
60 2 40 1 20
50
100
150
200 250 (b) nonadiabatic case
300
350
400
0
Figure 4.13: Temperature at t = 3.002
32
4
x 10 2
100
0
80
2
60
4
40
6 20 8 50
100
150
200 250 (a) Adiabatic case
300
350
400
4
x 10 4
100 80
3
60
2
40
1
20 0 50
100
150
200 250 (b) nonadiabatic case
300
350
400
Figure 4.14: Temperature at t = 3.142
33
5
x 10 14
100
12 80
10 8
60
6 40
4 2
20
0 50
100
150
200 250 (a) adiabatic case
300
350
400
4
x 10 10
100 80
8
60
6
40
4
20
2
50
100
150
200 250 (b) nonadiabatic case
300
350
400
0
Figure 4.15: Pressure at t = 3.002
34
4
x 10 1
100 80
0
60
1
40
2
20 3 50
100
150
200 250 (a) adiabatic case
300
350
400
4
x 10 10
100 80
8
60
6 4
40
2
20
0 50
100
150
200 250 (b) nonadiabatic case
300
350
400
Figure 4.16: Pressure at t = 3.142
35
CHAPTER 5
CONCLUSION
The main contribution of this research is to discuss the adiabatic and nonadiabatic acoustic waves. Based on the analysis several conclusions can be made:
i.
The adiabatic acoustic wave produced similar profile as nonadiabatic case for parameters stated in Table 4.1.
ii.
Due to the transfer of energy during collusions, the adiabatic wave velocity moved faster than the nonadiabatic wave velocity.
iii.
Vortex motion near membrane is present as with the case of a nonadiabatic model.
iv.
The temperature and pressure in adiabatic case at any instant is lower compared to nonadiabatic cases.
v.
For a chamber with no stack, the adiabatic system can be used to model an acoustic chamber as an alternative to a nonadiabatic system.
36
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