OPTIMIZATION OF FUEL-AIR MIXING FOR A SCRAMJET COMBUSTOR
GEOMETRY USING CFD AND A GENETIC ALGORITHM
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information
Vivek Ahuja
Certificate of Approval:
John Burkhalter Roy Hartfield, Chair
Professor Emeritus Professor
Aerospace Engineering Aerospace Engineering
Rhonald Jenkins Andrew Shelton
Professor Emeritus Assistant Professor
Aerospace Engineering Aerospace Engineering
George T. Flowers
Dean
Graduate School
Auburn Alabama
December 19, 2008
OPTIMIZATION OF FUEL-AIR MIXING FOR A SCRAMJET COMBUSTOR
GEOMETRY USING CFD AND A GENETIC ALGORITHM
Vivek Ahuja
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of Master of Science
iii
OPTIMIZATION OF FUEL-AIR MIXING FOR A SCRAMJET COMBUSTOR
GEOMETRY USING CFD AND A GENETIC ALGORITHM
Vivek Ahuja
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon request of individuals or institutions and at their expense. The author reserves all
publication rights.
Signature of Author
Date of Graduation
iv
THESIS ABSTRACT
OPTIMIZATION OF FUEL-AIR MIXING FOR A SCRAMJET COMBUSTOR
GEOMETRY USING CFD AND A GENETIC ALGORITHM
Vivek Ahuja
Master of Science, December 19, 2008
(B.S., University of Pune, 2007)
124 Typed pages
Directed by Roy Hartfield
A new methodology for the optimization of fuel-air mixing in a scramjet
combustor using integrated Genetic Algorithms and Computational Fluid Dynamics is
presented. A typical combustor design involving Mach 2 crossflow over a rearward
facing step with staged normal injection is considered for study and is optimized using
this method.
The CFD results are validated against experimental results prior to optimization to
allow for grid refinement and high accuracy of results. Quantification of typical
v
combustor performance and design parameters is discussed and adaptation for use with
CFD grids is presented. An integrated system of computers and software designed for fast
computation times has been created. Correlations between variations in physical
geometry and optimization of fuel-air mixing are presented.
vi
ACKNOWLEDGMENTS
The author would like to thank Dr. Roy Hartfield for providing the
encouragement, support and a free-thinking environment for conducting this research.
Thanks are also due to Dr. Rhonald Jenkins for helping improve the mathematical models
used in the study and to Dr. John Burkhalter, Dr. Andrew Shelton and Scott Thomas for
helping with the development of the integrated software used in the research. Thanks are
also due to my family members for their back stage support during a wonderful year of
work.
vii
Style manual or journal used:
American Institute of Aeronautics and Astronautics Journal
Computer software used:
FLUENT 6.2.17, GAMBIT, TECPLOT 10/360, FORTRAN, AUTOCAD 2008
viii
TABLE OF CONTENTS
LIST OF FIGURES .................................................................................................. ix
NOMENCLATURE ................................................................................................xii
1. INTRODUCTION ............................................................................................. 01
2. ANALYSIS AND RESEARCH ......................................................................... 04
2.1 Combustor Section Geometry and Design .................................................... 04
2.2 Grid Development and Refinement .............................................................. 07
2.3 Computational Methods and Solvers ............................................................ 14
2.4 Boundary Conditions .................................................................................... 21
2.5 Experimental Validation ............................................................................... 25
2.6 Flow without injection .................................................................................. 26
2.7 Flow with injection ...................................................................................... 33
2.8 Mass Fraction analysis ................................................................................. 42
2.9 Genetic Algorithm and CFD Integration ....................................................... 44
2.10 Design Parameters and Design Space ......................................................... 48
2.11 Design Goals .............................................................................................. 51
2.12 The Networked Environment ...................................................................... 56
3. THE RESULTS ................................................................................................. 61
3.1 Case Run-1................................................................................................... 61
3.2 Case Run-2................................................................................................... 75
3.3 Case Run-3................................................................................................... 82
3.4 Case Run-4................................................................................................... 89
4. CONCLUSIONS AND RECOMMENDATIONS ............................................. 96
REFERENCES ........................................................................................................ 98
APPENDIX ............................................................................................................ 101
ix
LIST OF FIGURES
Figure 2.1.1. Combustor section geometry in 3-D ...........................................................5
Figure 2.2.1. Flow field features for the non-injection case .............................................7
Figure 2.2.2. Two dimensional grid refinements .............................................................8
Figure 2.2.3. Grid refinement in greater detail and Zone interactions ..............................9
Figure 2.2.4. Combustor section grid for the non-injection case .................................... 11
Figure 2.2.5. Grid structure around the Injection surfaces ............................................. 13
Figure 2.2.6. 3D visualization of combustor section for the injection case ..................... 13
Figure 2.3.1. Residual plots for the Hybrid solver model............................................... 19
Figure 2.6.1. Pressure field comparison for the non-injection case ................................ 27
Figure 2.6.2. Static pressure profile comparison at the primary injector ........................ 28
Figure 2.6.3. Simulation results for the pressure contours in a 3D view ........................ 29
Figure 2.6.4. Comparisons for the temperature field for the non-injection case ............. 30
Figure 2.6.5. 3D visualization of the temperature field ................................................. 30
Figure 2.6.6. Static temperature profile comparison at the primary injector ................... 31
Figure 2.6.7. Comparisons for the velocity field for the non-injection case ................... 32
Figure 2.7.1. Flow features at the centerline section for the injection case ..................... 34
Figure 2.7.2. 3D visualization for the pressure field in the injection case....................... 34
x
Figure 2.7.3. Nature of the injectant flow through the combustor .................................. 35
Figure 2.7.4. Comparisons for the pressure field for the injection case .......................... 36
Figure 2.7.5. Comparisons for the static pressure profile at the primary injector ............ 37
Figure 2.7.6. Comparisons for the static pressure profile at the secondary injector ........ 37
Figure 2.7.7. Comparisons for the temperature field for the injection case..................... 38
Figure 2.7.8. Comparisons for the static temperature profile at the primary injector ...... 39
Figure 2.7.9. Comparisons for the static temperature profile at the secondary injector ... 40
Figure 2.7.10. 3D visualization of the temperature field...................................................39
Figure 2.8.1. Comparisons for the injectant mole fraction distribution.......................... 43
Figure 2.8.2. Mass Fraction distributions along the combustor length. ......................... 43
Figure 2.10.1. The Design Space for the optimization of the Scramjet Combustor ......... 50
Figure 2.12.1. An overview of the network structure at Auburn University .................... 57
Figure 2.12.2. A concept of operations view for a GA run ............................................. 58
Figure 2.12.3. Computational time versus number of processors .................................... 59
Figure 3.1.1. Total pressure ratio versus the generational best and worst performers .... 63
Figure 3.1.2. Mixing efficiency versus the generational best and worst performers ...... 64
Figure 3.1.3. Primary injector axial variation for the best and worst performers ........... 66
Figure 3.1.4. Secondary injector axial variation for the best and worst performers ....... 67
Figure 3.1.5. The primary and secondary injector variation for the best performers ...... 67
Figure 3.1.6. Optimizing the injector Axial (X) location .............................................. 69
Figure 3.1.7. Primary injector transverse variation for best and worst performers ......... 70
Figure 3.1.8. Secondary injector transverse variation for best and worst performers ..... 70
Figure 3.1.9. Effect of bringing the injectors closer to each other. .................................. 73
xi
Figure 3.2.1. Single Dimension Longitudinal Design Space for Case Run 2 ................... 76
Figure 3.2.2. Total Pressure Ratio for best and worst performers versus generations ...... 78
Figure 3.2.3. Mixing Efficiency for best and worst performers versus generations ......... 78
Figure 3.2.4. Movement of the primary and secondary injectors versus generations ....... 80
Figure 3.2.5. Relative distance between the injectors versus GA Generations ................ 81
Figure 3.3.1. Single Dimension Longitudinal Design Space for Case Run 3 ................... 84
Figure 3.3.2. Mixing Efficiency for best and worst performers versus generations ......... 85
Figure 3.3.3. Total Pressure Ratio for best and worst performers versus generations ...... 86
Figure 3.3.4. Primary and secondary injectors versus generation numbers ..................... 88
Figure 3.4.1. Mixing Efficiency for best and worst performers versus generations ......... 90
Figure 3.4.2. Total Pressure Ratio for best and worst performers versus generations ...... 91
Figure 3.4.3. Movement of the primary and secondary injectors versus generations ....... 92
Figure 3.4.4. Movement of the primary and secondary injectors versus generations ....... 94
Figure 3.4.5. Relative distances between the two injectors for Case Run 4 ..................... 94
Figure 3.4.6. Final results for the optimized injector locations ....................................... 95
xii
NOMENCLATURE
u Local velocity at any point in a grid
?g2206 Distance between two grid nodes in a grid space
?g2202 Time step
g2247 Courant number
N Total number of grid points in a grid surface
g2251s Average static density for a grid surface
g2251? Average total density for a grid surface
g2251?REF Reference total density
g2761g2201g4666g2191g4667 Static density at any nodal point ?i? in the grid surface
Ps Average static pressure for a grid surface
P? Average total pressure for a grid surface
P?REF Reference total pressure
g1790g2201g4666g2191g4667 Static pressure at any nodal point ?i? in the grid surface
Ts Average static temperature for a grid surface
T? Average total temperature for a grid surface
T?REF Reference total temperature
g1794g2201g4666g2191g4667 Static temperature at any nodal point ?i? in the grid surface
xiii
M Mach number
g2169g2206g4666g2191g4667 Mach number at any nodal point ?i? in the grid surface calculated using the
?X? axis component of velocity
g2204g2206 Velocity component of the fluid along the ?X? axis direction
g2204g2206g4666g2191g4667 Velocity component of the fluid at any grid point ?i' along the ?X? axis
direction
g2183g2174g2161g2162g4593 Reference sonic velocity of the fluid
g2237 Specific heat ratio for the fluid
R Universal gas constant
1
1. INTRODUCTION
For more than three decades extensive effort has been directed towards deploying
an operational Scramjet. However, an operational Scramjet is viable only if every
subsystem works at near optimum levels. Optimization of the subsystems is now possible
using Genetic Algorithms (GA) in conjunction with CFD solvers. This effort presents the
methodology for such optimization processes 1-6.
