EGFD 637 – Computational Fluid Dynamics Project 2 Fluent Simulation for the flow through a Convergent- Divergent Nozzle with Subsonic, Supersonic and Transonic Flow By Salah Soliman Sowmya Krishnamurthy Santosh Konangi Bhaskar Chandra Konala Mohamed Abdelaal In Aerospace Engineering Under Supervision of Dr. Kirti Ghia College of Engineering University of Cincinnati 1
82
Embed
Introduction - U-M Personal World Wide Web Server · Web viewEGFD 637 – Computational Fluid Dynamics Project 2 Fluent Simulation for the flow through a Convergent-Divergent Nozzle
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
EGFD 637 – Computational Fluid Dynamics
Project 2
Fluent Simulation for the flow through a Convergent-Divergent Nozzle with
Subsonic, Supersonic andTransonic Flow
BySalah Soliman
Sowmya KrishnamurthySantosh Konangi
Bhaskar Chandra KonalaMohamed Abdelaal
InAerospace Engineering
Under Supervision of
Dr. Kirti Ghia
College of EngineeringUniversity of Cincinnati
A research project submitted to the Universityof Cincinnati as the 1st project for the
Computational Fluid Dynamics Course
University of CincinnatiCincinnati, Ohio, USA
Winter 2009
1
Table of ContentsSubject PAGETable of Contents 2ABSTRACT 5CHAPTER 1 INTRODUCTION 6CHAPTER 2- Problem Description and Mathematical Formulation 112.1 Problem Description and Boundary conditions 112.2 Initial And Inlet Conditions 122.3 Governing Equations 12CHAPTER 3- Solution Mesh (Generated using Gambit) 143.1 Summarized Procedures for Mesh Generation Using Gambit 143.2 Grid Size, Zones and Quality 15CHAPTER 4- Fluent Solver Setup, Solution Strategy and Conversion Criterion. 164.1 Fluent Solver Setup and Solution Strategy 164.2 Convergence Criterion 18CHAPTER 5 -Fluent Results 205.1 Subsonic Flow through the CD nozzle (Case 1, Pback=585 kPa) 205.2 Supersonic Flow through the CD nozzle (Case 2, Pback=35 kPa) 205.3 Transonic Flow (shock wave) through the CD nozzle (Case 3,Pback=300 kPa) 21CHAPTER 6- Problem Exact Solution 326.1 Quasi 1D-Flow: Characteristics and Implications 326.2 Governing Equations for Quasi-1D Flow 326.3 Case 1: Subsonic Flow throughout the Nozzle (Pback=585 kPa) 346.4 Case 2: Supersonic Flow throughout the Nozzle (Pback=35 kPa) 356.5 Case 3: Normal Shock in the Diverging Section of the Nozzle (Pback=300 Kpa) 366.6 Summary of Exact Solution 37CHAPTER 7 -Comparison between Exact and CFD (Fluent) Results 43CHAPTER 8- Summary and Conclusions 46REFERENCES 49APPENDICES 50APPENDIX A – Height Versus Displacement 50APPENDIX B - Time Step calculation 51APPENDIX C - Convergence criteria 52
2
LIST OF TABLES
No. Description Page
1- Summarized boundary conditions defined using Gambit. 14
2- Exact solution Results for Case 1 (Pback = 585 KPa) 38
3- Exact solution Results for Case 2 (Pback = 35 KPa). 39
4- Exact solution Results for Case 3 (Pback = 300 KPa). 40
A-1 Height of C-D nozzle as a function of displacement x. 49
B-1 Table B.1- Time step calculation. 50
3
LIST OF FIGURESNo. Description Page1- Diagram of a de Laval nozzle, showing approximate flow velocity (v),
together with the effect on temperature (t) and pressure (p) 72- Convergent Divergent Nozzle Configuration 83- Flow regimes in CD nozzle for different Pback/Po 94- The Convergent Divergent nozzle and boundary conditions. 115- Grid 1 used for the subsonic case. 156- Grid 1 used for the supersonic and transonic cases. 157- Mach Contours for Pback=585 kPa (subsonic flow). 22 8- Static pressure Contours for Pback=585 kPa (subsonic flow). 22 9- Static temperature Contours for Pback=585 kPa (subsonic flow). 2310- Density Contours for Pback=585 kPa (subsonic flow). 2311- Stream function Contours for Pback=585 kPa (subsonic flow). 2412- Mach Contours for Pback=35 kPa (supersonic flow). 2413- Static pressure Contours for Pback=35 kPa (supersonic flow) 2514- Static temperature Contours for Pback=35 kPa (supersonic flow). 2515- Density Contours for Pback=35 kPa (supersonic flow) 2616- Stream function Contours for Pback=35 kPa (supersonic flow) 2617- Mach Contours for Pback=300 kPa (transonic flow). 2718- Static pressure Contours for Pback=300 kPa (transonic flow). 