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
A research project submitted to the University
of Cincinnati as the 1st project for the
Computational Fluid Dynamics Course
University of Cincinnati
Cincinnati, Ohio, USA
Winter 2009
Table of Contents
Subject PAGE
Table of Contents 2
ABSTRACT 5
CHAPTER 1 INTRODUCTION 6
CHAPTER 2- Problem Description and Mathematical Formulation 11
2.1 Problem Description and Boundary conditions 11
2.2 Initial And Inlet Conditions 12
2.3 Governing Equations 12
CHAPTER 3- Solution Mesh (Generated using Gambit) 14
3.1 Summarized Procedures for Mesh Generation Using Gambit 14
3.2 Grid Size, Zones and Quality 15
CHAPTER 4- Fluent Solver Setup, Solution Strategy and Conversion
Criterion. 16
4.1 Fluent Solver Setup and Solution Strategy 16
4.2 Convergence Criterion 18
CHAPTER 5 -Fluent Results 20
5.1 Subsonic Flow through the CD nozzle (Case 1, Pback=585 kPa) 20
5.2 Supersonic Flow through the CD nozzle (Case 2, Pback=35 kPa) 20
5.3 Transonic Flow (shock wave) through the CD nozzle (Case 3,Pback=300 kPa) 21
CHAPTER 6- Problem Exact Solution 32
6.1 Quasi 1D-Flow: Characteristics and Implications 32
6.2 Governing Equations for Quasi-1D Flow 32
6.3 Case 1: Subsonic Flow throughout the Nozzle (Pback=585 kPa) 34
6.4 Case 2: Supersonic Flow throughout the Nozzle (Pback=35 kPa) 35
6.5 Case 3: Normal Shock in the Diverging Section of the Nozzle
(Pback=300 Kpa) 36
6.6 Summary of Exact Solution 37
CHAPTER 7 -Comparison between Exact and CFD (Fluent) Results 43
CHAPTER 8- Summary and Conclusions 46
REFERENCES 49
APPENDICES 50
APPENDIX A – Height Versus Displacement 50
APPENDIX B - Time Step calculation 51
APPENDIX C - Convergence criteria 52
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
LIST OF FIGURES
No. Description Page
1- Diagram of a de Laval nozzle, showing approximate flow velocity (v),
together with the effect on temperature (t) and pressure (p) 7
2- Convergent Divergent Nozzle Configuration 8
3- Flow regimes in CD nozzle for different Pback/Po 9
4- The Convergent Divergent nozzle and boundary conditions. 11
5- Grid 1 used for the subsonic case. 15
6- Grid 1 used for the supersonic and transonic cases. 15
7- 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). 23
10- Density Contours for Pback=585 kPa (subsonic flow). 23
11- Stream function Contours for Pback=585 kPa (subsonic flow). 24
12- Mach Contours for Pback=35 kPa (supersonic flow). 24
13- Static pressure Contours for Pback=35 kPa (supersonic flow) 25
14- Static temperature Contours for Pback=35 kPa (supersonic flow). 25
15- Density Contours for Pback=35 kPa (supersonic flow) 26
16- Stream function Contours for Pback=35 kPa (supersonic flow) 26
17- Mach Contours for Pback=300 kPa (transonic flow). 27
18- Static pressure Contours for Pback=300 kPa (transonic flow). 27
1 9- Static temperature Contours for Pback=300 kPa (transonic flow). 28
20- Density Contours for Pback=300 kPa (transonic flow). 28
21- Stream function Contours for Pback=300 kPa (transonic flow). 29
22- Fluent result summery of Mach number for the three cases 29
23- Fluent result summery of P/P0 for the three cases 30
24- Fluent result summery of T/T0 for the three cases 30
25- Fluent result summery of /0 for the three cases 31
26- Mach Number Profile – Exact Solutions. 41
27- Pressure Profile– Exact Solutions. 41
28- Temperature Profile – Exact Solutions. 42
29- Density Profile – Exact Solutions. 42
30- Mach Number vs. Non-dimensional Nozzle Length. 44
31- Density Ratio vs. Non-dimensional Nozzle Length. 44
32- Pressure Ratio vs. Nozzle Length. 45
33- Temperature vs. Nozzle Length. 45
C.1- Convergence monitor (scaled residuals) for case 1 51
C.2- Convergence monitor (scaled residuals) for case 2 51
C.3- Convergence monitor (scaled residuals) for case 3 52
C.4 - Convergence monitor (mass flow rate) for case 1 52
C.5 - Convergence monitor (mass flow rate) for case 2 53
C.6 - Convergence monitor (mass flow rate) for case 3 53
C.7 - Convergence monitor (Heat transfer rate) for case 1 54
C.8 - Convergence monitor (Heat transfer rate) for case 2 54
C.9 - Convergence monitor (Heat transfer rate) for case 3 55
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.
CHAPTER 1
INTRODUCTION
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 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
· Supersonic: 1.2 < M < 5
· Hypersonic: M > 5
1.2 CD Nozzle Characteristics:
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.
Fig. 2- Convergent- Divergent Nozzle Configuration
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 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 (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.
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.
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)
(8)
(9)
(10)
The governing equations for steady quasi one dimensional flow are:
ρ1u1A1=ρ2u2A2 (11)
ρuA=ρ*u*A*=m (12)