The study of fuel-air mixing in a supersonic cross-flow has been looked at as
potential scramjet combustor geometry. With the development of computing technology,
it is possible to develop optimized preliminary designs for scramjet combustors using a
CFD solver and a GA 7. Experimental results from research conducted in the 1990s have
been used for the validation of CFD solutions 8. The experiments were highly focused on
developing accurate data sets for a single case flow situation 10-11. This effort builds on
this single validated case by considering geometric variations of the combustor design,
solving for the flow and arriving at a geometry which is optimized for mixing efficiency
with minimum total pressure loss.
2
This thesis includes validation efforts and the results obtained from the
optimization effort. The fuel-air mixing phenomena inside a scramjet combustor section
are modeled as a function of injector locations and geometric aspect ratio changes. The
GA is a tournament based, binary encoded FORTRAN code and the CFD solver is a
Reynold?s Averaged Navier Stokes code developed by FLUENT.
The combustor section is taken to be a rectangular cross-sectioned type with a
rearward facing step 8, 10-13. The inlet velocity at the entry plane of the combustor section
upstream of the rearward facing step is taken to be Mach 2 throughout the course of the
optimization effort, in accordance with experimental results 8.
Two injectors downstream of the rearward facing step which are moved in both
the axial (X) direction and the transverse (Z) directions independently during the course
of the optimization. In effect, this characterizes the fuel injection into the combustor as
being a ?Staged Normal Injection? type. For the purpose of demonstrating the viability of
the GA driven CFD approach, this is the only geometric variation considered during the
present effort.
The initial flow simulation results obtained from this effort were compared with
the results obtained from the experiments9-11 to validate the accuracy of the simulation
result. Upon completion of this validation effort, the next step involved the automation
and integration of the above process with a Genetic Algorithm. A centralized program
structure was created to distribute the computation effort over a large computer network
available at Auburn University. Further studies were then conducted to reduce the
3
computational time at both the CFD level as well as the external hardware level to make
this optimization effort feasible.
To summarize, the objective of this effort was to develop the methodology for
optimizing a scramjet combustor fuel-air mixing performance through the combined use
of a GA and CFD as enumerated below:
A) Development of the flow-field grid and the validation of the accuracy of the CFD
results using experimental data.
B) Integration of the CFD and GA software codes into a networked system of
computers and improving the robustness, reliability and computational efficiency.
C) Conducting case runs on the above system with standard scramjet combustor
geometry and presenting the optimization results.
D) Providing the interpretation of the GA results in terms of actual physical changes
in the combustor geometry to determine trends and correlations and also creating
the basis for future work.
4
2. ANALYSIS AND RESEARCH
2.1 COMBUSTOR SECTION GEOMETRY AND DESIGN
The combustor section in this optimization effort is based on the generic
rectangular flow cross-section design with a rearward facing step as shown in Figure
2.1.1(a) and Figure 2.1.1(b). The values of the various geometries of the combustor
design are presented in Table 2.1.1. The locations of the injection points are also shown
in Figure 2.1.1(a) and 2.1.1(b). Known as the primary and secondary injectors, they are
placed at three and seven step heights downstream of this step respectively.
The availability of experimental results dictated the initial values of these
dimensions to enable a comparative study of the simulation and experimental results 8.
The nature of the optimization objectives meant that these locations would thereafter
vary. These initial values represent a validated starting point for the optimization effort.
5
Figure 2.1.1(a) Combustor section geometry in 3-D
Table 2.1.1 Combustor Geometry
Injection surface diameter (D)
1.93 mm
Combustor section height
21.29 mm
Combustor section width
30.48 mm
Step height
3.18 mm
First injector location (Initial)
9.54 mm
Second injector location (Initial)
22.26 mm
Step location along axial direction
11.02 mm
6
Figure 2.1.1 (b) Combustor section geometry in the plan view
7
2.2 GRID DEVELOPMENT AND REFINEMENT
The first grid refinement objective was the refinement of a grid for the two
dimensional flow field using the axial (X) and vertical (Y) directions. To facilitate this
process, the first flow simulation was conducted using no fuel injection, and was used to
refine the grid in the X-Y plane in the region downstream of the rearward step where the
shear layer was expected to develop as a result of the rearward facing step. In the absence
of injection of fuel, this shear layer is seen to be symmetrical in all X-Y planes along the
transverse (Z) axis except in the near wall regions. Figure 2.2.1 presents the details of the
flow-field in the case of no injection.
Figure 2.2.1. Flow field features for the non-injection case 8
8
The grid was divided into two regions: above the step and behind the step. The
region above the step is where the expansion fan and the single oblique shock
downstream of this fan were expected to be seen. It was acknowledged that this is a
relatively simple flow field to be simulated in FLUENT, and that the grid need not be
highly refined. Hence, a relatively coarse grid was developed to maintain computational
efficiency.
A more refined mesh zone surrounding the expected region of the shear layer for
the given flow conditions was developed. The main effort at this stage was to achieve
good fidelity between the experimental data set and the simulations with a minimal
number of grid points.
The grid for the X-Y plane is shown in Figure 2.2.2. The refined region and the
zone segregations are shown in Figure 2.2.3(a) and Figure 2.2.3(b). This decentralized
grid structure for the X-Y plane is held constant for the remainder of the analysis.
Figure 2.2.2. Two dimensional grid refinements of the Combustor X-Y plane and
various nodal zones
9
Figure 2.2.3(a) (left) and Figure 2.2.3(b) (Right). Grid refinement in greater detail
and Zone interactions (Combustor X-Y plane)
The exact values of the nodes varied for each grid zone. Table-2.2.1 presents the
number of nodes for the 2-dimensional flow case for the various grid zones. As can be
seen, using double the number of zones as normally required helped to make the grid
very simple to construct and visualize. This concept was then carried into the 3-
dimensional cases as well.
One of the focal points during this effort was fixing the exact values of the node
numbers within the structured grid. Accuracy and computational time are always directly
proportional to each other. The higher the demands for accuracy, the higher were the
computational time for getting the CFD solutions. And the higher the computational
times, the higher was the run times. At the same time accuracy could not be sacrificed to
save on computational time. Unique methods were developed to reduce computational
time from the networked structure and this is discussed later. Suffice to say that high
10
accuracy could be maintained at this early stage of the effort. An incremental approach
was initiated to determine the value of the nodal densities beyond which no significant
change in accuracy was noticed. This value of the nodal density for the various zones was
then fixed and maintained throughout the analysis.
Table 2.2.1 Node points per zone
Zone
Number of Node points (X*Y)
1
275 (25x11)
2
2500 (25x100)
3
3000 (30x100)
4
2750 (55x50)
Given the rectangular nature of the flow field geometry, and the initial case of no
injection surfaces in the X-Z plane, the 3-dimensional grid was relatively simple to
develop and merely extended the existing grid network into the Z coordinates. This gave
the required 3-dimensional mesh as shown in Figure 2.2.4. All existing zones were
extended into their respective volumes and correspondingly meshed. Table-2.2.2 presents
the number of nodes for each of the volumes within this 3-dimensional environment. The
total number of nodes for the 2-dimensional case amounted to 8,525 for a single
11
combustor X-Y plane and those for the 3-dimensional case amounted to 170,500 for the
entire combustor volume.
Figure 2.2.4. Three dimensional visualization of combustor section geometry for the
non-injection case
At no point has the axial-length of the combustor section been stated. One of the
reasons for this was that this value needed to be determined from external data and is
very much a part of the optimization process and thus not arbitrary. However, not having
a fixed combustor axial length presented a unique problem during the construction of the
mesh. This problem was a result of the fact that an output value of the flow parameters
was required in terms of pressure and mass flow rate. However, this aspect is more
clearly discussed in later sections of this thesis.
12
Table 2.2.2 Nodes per grid volume
Volume
Number of Nodes
No. (X*Y*Z)
1
5500 (25x11x20)
2
50000 (25x100x20)
3
60000 (30x100x20)
4
55000 (55x50x20)
The second step in the grid development involved refinement around the injection
locations on the floor of the combustor in the X-Z plane and the results were again
compared with the experimental data. For this flow case, the simulations were done 3-
dimensions with the X-Y grid being maintained the same as the one achieved during the
first step of refinement discussed above.
The X-Z plane mesh was extensively altered once the treatment of the injection
surfaces was considered. The circular nature of these surfaces complicates the mesh and
is shown in Figure 2.2.5. The grid zone structures remained roughly the same, though the
minor differences are highlighted in Figure 2.2.6.
13
Figure 2.2.5. Grid structure around the Injection surfaces on the floor of the
Combustor (X-Z plane)
Figure 2.2.6. 3-Dimensional visualization of combustor section geometry for the
injection case with mesh distortions
14
2.3 COMPUTATIONAL METHODS AND SOLVERS
When considering the combined effects of supersonic flow over a rearward facing
step coupled with the shock interactions of a staged normal injection process, it quickly
became obvious from experimental data that the required flow-field was highly complex.
It was also clear that since the scope of this analysis moves far beyond the single case
runs with no geometrical variation, the flow field could not be predicted in great detail in
advance. This necessitated the need for a robust grid structure and a similarly robust CFD
solver model. High speed was also necessary for the large number of runs required by the
Genetic Algorithm.
Consideration of the available solvers showed that the two available choices with
the FLUENT software were as follows:
a) Segregated Solver Model
b) Coupled Solver Model
The segregated solver represented a fast solution model that fit well with regard to
the time constraint issues as discussed above, giving significantly faster solutions to a
given flow field as opposed to the coupled model (both explicit and implicit versions).
15
However, a significant disadvantage was the segregated solver?s inability to handle
highly complex flow fields as needed for the scramjet combustor analysis.
Indeed, initial trials even with the non-injection cases using this model resulted in
diverging solutions within a handful of initial calculations, necessitating a shutdown of
the analysis. The problem only got worse when injection cases were attempted.
On the other hand, the coupled solver proved to be a far more robust model at the
fundamental level as opposed to the segregated model. This model was better suited for
difficult and complex flow situations. The difference between segregated solvers and
coupled solvers is in their method of solving the continuity, momentum and energy
equations. While the segregated model solves these equations sequentially, the coupled
solver solves them simultaneously, thus maintaining superior control over solution
divergence. For these initial validation studies, the use of the implicit coupled model
(which solves for all variables in all cells at the same time) was employed.