271 9- Static temperature Contours for Pback=300 kPa (transonic flow). 2820- Density Contours for Pback=300 kPa (transonic flow). 2821- Stream function Contours for Pback=300 kPa (transonic flow). 2922- Fluent result summery of Mach number for the three cases 2923- Fluent result summery of P/P0 for the three cases 3024- Fluent result summery of T/T0 for the three cases 3025- Fluent result summery of 𝛒/𝛒0 for the three cases 3126- Mach Number Profile – Exact Solutions. 4127- Pressure Profile– Exact Solutions. 4128- Temperature Profile – Exact Solutions. 4229- Density Profile – Exact Solutions. 4230- Mach Number vs. Non-dimensional Nozzle Length. 4431- Density Ratio vs. Non-dimensional Nozzle Length. 4432- Pressure Ratio vs. Nozzle Length. 4533- Temperature vs. Nozzle Length. 45C.1- Convergence monitor (scaled residuals) for case 1 51C.2- Convergence monitor (scaled residuals) for case 2 51C.3- Convergence monitor (scaled residuals) for case 3 52C.4 - Convergence monitor (mass flow rate) for case 1 52C.5 - Convergence monitor (mass flow rate) for case 2 53C.6 - Convergence monitor (mass flow rate) for case 3 53C.7 - Convergence monitor (Heat transfer rate) for case 1 54C.8 - Convergence monitor (Heat transfer rate) for case 2 54C.9 - Convergence monitor (Heat transfer rate) for case 3 55
4
ABSTRACT
Flow through Convergent Divergent nozzle is solved using Fluent. The nozzle runs under a total inlet pressure (P0) and temperature (T0) of 600 kPa and 300 K respectively. The nozzle cross-section area (A) is specified as a function of the axial distance (x). The flow through the nozzle is investigated for three different back pressures that will result in subsonic (Pback=585 kPa), supersonic (Pback=35 kPa) and transonic flow (Pback=300 kPa). The grid used for the subsonic case was a coarse grid of 31 nodes in the axial directions and 11 nodes in the normal direction. To capture the thin shock in the transonic case a relatively finer mesh is used (121*41 nodes), also the finer grid was used in the second case with Pback=35KPa . A density based 2nd order implicit, inviscid and unsteady solver is selected in Fluent. Air is assumed as an ideal gas with a constant value of specific heat. A time step of 0.000143 seconds is used based on the CFL (courant) number of 0.5. The convergence criterion of continuity, x and y velocities is set to 10-4, while energy convergence criterion is set to 10-6. The flow field values obtained from fluent are averaged at each axial location over the area and compared with the exact 1D quasi-steady solver. The variation Mach, density, pressure and Temperature along the nozzle axis show good agreement with the exact 1D solution. For the subsonic case the relative difference between CFD results and the 1D results ranges between 5-21%, 1.1-17%, 0.1-4.6%, 0.14-11% for the Mach, P/Po, T/To and ρ/ρo respectively (depending on axial location). Also, for the supersonic case difference ranges between 11-25%, 4.8-25%, 1.4-19.6%, 3.5-15% for the Mach, P/Po, T/To and ρ/ρo respectively (depending on axial location). Finally, for the transonic case (pback=300kPa), the variation of Mach, P/Po, T/To and ρ/ρo along the nozzle axis also has a very good agreement with the exact 1D-quasi steady solution up to the point were the bow shock forms. A bow shock was expected because the real flow in the nozzle is a 2D (nozzle area increases sharply with axial distance). The relative difference between CFD results and the 1D results (up to that point) ranges between 0.13-17%, 0.5-15%, 0.6-14%, 1.3.14.6% for the Mach, P/Po, T/To and ρ/ρo respectively.
5
CHAPTER 1INTRODUCTION
The development of high speed digital computers along with the achievement of efficient numerical algorithms has enormously affected Computational Fluid Dynamics (CFD) to be advanced the last decades.
In this project work, the flow through a converging diverging nozzle will be computed through the commercial CFD program (FLUENT) in 3 different flow cases. Results will be compared with well established exact solutions of such cases.