The disadvantage of this model was efficiency due to a mathematical choke point
built into the model that provides both the robustness of the model and its slow run times.
To understand this more clearly, an understanding of the fundamental differences
between the two solution models is in order. Essentially, the iteration scheme consists of
the following steps:
a) Fluid properties are initialized for each calculation based on the previous solution
or the initial condition, whichever is the case.
16
b) The continuity, momentum and energy equations are solved simultaneously. Here,
the system of governing equations is cast in an integral, Cartesian form for an
arbitrary control volume ?V? with differential surface area ?dA? as follows:
(2.3.1)
Where the vectors ?W?, ?F? and ?G? are defined as:
(2.3.2)
and the vector ?H? contains source terms such as energy sources. All variables in the
above matrices are defined on a ?per unit mass? basis.
c) Equations for turbulence are solved based on the selected model.
d) A convergence check is conducted and the cycle is repeated.
The coupled set of governing equations as given above is discretized in time for
steady and unsteady calculations. In this case, with the flow situations being steady, the
time marching scheme proceeds until a steady state solution is reached. Temporal
discretization of the coupled equations is accomplished by an implicit time-marching
scheme.
17
The time-step is calculated from the CFL (Courant-Friedrichs-Lewy) condition (a
condition for algorithms solving partial differential equations to be convergent; named
after Richard Courant, Kurt Friedrichs and Hans Lewy who described it in their 1928
paper) 15, defined by the parameter:
(2.3.3)
This parameter is known as the Courant Number and is the critical parameter to
be determined when using the coupled solver. Since the coupled solver is a time
marching iteration technique, this number connects the refinement of the mesh with the
time iteration values (and hence solution divergence issues). Essentially this works
around the idea that for a time marching iteration technique, the time step must be less
than the time taken by an event to cross between two successive grid points. This
discussion involves the flow of fluids from one grid point to another grid point. This is an
extremely important issue for achieving convergence during the iterations since the value
of the Courant number varies for each grid.
For all grids the Courant number was determined after an incremental
advancement scheme to be 0.1, the maximum value to avoid solution divergence 12. One
note to be made here is that the grid developed previously did not have a uniform grid
structure. Indeed, there were several grid zones with each zone having its own nodal
densities. Some zones, such as the one handling the shear layer, possessed varying nodal
densities even within the zone thanks to the use of successive ratio schemes for
deliberately increasing nodal densities near the walls and reducing density further
18
downstream to allow the grid to interact with the other zones. With such a structure, the
Courant number must be set for the smallest grid distance for the entire grid.
To improve computational efficiency, a unique method was developed to combine
the segregated and coupled solvers into a single system designed to take advantage of the
robustness of the coupled solver with the speed of the segregated solver during various
stages of the calculation process. In essence, the idea was to use the coupled solver model
initially, and have it run through a number of time steps to allow the flow field to develop
around the rearward facing step; however, the coupled solver was not allowed to run
through the length of the combustor section. The flow section covered by the coupled
solver was the section of the combustor that the segregated solver had proven to be
incapable of solving.
Outside of this region, the flow field is relatively simple, and thus within the
capabilities of the segregated solver (at much higher speeds). It is here that the model is
shifted from the coupled solver to a segregated solver. In other words, the coupled solver
sets up the flow field to a point where the segregated solver can pick up and converge at
much faster speeds. This hybrid CFD solver model was found to effectively combine the
unique advantages of both solvers into a single system to allow converged solutions for
all geometries and boundary conditions in small time periods.
The above process is effectively visualized when reviewing the convergence data
plots as shown in Figure 2.3.1. It can be seen that the coupled solver initially engaged to
run through a set number of iterations (each of which represents a time step for the flow
19
field) provides very slow solution convergence owing to the manner in which the
building blocks have been set up. Once the model is shifted over to the segregated solver,
the combination of its building blocks and the ?initial guess? given to it by the coupled
solver allows it to converge almost instantly to very low values, and allows the
completion of the case run for post processing in a much smaller number of iterations
than would have been required had the coupled solver been used exclusively. Typical
numbers for single case runs for the scramjet combustor geometry are as included in
Table-2.3.1.
Figure 2.3.1. Residual plots for a single case run using the Hybrid solver model
20
Table 2.3.1 CFD Solver computational times
Solver
Time to solution convergence (<10-6)
for all flow parameters
(single case run, single processor)
Coupled solver only
Convergence achieved in 20+ Hours
Segregated solver only
Solution Divergence
Combined Coupled (Implicit)
and Segregated Solvers
Convergence achieved in 6+ Hours
21
2.4 BOUNDARY CONDITIONS
All fluid entry surfaces were designated as pressure-far-fields with all in-flow
boundary values fixed. All fluid exit planes were designated as pressure-outlets while all
remaining faces were designated as wall boundaries. The no-slip condition was applied
for all walls along with zero normal pressure and temperature gradients. The K-epsilon
Turbulence model was utilized with enhanced wall treatment and viscous heating effects
enabled.
The density model for the air was defined as the ideal-gas model. All operating
pressures were referenced against the standard sea-level atmospheric values for the center
point in the entry far-field plane for the air. Finally, at the injector, all dependant
variables are specified to be physical values of the temperature, pressure and velocity and
the injector surface itself was treated as planar. No wall radiation effects were simulated.
The mass flow rate of air was 0.20 kg/sec at nominal stagnation conditions. The
nominal Mach-2 cross-flow entry stagnation conditions were 300K and 274kPa, and
maintained at the entry plane (Y-Z). The injector stagnation conditions were 263kPa and
300K at Mach 1 with a uniform entry.
22
The injectant mass flow rate was maintained at 1.64 g/sec. These values conform
to those used for the experiments against which the initial simulations were validated 7, 8.
Mass flow rate tracking was enabled.
FLUENT uses the wall functions as defined by Launder and Spalding which are
quite commonly used throughout the industry. These are the default options with this
software with the additional option of Enhanced Wall Treatment options. This latter
option uses what are known as Enhanced Wall Functions. These functions extend the
applicability of throughout the near wall region including the laminar sublayer, buffer
region and fully turbulent outer region by formulating the law-of-the-wall as a single wall
law for the entire wall region.
As such it is possible to monitor the accuracy of the grid refinement process in
FLUENT using one of two ways: manual or automatic adaptations. In both cases fixing a
limit on the y+ values is extremely important. The Log-Law model is found to be
effective beyond values of y+ = 30 and less than y+ = 60. Below this range of values the
Enhanced Wall Treatment of the concerned region is necessary to accurately model the
flow.
The manual adaptation process involves plotting out the y+ values for a given grid
during a grid refinement process and adjusting the refinement process until the values
falls within the right range. This process is advantageous to use when single case runs are
being conducted. However, with the proposed GA driven CFD case runs with changing
geometric features, this process runs into limitations in that over-refining the grid might
23
be necessary to ensure that the wall region is being adequately modeled despite the
changing geometries.
The automatic grid adaptation process bypasses this problem by conducting
automatic, localized grid adaptations in the wall regions using the user-defined input
values for y+. This may involve coarsening the grid in some regions and refining the grid
in others. Overall, the process ensures good wall accuracy for a given geometry at the
cost of reduced grid uniformity.
Given the networked structure of this optimization effort, it was clearly
advantageous to use the self adaptation features of the CFD solver. However, it was
noticed that the reduced grid uniformity could lead to unexpected solution divergence
issues. This is a consequence of the hybrid solver setup as defined in the previous topic.
Since the coupled solver was only defined to resolve a certain region of the flow with the
given value of the courant number, which in turn is based on the grid refinement by its
definition, any changes initiated by the self adaptation scheme led to changes in the
requirement of the value of the courant number. Although the changes were usually
small, it often led to a situation where the solution diverged for a given case rather
unexpectedly because of the changes in the grid adaptation scheme.
This posed a problem that was overcome by using the grid zones structure defined
in Section 2.1. The zones were created with the issue of y+ values in mind so that only
the zones affected by the movement of the injector locations were tagged for the self
adaptation process and the value of the courant number was set to compensate for the
24
uneven changes in the grid. The Enhanced Wall Treatment option was not used. Given
the nature of the grid with different zones of varying nodal densities, the y+ values also
vary, but have been maintained to values of ~30.
25
2.5 EXPERIMENTAL VALIDATION
Hartfield, Hollo, Fletcher and McDaniel 10 extensively investigated non-reacting
staged transverse injection behind a rearward facing step. The simulation data was
compared against a spatially complete data set of the various flow parameters obtained
experimentally using optical techniques based on Laser-Induced-Iodine fluorescence and
supersonic wind tunnel facilities 8.
The validation efforts were divided into two parts:
a) Flow without injection
b) Flow with injection
The above was necessary since the small diameter injection holes on the floor of
the combustor meant that the shock interactions and flow perturbations as a result of the
injection process were unlikely to affect the nature of the flow along the entire width of
the combustor except in the immediate region near the injection surfaces and downstream
of the rearward facing step. As a result, regions remaining unaffected by the injection
process would continue to exhibit the nature of flow as if there were no injection
whereas, areas near the injection would exhibit complex and highly variable three
dimensional flow features.
26
2.6 FLOW WITHOUT INJECTION
The supersonic cross-flow over a rearward facing step meant that the focus in this
section was on developing the accuracy of the expansion fan, the shear layer and the
oblique shock downstream of the step. The experimental flow field can be characterized
by using the flow diagram shown in Fig.2.2.1.
The typical flow from upstream of the rearward step is roughly maintained except
for the boundary conditions and the flow expands over the step leading to the formation
of the classic expansion waves. The presence of the step causes the flow to behave as if it
is a mixing case of two parallel flow fields having different velocities. As a result, with
the flow above the step moving at Mach 2, the flow just behind the step is characterized
as a low speed recirculation. The formation of a shear layer having an axial length of
multiple step heights takes place. Figure 2.6.1 compares the experimental results for the
static pressure with those from the simulations. As can be seen, a close compatibility is
maintained between the theoretical calculations, the experimental results and the
simulation results for all flow features.
27
Figure 2.6.1. Experimental 7 and simulation comparisons for the pressure field for
the non-injection case
The above plots are for the X-Y plane at the centerline axis of the combustor.
Nevertheless, for the non-injection cases, the symmetry of the flow is maintained in all
subsequent X-Y planes except for those near the wall edges where the boundary wall
effects cause some perturbations and hence an a slight asymmetry in the X-Y flow fields.
In order to be able to compare the results more clearly, a comparison of the
experimental and simulation results for the pressure profile has been shown in Figure
2.6.2. A close compatibility of the simulation and experimental results is clearly seen.
28
Figure 2.6.2. Comparison of the experimental 7 and simulated static pressure profile
along the line from the combustor floor to the top of the combustor at the center of
the primary injector
One prominent effect noted in the comparison of the experimental and simulated
results was the inability of the CFD solvers to accurately depict the expansion fan region
in the vicinity of the upper edge of the rearward step. Experimentally, the very low
pressure region just downstream of the rearward step is seen to provide a suction effect
that causes the expansion effects to occur upstream of the physical step location 8.
However, repeated CFD efforts failed to reproduce this effect to the extent as that seen in
the experimental data even after the use of localized ultra-high resolution grid domains
near the step location. This is found to concur with other independent efforts using the
SPARK three dimensional Navier-Stokes codes wherein the same problem was
encountered and remained unresolved 13.
0
0.2
0.4
0.6
0.8
1
1.2
Sta
tic
Pr
es
su
re
(P/
Pin
f)
Experimental Simulation
29
Figure 2.6.3. Simulation results for the pressure contours in a 3-Dimensional view in
the FLUENT display screen
Similarly, the static temperature results are shown in Figure 2.6.4. Here the
results compare equally well as far as the flow field was concerned. However it was
noticed that the temperatures obtained experimentally were somewhat higher than those
obtained via calculations in the recirculation region behind the shear layer. Figure 2.6.5
displays the wall temperature distribution for the entire combustor section as a result of
the flow and effectively shows the high temperature region on the top of the combustor
section where the stagnation conditions are nearly reached.
30
Figure 2.6.4 Experimental 7 and simulation comparisons for the temperature field
for the non-injection case
Figure 2.6.5. Three dimensional visualization of the temperature field in the
FLUENT display mode for the test section
31
Figure 2.6.6. Comparison of the experimental 7 and simulated static
temperature profile along the line from the combustor floor to the top of the
combustor at the center of the primary injector
One of the main points of focus for the validation was the accuracy with which
the shear layer was modeled immediately downstream of the rearward facing step. This
was the driving factor behind the grid development. The details of this region are shown
in Figure 2.6.7. The close compatibility between the results obtained through the
simulation and the experimental results is clearly visible in these two images.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 5 10 15 20
Sta
tic
Te
mp
era
tu
re
(T/
Tin
f)
Experimental Simulation
32
Figure 2.6.7. Experimental 7 and simulation comparisons for the velocity field for
the non-injection case
The recirculation region is reproduced accurately from a geometrical standpoint.
The length of the shear layer along the axial (X) direction is seen to conform to
experimental data as does the near planar nature of the shear plane itself.
33
2.7 FLOW WITH INJECTION
The flow with injection must be evaluated at several sections for different
theoretical explanations for the flow features. Nevertheless, some distinct features remain
constant throughout. The main features of this flow are the formation of detached bow
shocks upstream of the injection points. These are a result of the injection stream acting
as a blunt body being moved through a supersonic cross-flow as shown in Figure 2.7.1
and the under expanded jet cores of the two injectors.
The shock is detached in all three dimensions and this can be visualized in Figure
2.7.2. The shock essentially moves around the disturbance created by the injection. The
flow upstream of the first injection point is uniform causing the detached shock to be
very clearly defined. The same is not true of the second injection point, which faces
severe disturbances in the upstream flow as a result of the first injection point. This can
be seen in Figure 2.7.3.
34
Figure 2.7.1 Flow features at the centerline section for the injection case 7
Figure 2.7.2. Visualization for the pressure field in the injection case to highlight the
3-Dimensional nature of the shock patterns
35
The second injector is seen to penetrate further into the free-stream than the first
due to the disturbed nature of the free-stream downstream of the first injector. This
entails lower Mach numbers and higher pressure fluctuations. More shear layers are seen
to develop in the region downstream of the second injector. Streamwise vortex
formations are seen in the Y-Z planes.
The nature of the flow of the injectant along the length of the combustor
downstream of the injection locations is shown in Figure 2.7.3. The injectant plane is
seen to be in the shape of long cylindrical sections with low dispersion into the cross flow
even at substantial distances downstream of the injection locations. Notice the long,
narrow, cylindrical shape of the mass fraction distribution of the fuel. The divergence is
achieved mainly in the immediate downstream region of the injection locations.
Figure 2.7.3 Nature of the injectant flow through the combustor.
36
Further, the expansion fan faces severe disturbances as a result of this injection as
well. The detached shock is seen to effectively destroy the lower expansion shocks (near
the bottom of the fan) upstream of the injection location while the overall angle of the
expansion fan is reduced as a result of reduced pressure gradients upstream and
downstream of it. The experimental and simulated results compare favorably in all these
aspects, as shown in Figure 2.7.4 for the pressure contours.
Figure 2.7.4 Experimental 7 and simulation comparisons for the pressure field for
the injection case
37
Figure 2.7.5 Comparison of the experimental 7 and simulated static pressure
profile along the line from the combustor floor to the top of the combustor at the
center of the primary injector
Figure 2.7.6 Comparison of the experimental 7 and simulated static pressure
profile along the line from the combustor floor to the top of the combustor at the
center of the secondary injector
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20
Sta
tic
Pr
es
su
re
(P/
Pin
f)
Experimental Simulation
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
Sta
tic
Pr
es
su
re
(P/
Pin
f)
Experimental Simulation
38
Figure 2.7.4 also illustrates the formation of the barrel shocks just above the
injection surfaces. This shock pattern is seen to be deflecting in the downstream
direction. A better realization of this is achieved by considering the temperature plots
shown in Figure 2.7.7.
Figure 2.7.7 Experimental 7 and simulation comparisons for the temperature field
for the injection case
39
Figure 2.7.8 and Figure 2.7.9 shows the comparison between the experimental
and simulated results in the form of line profiles for the primary and secondary injector
centers. Note that these lines are perpendicular to the floor of the combustor and move
vertically upwards towards the roof of the combustor. Along this line various data points
have been extracted for comparison.
Figure 2.7.10 shows the complex nature of the three dimensional temperature
distributions in the combustor cross section and the much higher temperature gradients
obtained in the injection case as compared to the non-injection case.
Figure 2.7.8 Comparison of the experimental 7 and simulated static
temperature profile along the line from the combustor floor to the top of the
combustor at the center of the primary injector
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20
Sta
tic
Te
mp
era
tu
re
(T/
inf
)
Experimental Simulation
40
Figure 2.7.9 Comparison of the experimental 7 and simulated static
temperature profile along the line from the combustor floor to the top of the
combustor at the center of the secondary injector
Figure 2.7.10 Three dimensional visualization of the temperature field in the
FLUENT display screen
The velocity vector plots shown in Figure 2.7.11 also present a favorable
comparison between the experimental and simulation results. Some unique flow features
are revealed in the velocity vector plots. As the streamlines show, the flow immediately
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 5 10 15 20
Sta
tic
Te
mp
era
tu
re
(T/
Tin
f)
Experimental Simulation
41
above the injectors is seen to be compressing in the area upstream of the injection point
while in the downstream area, the flow is seen to be expanding.
This is also consistent with the deflection of the barrel shock structures discussed
previously. Note that the streamlines shown in the figures are shown in a two-
dimensional plane, and does not fully represent the three dimensional velocity fields.
Figure 2.7.11 Experimental 7 and simulation comparisons for the velocity field for
the injection case
42
2.8 MASS FRACTION ANALYSIS
Figure 2.8.1 compares the experimental and simulation results for the mole
fraction distribution of the injectant in the X-Y plane (single diameter transverse offset).
It should be noted that the simulation studies were compared with experimental data that
dealt with injecting air into air. In the case of the CFD simulations, the injectant air was
?tagged? and tracked in the flow-field as if it were a different fluid altogether. As will be
described in later sections of this thesis, the fuel-air mixing phenomenon was suitably
modified as well. Note also that since air is being injected into air, the mole and mass
fraction distributions are the same.
Good accuracy is seen to be maintained for the mole fraction distributions. Figure
2.8.2 presents a three dimensional plot of the mass fraction distribution for the injectant
along the length of the combustor and the evolution of the distribution field at the exit of
the combustor can be clearly visualized.
43
Figure 2.8.1 Comparison of the experimental 7 and the simulation results for the
mole fraction distribution of the injectant along the centerline axis
Figure 2.8.2. Simulation results for the Mass Fraction distribution for various Y-Z
planes along the combustor length.
44
2.9 THE GENETIC ALGORITHM AND CFD INTEGRATION
A Genetic Algorithm is an evolutionary biologically inspired optimization
technique which uses various types of evolutionary mechanisms such as mutation,
selection, inheritance etc, to move toward an optimized solution for a given function or
mathematical process 1-6. For this effort, the IMPROVE V2.8 binary GA based on the
Fortran 77 software base was used. A more detailed description of the operational
procedure is discussed in the results section given the direct connection between the
results obtained and the GA setup. In this section a brief overview of a typical run is
presented for creating a fundamental base for the more detailed discussions that follow.
IMPROVE is the acronym for Implicit Multi-Objective PaRameter Optimization
Via Evolution 16. Here the term ?evolution? is a misnomer given that the GA is more
adaptive than evolutionary in the absolute biological sense. The GA adapts existing genes
to give better performance but does not create new genes and is thereby restricted to a
given gene pool as defined by the design space. For complex problems with non-
definable objective functions, GA?s have been found to be much better at estimating the
global maximum or minimum than gradient based methods that are prone to finding local
maxima or minima.