In Chapter two, cd nozzle problem is defined and the formulation of the problem is presented. In chapter Three, Generation of mesh is discussed for all cases of flow. In Chapter Four, FLUENT solver setup will take place. The Fluent results will be presented and discussed in Chapter Five. In Chapter Six, the problem will be solved using the 1-D approach in exact form. Next Chapter, which is Seven, will deal with the comparison between the results and the exact solution. Finally, Summary and conclusion for this work are presented.
1.1 CONVERGENT- DIVERGENT NOZZLE:
Any fluid-mechanical device designed to accelerate a flow is called a nozzle. A de Laval nozzle (or convergent-divergent nozzle, CD nozzle ) is a tube that is pinched in the middle, making an hourglass-shape. It is used as a means of accelerating the flow of a gas passing through it to a supersonic speed. It is widely used in some types of steam turbine and is an essential part of the modern rocket engine and supersonic jet engines.
Its operation relies on the different properties of gases flowing at subsonic and supersonic speeds. The speed of a subsonic flow of gas will increase if the pipe carrying it narrows because the mass flow rate is constant. The gas flow through a de Laval nozzle is isentropic (gas entropy is nearly constant). At subsonic flow the gas is compressible; sound, a small
6
pressure wave, will propagate through it. At the "throat", where the cross sectional area is a minimum, the gas velocity locally becomes sonic (Mach number = 1.0), a condition called choked flow. As the nozzle cross sectional area increases the gas begins to expand and the gas flow increases to supersonic velocities where a sound wave will not propagate backwards through the gas as viewed in the frame of reference of the nozzle (Mach number > 1.0).
Fig.1- Diagram of a de Laval nozzle, showing approximate flow velocity (v), together with the effect on temperature (t) and pressure (p)
The Mach number is a non-dimensional velocityand it is equal to the velocity of the fluid relative to the local speed of sound.
(1)
The regimes of flow depending on the value of Mach number are:
Subsonic: M < 1 Sonic: Ma=1 Transonic: 0.8 < M < 1.2
The configuration of a converging diverging nozzle (CD) is shown in the figure. Gas flows through the nozzle from a region of high pressure (usually referred to as the chamber) to one of low pressure (referred to as the ambient or tank). The chamber is usually big enough so that any flow velocities here are negligible. The pressure here is denoted by the symbol pc. Gas flows from the chamber into the converging portion of the nozzle, past the throat, through the diverging portion and then exhausts into the ambient as a jet. The pressure of the ambient is referred to as the 'back pressure' and given the symbol Pback.
To understand the flow behavior in a CD nozzle let’s assume that the pressure at the exit of the nozzle is reduced than the inlet total pressure. Consequently, the mass flow increases through the nozzle. But if the back pressure is lowered too much then the flow rate suddenly stops increasing all together. This condition is called ‘Choking’. The reason for this behavior has
8
to do with the way the flow behaves at Mach 1, i.e. when the flow speed reaches the speed of sound. A good number to keep in mind while conducting experiments is that approximately 50% pressure ratio will result in a chocked nozzle.
The flow regime of a convergent divergent nozzle depends on the pressure ratio across the nozzle. That is, the value of Pback/Po (back pressure to total pressure ratio) will imply wither the flow will be supersonic or subsonic. A common plot that shows different flow configuration based on the pressure ratio is shown in Fig. 3 [3].
Fig. 3- Flow regimes in CD nozzle for different Pback/Po [3].
Analysis of above figure we can summarize the following:
• Case a: When the nozzle isn't choked, the flow in both sections of the nozzle is subsonic.
• Case b: As the back pressure is lowered the flow speed increases everywhere in the nozzle and eventually reaches the sonic speed
9
(Mach 1) at the throat, but continues to be subsonic in the divergent part.
• Case c: further reduce in pressure will result in a supersonic flow in the divergent part, but as the back pressure is not low enough the supersonic acceleration is terminated by a normal shock wave after which the flow will be subsonic.
• Case d: further decrease in the back pressure will result in the movement of the normal shock to the nozzle exit.
• Case e: That is an under expanded nozzle.
• Case f: The design point of the nozzle. The back pressure is low enough such that the flow will be fully expanded and supersonic flow is generated.
• Case g: That is an over expanded nozzle.
10
Chapter Two
Problem Description and Mathematical Formulation.
2.1- Problem description and boundary conditions.
A quasi 1-D inviscid compressible flow through a converging-diverging nozzle
(CD) is assumed to take place. The area of the nozzle varies according to equation (2).
The nozzle throat is located at x = 1.5 and the convergent section occurs for x <1.5 and
the divergent section occurs for x > 1.5. The nozzle is shown in Fig. 4
(2)
Three cases discussed in this project are:(a) Subsonic flow at the exit, using back pressure of 585kPa.
(b) Supersonic flow at the exit, using back pressure of 35kPa.
(c) Supersonic flow with a shock leading to subsonic flow at the exit,
using back pressure of 300kPa
Fig. 4- The Convergent Divergent nozzle and boundary conditions.
11
12
The Grid is to have 31 point in the x direction (dx=0.1) and 11 points in the y
direction. The problem is to be solved as an unsteady problem with the initial values of ρ,
T and u defined by equations 3, 4 and 5 respectively. The problem is to be solved as a
symmetrical problem (to reduce computation al time). The total pressure and temperature
are known at the inlet are:
(3)
(4)
(5)
The time step (dt) is to be evaluated from the Courant number (C) condition using
equation (6);
(6)
Where: a is the speed of sound and u is the velocity in x direction velocity
2.2 Initial And Inlet Conditions:
The following are the inlet conditions applied to the CD nozzle-
P0= 600kPa
T0= 300K
The above parameters are used for modeling using GAMBIT and solving the
problem using FLUENT software.
Using the above mentioned parameters, the nozzle was modeled using the popularly
used software GAMBIT, and eventually all parameters in the flow field were determined
using the commercially available code, FLUENT.
2.3- Governing Equations.
The continuity equation (7), Navier stokes equations (8, 9) along with the energy equation (10) will be solved using FLUENT codes
(7)
13
(8)
(9)
(10)
The governing equations for steady quasi one dimensional flow are:ρ1u1 A1=¿ ρ2u2 A2¿ (11)ρuA=ρ¿u¿ A¿=m (12)
Where, 1 and 2 denote the inlet and the outlet sections of the nozzle and conditions denoted with an * at sonic speed i.e. at M=1.Area Mach number relation:
( AA ¿ )
2= 1M 2 [ 2
γ+1 (1+ γ−12M 2)]
γ +1γ−1
(13) The above equation tells us that the Mach number M is a function of the ratios of the areas (A/A*). i.e. the Mach number at any location is the ratio of the local duct area to the sonic duct area. Other relations used to determine the required values analytically are given below.
pop2
=(1+ γ−12M 2
2)γγ−1
(14)a=√γ RTT 2
T o=( pepo )
γ−1γ
(15)
(16)
14
Chapter Three
Solution Mesh
GAMBIT is a software package designed to help analysts and designers build and
mesh models for computational fluid dynamics (CFD) and other scientific applications.
GAMBIT receives user input by means of its graphical user interface (GUI). The
GAMBIT GUI makes the basic steps of building, meshing, and assigning zone types to a
model simple and intuitive, yet it is versatile enough to accommodate a wide range of
modeling applications.
3. 1- Summarized Procedures for Mesh Generation using Gambit.
Procedure for creating the mesh is briefed as follows and is detailed in Appendix A:
In order to create the geometry, the equation
is solved using Excel.
The number of data points used is 61 and the unit cross sectional area at each step
along the nozzle (x-direction) is calculated. Data points are also given in
Appendix A.
Mesh is divided into 30 divisions in the x-axis and 10 divisions in the y-axis.
Boundary conditions are specified as summarized in Table 1.
2D mesh is exported.
Edge Position Name TypeLeft inlet PRESSUR_INLETRight outlet PRESSURE_OUTLETTop wall WALLBottom centerline SYMMETRY
Table 1- Summarized boundary conditions.
15
3. 2 Grid Size, Zones and Quality.
Two grids were used in the current research. First, a coarse gird (30*10) for the
simulation of the subsonic flow case 1. Second, a relatively finer grid for the simulation
of the supersonic and transonic case (120*40). Both grids are shown in Fig. 5 and 6
respectively.
Fig. 5- Grid 1 used for the subsonic case.
16
Fig. 6- Grid 2 used for the supersonic and transonic case.
17
Chapter FourSolver Setup, Strategy and Convergence Criterion
We are required to solve the problem defined in Chapter two for three different
cases. Namely; subsonic, supersonic and flow with a normal shock. The solver setup
will be the same for all three cases, but the flow field initialization will be different. The
reason is that the flow field characteristics are different. The convergence criterion that
will be used is similar for all three cases as we will discuss later.