45
A binary GA operates on the binary encoded input parameters. In this sense the input
must include a range of values for each parameter that defines the design space as well as
a field resolution value for the said parameter. Since the length of a chromosome in the
binary GA is defined by the number of bits for the given design space (sum of the
number of bits for each design parameter), the resolution value, the parameter design
space and the number of parameters define the size of the optimization problem as shown
in equation 2.9.1.
g1840g1873g1865g1854g1857g1870 g1867g1858 g1854g1861g1872g1871 = uni2211 [g2922g2924g4672
g3171g3159g3182g4666g3284g4667g3127 g3171g3167g3172g4666g3284g4667
g3293g3280g3294g3290g3287g3296g3295g3284g3290g3289 g4673
g2922g2924g4666g2870g4667 ]
g3015g3048g3040g3029g3032g3045 g3042g3033 g3017g3028g3045g3028g3040g3032g3047g3032g3045g3046
g3036g2880g2869 (2.9.1)
A typical run with this version of the GA therefore involves conversion of the
input parameters into binary forms and the generation of an initial population of members
whose size is fixed by the user before the activation of the run. This initial ?generation? is
composed of a random selection of parameters from the design space. The design values
are then passed to a code to evaluate the performance of a given engineering system (in
our case the initiation and evaluation of CFD data) and allows the GA to stack the
members of the population in order of their individual performances.
After the performance evaluation, the adaptation process begins with a
tournament selection process in which the members of the population are chosen for
reproduction. This is followed by processes such as probability-based crossover and
mutation to develop the next generation of members for evaluation. Since these members
have been constructed based on the results of the previous generation best performers, the
46
best performers of the new generation are statistically found to be an improvement over
those of the previous generation.
This effort was a multi-goal analysis which required a critical review of the
various GA fitness stacking selection schemes such as strict, relaxed and apportioned
Pareto methods. The use of these schemes evolved with the analysis of each run and their
description fits better within the results section. However, there are other crucial GA
schemes like micro GA and steady state GA versus conventional GA selection, creep
selection, uniform versus non-uniform crossover etc. that defined the final results but
unlike the goal evaluation schemes, were common throughout the various runs and can be
described here in detail.
The steady-state GA selection is referred to as a population elitist strategy in that
a steady state GA keeps only the best performers from the current generation and the
previous generation. As such, it can lead to a loss of genetic diversity. It also means that
this process can be used with a higher mutation rate. The mutation rate defines the rate at
which mutation occurs within the chromosome while crossover probability is the
probability of crossover when two parents come together after the end of the tournament
selection process. Higher values of these schemes along with the steady-state GA can
lead to quicker solution convergence.
Elitism was another scheme that required usage for cases when the total pressure
ratio was being optimized (discussed later). It is used to preserve the best performer for
each goal. Creep mutation is used to ?fine tune? the GA once significant hyperplanes have
47
been found by randomly varying the least significant bit in a random parameter. Uniform
Crossover was kept inactive to allow for random matchup of parent genes within the
children members of the next population.
As a result of the above procedure, the following requirements must be met when
attempting to use the GA in coordination with CFD solvers for the optimization of the
Scramjet Combustor Geometry:
a) Selecting the variable design parameters
b) Defining the design space
c) Selecting the optimization goals
d) Automating the grid generation process
e) Automating the setup of the grid and the case parameters within the CFD software
f) Integrating the CFD software with the GA code
g) Modification of the GA to allow parallel processing of numerous members of a
generation over multiple computer processors simultaneously to reduce
calculation time
h) Modification of the CFD solver setup to allow parallel processing of each given
case over multiple computer processors for each member of the generation.
i) Transfer of the above structure to a networked system of computers
48
2.10 THE DESIGN PARAMETERS AND DESIGN SPACE
The overall geometry and boundary conditions of the scramjet combustor for this
analysis remains fixed throughout the course of the optimization study. The physical
aspect of the combustor varied in the optimization is the location of the two injectors on
the combustor floor (X-Z plane). This can then be subdivided into four variables as
discussed below:
a) The axial (X) location of the first injector
b) The transverse (Z) location of the first injector
c) The axial (X) location of the second injector
d) The transverse (Z) location of the second injector
Having fixed the parameters which are varied during the course of the optimization, the
range and resolution within which the GA can select the values must be specified based
on the concept mentioned in the previous topic. It should be noted here that this design
space is not fixed as further runs were conducted based on the evaluation of trends and
analysis of results from the initial results.
49
As such, the design space evolved or was restricted depending on the requirement of
further runs as will be discussed in the results section. Therefore, the values mentioned in
this section correspond only to the initial runs and are designed to give an idea of the
typical design space and values that will be used throughout the rest of this analysis.
The range of values for each of the above parameters was fixed to be within one
step height of its original experiment values and to have a resolution of selection of 0.1.
The step height is 3.18 mm, and therefore the value for each of the parameters can change
from +3.18 to -3.18 mm beyond the original position.
The overall design space then is the area swept by these limiting values and in this
case takes the form of a square for each of the two injector locations as shown in Figure
2.10.1
50
Figure 2.10.1. The Design Space for the optimization of the Scramjet Combustor
51
2.11 DESIGN GOALS
The primary performance parameter of interest in this effort are the total pressure
loss and the mass fraction of the injectant, with both being defined for a cross sectional
(Y-Z) plane of the combustor. The total pressure loss represents overall propulsion
performance degradation while the mass fraction plots are considered as a means of
quantifying the mixing phenomena 17. Generally a tradeoff exists between the total
pressure decrease and the degree to which injectant is mixed into the free stream.
The optimization includes the total pressure loss versus what is referred to as the
mixing efficiency 18. The term ?mixing efficiency?, as defined by Anderson 18, is an
empirical, one dimensional measure of the degree of mixing completeness which takes
into account both the near-field mixing (initial macro-mixing phase) and far-field mixing
(molecular diffusion/micro-mixing). For an overall fuel-lean mixing location, it is ?the
amount of fuel that would react if complete reaction occurred without further mixing
divided by the amount of fuel that would react if the mixing were uniform? 18.
52
In order to be able to define the mixing efficiency at any given Y-Z plane
downstream of the second injector location, the mixing phenomena must be quantified
against a reference standard. In this context, the mixing efficiency is defined as the ratio
of the number of cells on the required Y-Z grid surface which possess a fuel-air ratio
within a given range of values where the combustion process can occur to the total
number of cells on the entire surface.
In other words, beyond the standard stoichiometric fuel-air ratio, there exists a
fuel-rich value and a fuel-lean value which represent the practical range of values for
proper combustion. Any value of the fuel-air ratio above or below this range will have
insufficient reactants to combust properly and thus is disqualified from the selection
procedure. Any value in between these two is selected as having desirable combustion
conditions as far as mixing of the reactants is concerned.
The total grid nodes are fixed to be the same for a given Y-Z plane at the end of
the combustor where all calculations for the exit conditions are conducted. This Y-Z
plane exists at an axial distance of ten step heights beyond the second injector location in
the original setup and thereafter remains fixed at that distance regardless of the change in
the second injector location during the optimization process. This plane is therefore
located at a distance of 65.08 mm downstream of the inlet of the combustor for all
calculations. The number of cells for this exit plane of the combustor is also fixed at
1890.
53
The flammability limits for the fuel-air ratio above and below the stoichiometric
ratio must be defined in order to define the selection criteria for any given cell in the exit
plane. However, so far all validation studies discussed here have involved injection of air
into air. Therefore there is no fuel involved. To move beyond this condition would
involve changing the injectant from air to some fuel. In order to provide some use for the
validation studies involving gaseous injectant discussed in previous sections, hydrogen
was the obvious choice as the fuel. It was further determined that the flammability limits
in terms of the fuel-air ratio of hydrogen to air per mass basis varies from 0.0520 (fuel
rich) to 0.002774 (fuel lean). The stoichiometric value is 0.0294.
A validation study of hydrogen injection into air was initially bypassed and the
simulations were conducted using the previous limits for hydrogen while keeping the
injectant as air. In this way the flow field retains the nature discussed in the validation
study in previous sections and yet provides some realistic values for the calculation of the
mixing efficiency. The mixing efficiency at this point is therefore defined as the ratio of
the number of cells in the exit plane having a fuel-air ratio between the flammability
limits of hydrogen mentioned above to the total number of cells for the entire exit plane.
The total pressure ratio in three dimensional flow fields must also be quantified.
The concept of single dimension analysis employed here treats the total pressure ratio as
the ratio of the total pressure at the exit of a component to that at the entry of the
component. It logically follows that in order to do the same with regard to entry and exit
planes in a three dimensional flow case, one must determine the averaged total pressure
for both the entry and the exit planes of the given scramjet combustor. At the same time,
54
the validity of the governing equations of the flow must be maintained. These
requirements are met with the development of the averaging equations using the mass-
momentum averaging technique which is a modification of the Stewart mixing analysis23.
The equations used in this analysis have been customized to handle rectangular
geometries and meshed surfaces.
This derivation for the mass-momentum averaging technique for CFD grids is
explained in detail in Appendix A. The resulting equations needed for quantifying the
total pressure ratio for the scramjet combustor are given in equations 2.11.1 to 2.11.4.
The average total pressure across a grid plane in a flow field is found to be:
g1842? =
g4678g3330g3267g3269?g3276
g3267g3254g3255g4594
g4679uni2211 g3017g3294g4666g3036g4667g3436g3262g3299g4666g3284g4667g3118 g3297g3299g4666g3284g4667 g3440g3002g4666g3036g4667g3263g3284g3128g3117
g3344g3294
g3344?
g3297g3299
g3276g3267g3254g3255g4594 g3002
(2.11.1)
Where all parameters are defined in the nomenclature.
As can be seen, except for the denominator terms, the entire equation is composed
of terms that can be determined from the CFD grid data. As for the denominator terms,
three additional equations are required to quantify them in terms of the grid data as well.