4.1- Fluent Solver Setup and Solution Strategy.
The setup of the solver is summarized as follows:
Precision: a 2D double precision is used as it permits a higher precision than 2d
single precision solver, but larger memory is required [1].
Solver: Density based solver is recommended as the flow is compressible.
Time: Unsteady formulation. A time step of 0.000143 seconds based on a CFL of
0.5 using the equation 6. A complete calculation of time step is shown in
Appendix B.
Space: 2D problem. The center line of the CD nozzle is taken as symmetric to
reduce computational time.
Solver Formulation: Implicit formulation is used which is more stable than the
explicit formulation. It depends on time step and it is well known that implicit
formulation is less sensitive to the time step selected than Explicit formulation.
Unsteady Formulation: 2nd Order Implicit which is more accurate than 1st order
implicit.
Energy: Energy equation is involved in the computations as the flow is
compressible and temperature will change in a sensitive manner, and
consequently the there will be an effect the density. (Pressure, density and
temperature are related through the ideal gas equation).
Viscous: Inviscid formulation. It is a good approximation when solving CD
nozzles. That means that we are dropping the viscous terms from the Navier
Stoke’s equation .
18
Materials: Air is chosen as an ideal gas.
Operating Conditions: Operating pressure in the CD nozzle to be zero as
recommended by the Fluent Manuals [1].
Boundary Conditions: For all three cases the inlet total pressure is set as 600,000
Pascal. The inlet static pressure is 585,000 Pascal. The inlet total temperature is
300 K. However at the exit we define the following for the three different cases:
Case one (Subsonic) Back pressure is 585,000 Pascal.
Case two (Super Sonic) Back pressure is set to 35,000 Pascal. Once Fluent
determines that the flow is Supersonic it will ignore the defined exit back
pressure and will calculate the pressure at the exit from adjacent cells [2].
Case Three (Normal shock) Back pressure is set to 300,000 Pascal.
Solution Initialization: The flow should be initialized in a manner which is
physically correct. For example, a supersonic flow through the CD nozzle will
have the values of pressure, density and temperature continuously decreasing
throughout the nozzle. Then, it is advisable to use an initial profile that is
consistent with the expected profiles to reduce the convergence time steps
required [2]. While in the case of subsonic flow that is not the case and initializing
the flow in the same manner will result in a solution that needs a long period of
time to converge and reach steady solution. Based on that the flow field was
initialized as follows:
Subsonic Case and Transonic case:
The solution was initialized from the inlet conditions. Fluent computes the
inlet velocity and temperature based on the defined total pressure (600
KPa), static pressure (585 KPa) and total temperature (300 K).
Supersonic:
The solution was initialized using the equations defined previously in
chapter 1. However, Fluent doesn’t enable the initialization of the density
in the flow field, but rather enables pressure initialization (using ideal gas
equation). Also, Fluent uses dimensional values and not dimensionless
values. The way we can initialize the flow filed using equations like 1.2,
1.3 and 1.4 is by using the Patch function in Fluent. However the first step
19
will be defining the equations using User Defined Function (UDF) in
Fluent. In summary here are the steps to initialize the flow field:
A. We define the following equations using UDF:
(17)
(18)
(19)
B. We use Patch, Zone, and User Defined Function to initialize
temperature, velocity and pressure respectively.
4.2 Convergence Criterion.
As it is usually the case in Fluent three convergence Criterion are used to ensure
convergence of the solution:
4.2.1- Residuals:
Usually Residuals plots are used to monitor how the solution is conversing. Also, a
certain value is assigned for residuals such that once this value is reached that is an
indication that solution converged. The first question is what are the residuals? To
answer that question we choose for example the velocity residuals to talk about.
When we are solving a problem using CFD variables like velocity change changes as
iterations proceed. An indication of convergence is that this change keeps on
decreasing until it becomes insignificant. Therefore, Residuals of velocity is the
average value, though the domain, by which the velocity is changing from one
iteration to the other. The values we used to ensure convergence are 1*10 -4 for
continuity and velocity. We used a value of 1*10-6 for energy as recommended by
Fluent for the density based solver [3]. Convergence monitors for the three cases are
presented in figures listed in Appendixes C.
4.2.2 – Surface variables:
20
Usually variables at different surfaces are monitored through the solution iterations to
ensure that they will reach a steady value. Usually that is done to derived values as
forces, but we will use in our case velocities at inlet and exit.