These equations are:
g1855 =
g3437 g3211RTg4594
g3276g3267g3254g3255? g3118
g3441uni2211 g3427g3017g3294g4666g3036g4667g3014g3299g3118g4666g3036g4667g3002g4666g3036g4667g3431g3263g3284g3128g3117 g2878 g4672g3211g3126g3117g3118g3211 g4673uni2211 [g3017g3294g4666g3036g4667g3002g4666g3036g4667]g3263g3284g3128g3117
g4678g3330g3267g3269?g3276
g3267g3254g3255g4594
g4679uni2211 g3017g3294g4666g3036g4667g3436g3262g3299g4666g3284g4667g3118 g3297g3299g4666g3284g4667 g3440g3002g4666g3036g4667g3263g3284g3128g3117
(2.11.2)
55
g3096g3294
g3096? = g3428g883 g3398 g4672
g3082g2879g2869
g3082g2878g2869g4673g484
g3049g3299g3118
g3028g3267g3254g3255? g3118
g3432
g4672 g3117g3330g3127g3117g4673
(2.11.3)
g3049g3299
g3028g3267g3254g3255g4594 = g4672
g3082g3030
g3082g2878g2869g4673g3398 g3495g4666
g3082g3030
g3082g2878g2869g4667
g2870 g3398 g883 (2.11.4)
These equations then allow for the accurate calculation of the total pressure ratio
over the inlet and outlet planes (Y-Z) of the scramjet combustor and then take the ratio of
the total pressure at the outlet to that at the inlet as a goal for the optimization. With the
mixing efficiency and the total pressure ratio now defined based on the flow parameters
at the inlet and the outlet, the optimization goals for the GA can be evaluated.
56
2.12 THE NETWORKED ENVIRONMENT
Automation is an important requirement for optimization studies. Another
requirement was that the entire program code be initiated on as many processors as
possible to allow for rapid case runs. This meant that the CFD solver and the GA must
both run in parallel. The computer network available at Auburn University consisted of a
total of sixty available processors for CFD runs on thirty computer nodes. All processes
are initiated on a central node referred to as the head node. In addition, a total of four
CFD solvers can be run simultaneously on the above cluster. This means that there were a
total of ten processors available for each CFD run for use as parallel processors. All CFD
simulations are run through the FLUENT software, whereas all grid generation
requirements are met by the use of GAMBIT software with either software being
remotely initiated and controlled in a non GUI environment by the GA.
Figure 2.12.1 shows the overview of how GAMBIT as a grid generator is
integrated into the code to allow for automated grid generations and also its interaction
with FLUENT. GAMBIT can be initiated in a batch mode with the creation of a journal
file from a FORTRAN subroutine. This is done only on the head-node of the cluster to
prevent increase in computational time on the nodes for the CFD runs. Once initiated in a
non-GUI mode on the head-node, GAMBIT creates a mesh file using the set of
57
commands written in the journal file and this is combined with a similar data file for
FLUENT. This allows a mesh file and a command data file to be passed to the nodes
where the CFD cases are being run.
Figure 2.12.1. An overview of the network structure involving GAMBIT and
FLUENT on the Computer cluster at Auburn University
Once the CFD run for a given case is completed, case and data files are
automatically written based on the commands in the data file (.dat) and passed back to the
head-node. Here they are reopened in a non-parallel mode by the Fortran program and an
ASCII file is written containing the required flow field parameters. This is necessarily
done on the head-node since FLUENT cannot execute the formation of ASCII files
(which are exported) while in parallel mode on the cluster nodes.
58
Once the formation of the ASCII file is complete, it is opened by the FORTRAN
subroutine designed to process the flow field parameters for the quantification of the
required goal parameters and used by the Genetic Algorithm to proceed forward.
Figure 2.12.2 shows the nature of the interaction of the head-node and the cluster
nodes. In effect, the mesh files generated for each member of a generation are passed on
from the head node to each node on the cluster as per the requirement. Since the amount
of time required for the case runs is usually large, the time required for the generation and
transfer of the mesh files sequentially and the transfer of the calculated data back to the
head-node does not significantly affect the overall computational time adversely.
Figure 2.12.2. A generalized concept of operations view for a given generation in a
GA run
59
An important issue is the computational time versus number of processors being
used. The maximum number of processors available for each CFD run on the network is
ten. With a single processor, the time required for each case run is several hours, but as
the number of processors is increased, the time for overall calculation decreases
dramatically from around six hours for a single processor to around two hours for four
processors or about one and a half hours for six processors. Figure 2.12.3 shows the
nature of the curve for calculation time versus the number of processors.
It was noted that after a maximum of around eight processors, the curve levels off,
and further decrease in calculation time by adding even more processors is offset by the
increased I/O time between the large numbers of processors.
Figure 2.12.3. Computational time for a single FLUENT case run versus number of
parallel computer processors used
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Based on the above results, the structure for this optimization analysis conducted four
simultaneous CFD case runs (i.e. four members of a generation in parallel mode) with a
parallel processor structure involving a total of eight processors leading to the overall use
of 32 processors at any given time. The calculation time for each case run for such a
structure was found to be around one hour on average. This meant that an entire
generation of 20 members of a GA run with full CFD results could be completed in as
little as five to six hours allowing for the completion of around four generations per day
and the completion of a single optimization run to be completed in less than a week.
61
3 THE RESULTS
3.1 CASE RUN - 1
The first optimization run was conducted using the following parameters:
a) Optimization goals: maximize total pressure as primary goal and maximize
mixing efficiency as secondary goal using the strict pareto methodology. Both
parameters are calculated from the CFD data as discussed above.
b) Design space: movement of the two injection surfaces in the X-Z direction on the
floor of the combustor within one step height of their original experimental
location with a resolution of 0.1mm.
c) Genetic Algorithm parameters:
a) A 20 generation run
b) Each generation has 20 members
c) Two goal optimization run
d) Total pressure ratio has priority over mixing efficiency as optimization goals
62
A brief discussion of the pareto methodology is presented here to explain the use
of the strict pareto scheme used for this run. The Pareto scheme, in general, differs from
the conventional GA setup when multiple goals (>1) are to be simultaneously optimized.
The goals are optimized individually based on the input parameters common to all goals
but the fitness selection differs in the way goals are classed above or below one another.
This is known as ?goal domination? and defines the selection process in that one goal set,
which is a collection of individual goal performances for the said input, must dominate
the others if the said parameters are to survive.
In this way all goal sets are compared to determine which parameter set is chosen
over the other sets. A ?strict pareto? is one that conducts this domination statistics
analysis but with a relative measure of significance between the goals themselves in that
the total pressure ratio was stacked as higher priority than the mixing efficiency.
By default, the GA stacks the results for all members of a given generation in the
order of increasing performance (looking top to bottom), with the performance of the
priority goal taking precedence over the secondary goal. As a result, for each generation
there exists a best performer and a worst performer in its list of members. All results
presented in this paper include plots of the best and worst performers of a given
generation versus the optimization goals or design parameters over the entire number of
generations available. This allows one to follow the trend that the optimization efforts
tend to take and provide valuable insights into the inner workings of the optimization
process.
63
Figure 3.1.1 shows the results obtained for the total pressure ratio over the course
of the GA run. It is seen that the best performer of each generation continually improves
over that of the previous generation until a period of stabilization is achieved after which
the best performer does not practically improve further.
Another feature visible from Figure 3.1.1 is that the worst performer undergoes a
lot of fluctuations in its values over the course of the optimization process as each worst
performer is replaced with a new result based on the previous generation as the GA tries
to improve performance with each generation. Occasionally a result of such
manipulations is that a member having better performance than the best performer of the
previous generation, in which case the latter gets replaced by the new best performer and
the GA then builds up on it.
Figure 3.1.1 The total pressure ratio versus the generational best and worst
performers
0.775
0.78
0.785
0.79
0.795
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0.805
0.81
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64
Figure 3.1.2 shows the results obtained for the mixing efficiency over the course
of the GA run. It is clearly visible that the mixing efficiency has been treated as a
secondary goal by the GA over the pursuit of the total pressure ratio optimization. In
other words, when referring to the results for the mixing efficiency, it is clearly seen that
the absolute best performer in terms of just the mixing efficiency has been classed as the
worst performer for a given generation because of the total pressure loss. Clearly, the best
performer in terms of the overall analysis objective is not necessarily the best performer
in terms of the mixing efficiency alone, and vice versa.
Figure 3.1.2. The mixing efficiency versus the generational best and worst
performers
0.175
0.18
0.185
0.19
0.195
0.2
0.205
0.21
0.215
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65
It is seen that the total pressure ratio (as well as mixing efficiency) can be
compared to a number of reference points within the system. The reference points are the
best and worst performers of the first generation. That is, the best performer of the last
generation can be compared to the best and worst performers of the first generation to
compare the overall optimization effort. However, in this analysis, we have an
experimental reference point for the two injection locations and they will serve as the
?global? reference points for the entire analysis. The value of the total pressure ratio
obtained for the experiment based setup was found to be 0.8043 (or 80.43%) based on the
equations set up previously and the mixing efficiency was found to be 0.1852 (or
18.52%).
As such then one can calculate the improvement in both the total pressure ratio
and the mixing efficiency against the above mentioned values. For the total pressure
ratio, this value comes out to be 0.11% (80.52% from 80.43%) and for the mixing
efficiency there is an associated decrease in value from 18.52% to 18.50% for a total loss
of 0.10%. the improvement of total pressure ratio can be justified against this loss in fuel-
air mixing. However, this is the first case run, and as we shall see, the results obtained
with improvements in our understanding of what the system trends are will lead to
improved results.
The plots shown in Figure 3.1.3 and Figure 3.1.4 are similar in nature to that
presented previously in that all results are shown versus the generation number on the X-
axis and design range for each parameter in the Y-axis of the plot. Both the best and the
worst performers for each generation are presented.
66
Figure 3.1.3 and Figure 3.1.4 present the results for the combustor axial (X)
variation of the two injector locations. It is immediately seen that the GA moves towards
the inner limits of the design space for these two parameters beyond generation number
six.
Figure 3.1.3 The primary injector axial (X) variation for the generational best and
worst performers
2
2.2
2.4
2.6
2.8
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3.4
3.6
3.8
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Best Performers Worst Performers
67
Figure 3.1.4 The secondary injector axial (X) variation for the generational best and
worst performers
Figure 3.1.5 The primary and secondary injector variation for the best performers
6
6.2
6.4
6.6
6.8
7
7.2
7.4
7.6
7.8
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Best Performers Worst Performers
68
Figure 3.1.5 shows the physical presence of both of the injectors and their axial
(X) locations within the design space. It explains more clearly the idea that the design
space is seen to be limiting the results obtained as far as the axial variation of the
injectors is concerned. This is related to the type of definition used for the mixing
efficiency and the physical nature of the injectant flow. From Figure 2.5.10 it can be seen
that the nature of the injectant flow after entering the combustion chamber is in the form
of a thin long frustum cone.
Any section in the Y-Z plane of the combustor taken from this conical flow
structure has a different fuel-air dispersal contour set. In order for the maximum dispersal
to take place, the conical structure must be pulled as far upstream within the combustor as
possible as shown in Figure 3.1.6. This is exactly what the GA tries to do. This suggests
that either the design space should increase along the axial direction of the combustor or
a different definition of the mixing efficiency which is independent of the length of the
combustor is needed.
69
Figure 3.1.6 optimizing the injector Axial (X) location
Figure 3.1.7 and Figure 3.1.8 display the results corresponding to the best and
worst performers for the variation in the combustor transverse (Z) direction versus the
generation numbers. Here the results are more promising in that optimized results are
obtained well within the current design space. One very early conclusion suggested is that
the design space for transverse variation of the injector locations does not need to be
increased beyond the current range for any future runs with the current setup. However,
using different fuels or inlet conditions will render this assumption invalid.
70
Figure 3.1.7. The primary injector transverse (Z) variation for the generational best
and worst performers
Figure 3.1.8. The secondary injector transverse (Z) variation for the generational
best and worst performers
-3
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-1
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Best Performers Worst Performers
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71
For both plots of injector transverse variations, it is seen that the location for the
injectors in the combustor ?Z? direction is almost constant after the fifth generation.
However, the worst performers show more disturbances even after the fifth generation
and it is these disturbances that provide some very useful insights into the optimization
process.
From Figure 3.1.8 it can be seen that the secondary injector has its transverse
position nearly fixed for both the best and worst performers after the fifth generation with
very few perturbations between the best and worst performers. At the same time,
however, the primary injector displays large fluctuations for the worst performers even
after the fifth generation.
Here an interesting trend that is noticed is that all fluctuations for the first injector
seem to take place closer towards the centerline of the combustor floor and almost
entirely fail to cross the best performer position. That is, once the GA has started to
optimize the mixing efficiency after having finished the optimization of the total pressure
ratio to the best available value, there is a trend towards moving the two injectors closer
towards each other in the transverse direction.
After the fifth generation, even the axial variations of the two injectors were
limited to their best possible locations within the design space as discussed previously.
As a result, with the second injector remaining relatively fixed in the transverse direction
as well, the only parameter varying noticeably after the fifth generation is the ?Z?
variation of the first injector.
72
From Figure 2.5.10 and in light of the above discussions, some interesting results
are observed. There appears to be a direct relationship between the mixing efficiency for
the worst performer and its corresponding transverse location around the centerline of the
combustor. Even more interesting is that the best result for mixing efficiency is obtained
by bringing the two injectors closer to each other along the transverse direction rather
than spreading them out.
This has a lot to do with the type of fuel used and its flammability limits given the
current definition of the mixing efficiency. Figure 3.1.9(a), Figure 3.1.9(b) and Figure
3.1.9(c) represent this more visually with a simple example of concentric mass fraction
contours. As is seen from these figures, the nature of the movement clearly affects the
mixing efficiency directly, and is therefore dependent on the type of fuel and the values
used.
While it is clear from the above discussion that, given the current type of fuel and
the corresponding flammability limits, the mixing efficiency is seen to improve as the
two injectors are brought closer. The overlap of the mass fraction contours clearly
presents an advantage initially to try and increase the mixing efficiency.
It is also clear that bringing them too close to each other creates a reverse effect
as seen in Figure 3.1.9(c). Once the two injectors are brought nearly in line with each
other, with little or no transverse distance between them, the initial effect of increasing
mixing efficiency is lost.
73
Figure 3.1.9(a) (left) and 3.1.9(b) (right). Effect of bringing the injectors closer to
each other.
Figure 3.1.9(c). Effect of bringing the injectors closer to each other.
74
This result is also seen from Figure 3.1.7 and Figure 3.1.8 at generation 18. At
this point, the two injectors are closer to each other than at any other point. It shows both
a drop in the total pressure ratio and a drop in the mixing efficiency. This suggests that
there exists a certain optimum location between these two transverse location values
depending on the values put forward for a given fuel?s flammability limits.
75
3.2 CASE RUN - 2
One of the interesting results obtained from the first case run was the movement
of the injector locations in the longitudinal direction. It was clearly seen that the GA had
been restricted in its movement of the two injector locations in its effort to improve the
mixing efficiency without losing out on total pressure ratio.
Further, by considering the worst performers for that run, it can also be seen that
the movement of the two injector locations closer to each other might yield a better total
pressure ratio and mixing efficiency. However, it should be noted that moving the
injectors too close to each other would prove detrimental to total pressure ratio but prove
highly beneficial to mixing efficiency thanks to the turbulence created within the
combustor due to close Mach 1 injections. In this sense there was clearly a need to further
explore this longitudinal variation in more detail and a more aggressive design space.
This therefore formed the basis for Case Run 2.
76
Figure 3.2.1 The Single Dimension Longitudinal Design Space for Case Run 2
Several changes were made to design space based on the trends observed from
Case-1. The transverse variations were completely removed and gave way to a single
dimension design space along the centerline of the combustor floor as shown in Figure
3.2.1. Further, the design space was extended for the primary injector to bring them
closer to the rearward facing step in accordance with previously observed data. The new
design space therefore allowed a variation in step heights from 1.0 to 4.0 for the primary
injector and 6.0 to 8.0 for the secondary injector. The strict pareto scheme was removed
and the number of goals reduced to one by combining the two goals (both of same
77
magnitude) with suitable domination factors. For this case run, the total pressure ratio
was again kept dominant over the mixing efficiency.
The results obtained are shown in Figure 3.2.2 and Figure 3.2.3 for the total
pressure ratio and the mixing efficiency respectively. The GA performs as expected and
the plots are typical of GA optimization results. The total pressure ratio has clearly been
improved over the course of the optimization and so has mixing efficiency, though the
latter is not close to its full potential independently as observed from the worst performer
data in Figure 3.2.3.
Given that the transverse variation in the injector movements have been
eliminated for this run, it is easy to compare the results obtained from the CFD data with
those obtained from the experimental data. Namely, the total pressure ratio has been seen
to improve by 0.11% (80.52% from 80.43%) while the mixing efficiency has improved
by 10.69% (20.5% from 18.52%). These numbers are encouraging but not especially
impressive. The trends are clear in that the mixing efficiency has a potential to improve
dramatically given the chance to be dominant in the optimization even without the need
for the transverse variation improvement discussed in Case-1 results. That is encouraging
enough to be the core of the setup for the next case run, i.e. Case-3.
78
Figure 3.2.2 The Total Pressure Ratio for best and worst performers for Case Run 2
versus GA Generations
Figure 3.2.3 The Fuel-Air Mixing Efficiency for best and worst performers for Case
Run 2 versus GA Generations
0.8025
0.803
0.8035
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79
Further analysis of the results of Case-2 helps validate the suspected trends as
noticed during Case-1. Figure 3.2.4 shows the details of injector movement for the best
performer data along the combustor floor. Without any transverse movement, this
analysis is relatively simple to visualize. It can be clearly seen that the GA attempts to
find an optimum variation between the two injectors as also their global location on the
combustor floor. As such, the analysis can be broken up into two parts: global injector
locations and relative separation between the two.
In the global sense, the two injectors are seen to move modestly closer towards
the base of the rearward facing step. It is especially interesting to note that despite the
increased design space that moved the possible primary injector location within one step
height of the base of the rearward step and the trend noticed in Case-1 regarding the
global movement of the two injectors and their movement upstream, the GA has found a
result that does not move beyond our initial design space for Case-1.
This result makes sense if one notices the pressure contours as presented in the
CFD grid validation studies at the beginning of the thesis. Given a set of inlet conditions
for the combustor and a fixed geometry of the combustor section, the formation of the
expansion fan at the edge of the rearward facing step is fixed despite the movement of the
injectors, within certain limits. It is this ?limit? that gets broken if the primary injector
were to move further upstream. In other words, if the primary injector were to move
upstream, the associated bow shock would move into the expansion fan leading to
increased total pressure losses.
80
Case-2 was set up to preserve total pressure ratio, and the GA successfully
achieves that objective by limiting the location of the primary injector location. In
retrospect the Case-1 setup was not necessarily restrictive as earlier thought for the goal
of preserving total pressure ratio.
Figure 3.2.4 The movement of the primary and secondary injectors versus
Generation numbers for Case Run 2
Figure 3.2.5 displays the results for the relative distances between the two
injectors for best performers of Case-2. Again the trends observed in Case-1 are
validated in that it can be clearly seen that the general trend in preserving total pressure
ratio by moving the two injector locations away from each other but closer to the forward
edge of the design space (just like in Case-1) provides increased total pressure ratios with
lower mixing efficiencies.
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Figure 3.2.5 The relative distance between the two injectors for Case Run 2 versus
GA Generations
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3.3 CASE RUN - 3
The results of Case-1 and Case-2 have suitably established some trends regarding
the optimization of total pressure ratios for the given combustor geometry. Based on the
discussion highlighted in the previous topic it was noticed that extending the design space
in the longitudinal direction provided no visible improvement (and therefore the need for
the extension) as far as the total pressure ratio was concerned. It was also noticed that
despite efforts to create a suitable enviornment for improving the total pressure ratio
(including the high amount of damage to mixing efficiency), the amount of improvement
in the value of the total pressure ratio above the experimental values was hardly
justifiable for the losses incurred in other goals.
It was also noticed that the mixing efficiency was showing great potential for
improvement, given the chance. Three separate trends had been noticed that individually
provided improvement in the mixing efficiency almost independently of each other. It
was therefore possible that if combined, the overall improvement in mixing efficiency
might completely sweep away the accompanying losses in the total pressure ratio.
83
But before moving on the case run with a full two dimensional design space with
mixing efficiency as the dominant goal, it was necessary to validate the claims seperately.
In conjunction with the results of Case-1 where the trends in the transverse direction
regarding the mixing efficiency optimization were distinctively shown, it was necessary
to analyze the longitudinal optimization potential for mixing eficiency. This then formed
the basis for Case-3.