4.2.3 - Flux balance:
It is important to check the mass flow rate balance and energy balance through the
control volume (or solution domain) to ensure convergence. Indeed, if convergence is
reached the mass flow rate in should be equal to the mass flow rate out (steady state is
reached) and same applies to energy.
To sum up, the three different convergence criterion discussed previously will be
used in combination to ensure that we do have a converged solution. As listed in
Appendixes C.
21
Chapter Five
Fluent Results
As it was mentioned previously we are solving the CD nozzle for three different
back pressures. Firstly, a back pressure of 585 kPa will result in a subsonic flow through
the nozzle. Secondly, a back pressure of 35 kPa will result in a supersonic flow through
the nozzle with complete expansion. Finally, a back pressure of 300 kPa will result in a
flow with a shock wave. The 30*10 grid was good enough for the subsonic case, but it
was required to have a finer mesh to capture the shock for the transonic case and we use
it also for the second case (supersonic flow) because of the dramatic changes in flow
parameters. The mesh was adapted for that reason and we used a 120*40 grid. However,
we would like to mention that the mesh we are solving for is a coarse mesh and that will
be the main reason that the solution contours are not smooth enough.
5. 1 Subsonic flow through the CD nozzle (Case 1, Pback=585 kPa).
The Mach, static pressure, density and static temperature contours are shown in
Fig. 7, 8, 9 and 10 respectively. As expected, the flow accelerates in the convergent
section until it reaches the maximum velocity at the throat and then decelerates again in
the divergent section. Consequently, pressure decreases until the throat is reached and
then increases again in the divergent section until the exit is reached. Temperature and
density behaves in the same manner as pressure. Both of them are inversely proportional
to the fluid velocity. The total pressure, density, and temperature are almost constant
through the flow for the shock free inviscid flow configuration we are solving. The
Stream function is plotted in Fig. 11. Again, particles path and regions of high and low
velocity are easily observed. Appendix C shows the convergence history of the Fluent
simulation.
5. 2 Supersonic flow through the CD nozzle (Case 2, Pback=35 kPa).
The Mach, static Pressure, Density and static temperature contours are shown in
Fig. 12, 13, 14 and 15 respectively. In the convergent part the flow behaves in the same
22
manner as the previous case, but reaches sonic at the throat. Then, through the divergent
the flow continues as a supersonic flow were Mach number and velocity keep increasing
until the flow exits from the CD nozzle. Pressure, temperature and density will be
decreasing in the divergent part because of the increase in velocity, The Stream function
is plotted in Fig. 16. The flow field looks as expected!
5. 3-Transonic flow (shock wave) through the CD nozzle (Case 3,
Pback=300 kPa).
The Mach, static Pressure, Density and static temperature contours are shown in
Fig. 17, 18, 19 and 20 respectively. The shock region is noticed in all figures (region of
sudden change from super sonic to subsonic). The mesh we used to capture the shock
has 120 intervals (dx=0.025 m) in the x axis direction and 40 intervals in the y axis
direction. To produce a better solution for the normal shock case we even need a finer
mesh, but the computational time will increase significantly.
From the figures we notice that the Mach number increase until it reaches the
shock region and then changes to a subsonic flow. We can easily note that pressure,
temperature, velocity and density jumps around the shock region as the flow changes
from subsonic to supersonic.
There are two reasons for why we see a thick bow shock rather than a thin normal
shock in the simulated flow. First, the mesh we used is not fine enough. If the mesh size
was to be reduced less than 1 mm we would have been able to obtain a better thin shock.
However, as the mesh becomes finer the simulation time will increase drastically.
Second, the nozzle we are simulating has a profile that is increasing sharply with the x
axis (area gradient is sharp). From the previous results we notice that the stream lines are
not straight, but rather they do follow the contours of the nozzle. Then, the shock is still
normal to the flow, but not the axis of the nozzle. That is clear from the stream function
plot Fig. 21.
23
Contours of Mach Number (Time=2.8000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)
Jan 22, 2008
6.85e-016.55e-01
6.24e-015.94e-01
5.63e-015.32e-015.02e-01
4.71e-014.41e-014.10e-01
3.79e-013.49e-013.18e-01
2.87e-012.57e-012.26e-01
1.96e-011.65e-01
1.34e-011.04e-017.31e-02
Fig. 7- Mach Contours for Pback=585 kPa (subsonic flow).