The changes in the GA setup for Case-3 relative to Case-2 were minor. The same
longitudinal single dimension design space was maintained as that for Case-2. However,
the domination factors were changed to make the mixing efficiency as the dominant goal
over the total pressure ratio. Figure 3.3.1 shows the details of the design space for Case-3
optimization run and is in fact indentical to that for Case-2.
84
Figure 3.3.1 The Single Dimension Longitudinal Design Space for Case Run 3
The results obtained are shown in Figure 3.3.2 and Figure 3.3.3 for the mixing
efficiency and the total pressure ratio for the best and worst performers of the run
respectively. The mixing efficiency has shown a drastic improvement over that of Case-1
and Case-2 results and the experimental results. At a current best value of 24.82% it has
improved by 34.01% over the experimental setup. Clearly this was the dominant result
beyond Generation-3 as seen in Figure 3.3.2. The reason why this value remains the same
beyond Generation-3 all the way to the final Generations is explained by the setup of the
GA schemes.
85
It was mentioned previously in this thesis that the steady state scheme was used.
This scheme is basically an elitist population scheme and therefore preserves the best
performers to the detriment of reducing the gene diversity. In this case the steady state
scheme has preserved this parameter set throughout and indeed the diversity of the
parameter sets was seen to reduce dramatically as the optimization took its course. The
majority of the members for the final generations showed very little deviation from this
best value towards the end of the run.
Figure 3.3.2 The Fuel-Air Mixing Efficiency for best and worst performers for Case
Run 2 versus GA Generations
The total pressure ratio obviously took a hit as a result of the improvement in the
mixing efficiency. As can be seen from Figure 3.3.3, the value of the total pressure ratios
corresponding to the best performers for Case-3 clearly show reduced values as compared
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86
to the total pressure ratio of the worst performers in a trend similar to that observed for
Case-1 and Case-2 with respect to the Mixing efficiency. The reduction in the total
pressure ratio for Case-3 is 0.11% (80.34% from 80.43%). Given the enormous increase
in mixing efficiency, these losses can be best considered negligible.
Figure 3.3.3 Total Pressure Ratio for best and worst performers for Case Run 2
versus GA Generations
However, the physical trend behind this drastic increase in fuel-air mixing
efficiency still needs further investigation. Figure 3.3.4 is a plot of the injector locations
in the global setup on the floor of the combustor as a function of step heights away from
the rearward step through the course of the Case-3 run (best performers only).
0.8028
0.803
0.8032
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87
It can be clearly seen that the dominant trend is to push the primary injector as far
forward as possible and to push the secondary injector as far back as possible. This is
consistent with the discussion for the Case-2 injector longitudinal movements in that
moving the primary injector as far forward as possible leads to the destruction of the
expansion fan region and leads to increase in turbulence in the flow at the cost of total
pressure ratio. While this led to the GA not using it for Case-2 in order to preserve total
pressure ratio, the GA did in fact use it in Case-3 to improve the mixing efficiency
drastically.
Further, it should be noted that from the analysis of the experimental validation
studies, that the recirculation region just downstream of the rearward facing step was an
interesting possibility in terms of spreading the injectant further along the transverse
direction. This has been noticed by other researchers as well 12. Indeed, it should be
mentioned that even with the experimental validation studies using CFD, it was noticed
that a small quantity of the fuel from the primary injector had a tendency to get dragged
into this recirculation region and get spread out along the floor of the combustor.
In Case-3 runs, this phenomenon has been exploited by the GA to further improve
mixing efficiency. As the primary injector was moved forward with the aim of destroying
the expansion fan near the rearward facing step, it also moved the injector well within the
recirculation region which in conjunction with the increased turbulence and erratic fluid
motions caused a significant portion of the fuel from the primary injector to be dispersed
far more effectively than would have been possible otherwise.
88
Figure 3.3.4 The movement of the primary (blue) and secondary (red) injectors
versus Generation numbers for Case Run 3
It can now be appreciated that the trends observed in the Case-3 run regarding
improvement of mixing efficiency are relatively independent of the transverse movement
trend of Case-1 to the level that if combined, they provide a possibility for further
increase in the mixing efficiency of the combustor. This forms the basis of Case-4, which
is then our final combination study for the trends of Case-1 and Case-3. This is discussed
in detail in the next topic.
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3.4 CASE RUN ? 4
As mentioned previously, Case-1 and Case-3 provided the basis for the setup for
Case-4. The modification for this case run was the change from the one dimensional
design space back to a two dimensional design space allowing both transverse and
longitudinal variation in the injector locations. The mixing efficiency was now set as the
dominant goal just like for Case-3. The remaining setup is the same as that for Case-3.
Figure 3.4.1 shows the results obtained for the mixing efficiency versus the
generation numbers. The mixing efficiency is found to increase above the experimental
mixing efficiency (which was found to be 18.52% as previously explained) by 37.70% to
a value of 25.55%. More interestingly, the value of the Case-4 best performer has
increased beyond that of the Case-3 Best performer by 2.74%.
90
Figure 3.4.1 The Fuel-Air Mixing Efficiency for best and worst performers for Case
Run 4 versus GA Generations
Figure 3.4.2 shows the results for total pressure ratio versus the generation
numbers for the Case-4 best and worst performers. There are two observations that are
made for Figure 3.4.2. Firstly, it is found that the total pressure ratio for the best
performers has been maintained at very nearly the same value as that of the Case-3 best
performers (a reduction of 0.09% has been noted) while improving the mixing efficiency
by 2.74% as mentioned previously.
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Figure 3.4.2 The Total Pressure Ratio for best and worst performers for Case Run 4
versus GA Generations
Secondly, it can be seen from Figure 3.4.2 that as the Case-4 run has progressed,
not only has the mixing efficiency been improved beyond that of the generational worst
performers as expected, it has also improved the total pressure ratio beyond that of the
worst performers. This is significant in that in all previous runs, the total pressure ratio
and the mixing efficiency had always been found to be inversely proportional to each
other so that if one of these two goals was made dominant, the other goal almost always
performed poorly to the level that the generational worst performers for that goal
outperformed the best performers.
0.8026
0.8028
0.803
0.8032
0.8034
0.8036
0.8038
0.804
0.8042
0.8044
0.8046
0.8048
0 5 10 15 20
To
tal
Pr
es
su
re
Ra
tio
Generations
92
Figure 3.4.3 The movement of the primary (blue) and secondary (red) injectors with
respect to the rearward facing step versus Generation numbers for Case Run 4
Figure 3.4.3 displays the longitudinal movement of the two injectors on the
combustor floor for the best performers and the result is almost exactly the same as that
obtained in Case-3. This validates the idea of the independency of the trends of Case-1
and Case-3 as originally expected. Figure 3.4.4 displays the relative transverse
movements of the two injectors on the combustor floor for the best performers of each
generation.
Figure 3.4.4 displays the movement of the two injectors versus the run progress
expressed in generation numbers. Clearly the trend is not entirely clear from this plot and
0
1
2
3
4
5
6
7
8
0 5 10 15 20
Dis
tan
ce
in
st
ep
he
igh
ts
fro
m
th
e R
ea
rw
ard
ste
p
Generations
93
the data seems to be behaving erratically. In order to find the underlying trend behind this
data the relative distances between the two injectors are presented in Figure 3.4.5.
It can be seen in Figure 3.4.5 that the GA attempts to find the optimum transverse
difference between the two injectors as the run proceeds. Here it is noticed that the data
seems to be oscillating about the expected optimum position when the data is compared
to the minute differences in the mixing efficiency values from Figure 3.4.1. As the run
proceeds, however, the oscillations are seen to be reducing and the GA is seen to be
converging on the optimum location. Figure 3.4.6 shows the final locations of the
optimized injector locations at the end of Case-4.
This is in accordance with the trend from Case-1 where the GA finds the optimum
transverse distance between the two injectors based on the value of the flammability
limits of the fuel being injected into the combustor. The results from Case-1 to Case-4
therefore allow one to evaluate the optimization trends observed for the staged normal
injection combustor very effectively.
94
Figure 3.4.4 The movement of the primary (blue) and secondary (red) injectors with
respect to the Combustor Centerline versus Generation numbers for Case Run 4
Figure 3.4.5 The relative distance between the two injectors for Case Run 4 versus
GA Generations
-4
-3
-2
-1
0
1
2
3
0 5 10 15 20
Dis
tan
ce
in
m
m
ab
ou
t c
en
ter
lin
e
Generations
0
0.5
1
1.5
2
2.5
0 5 10 15 20
Tra
ns
ve
rse
di
sta
nc
e b
etw
ee
n i
nje
cto
rs
in
ste
p h
eig
ht
s
Axis Title
95
Figure 3.4.6 The final results for the optimized injector locations
96
4 CONCLUSIONS AND RECOMMENDATIONS
This effort involved the development of a unique way of using Genetic
Algorithms and Computational Fluid Dynamics in an automated and integrated
optimization structure designed to optimize a scramjet combustor for improved total
pressure ratio and mixing efficiency by varying the injector locations on the floor of the
combustor.
Validation studies were designed to utilize the available experimental data for
single case runs to develop the geometry to conduct the optimization analysis. Fuel
injection was simulated so that actual values of hydrogen flammability limits were
chosen for the calculations but the analysis remained close to its experimental origins by
simulating air-on-air injection as fuel-on-air injection by tagging the injectant flow in the
simulations.
Also presented was the description of the use of the binary GA with the
tournament selection method and its use in a multi-goal optimization process. A design
space was created and the design parameters were specified. Also specified was the
determination process for the design goals from the available three-dimensional flow
simulation data.
97
The overall structure behind the automation of the entire process was presented.
Computational times involved and the numerous methods used over the course of various
sections of the research were discussed. Successful results were obtained in the above
efforts and the current hardware and software setup available at Auburn University is
now a fast, proven, robust and effective system and one that was used in the optimization
process successfully.
Finally, discussions of the results obtained from the optimization runs were
presented. Successful results were obtained for the total pressure ratio and the mixing
efficiency and the methodology was proven as being effective. Improvements of mixing
efficiency in excess of 30% above those obtained in experiments with minimum losses in
total pressure ratio effectively established the potential for the use of GA in propulsion
system designs.
Future work should include moving towards more practical situations by
simulating combustion between the fuel and air within the same combustor, variations in
fuel input to the system and variations in the input flow parameters. In addition, CFD
methods could be made more robust with the use of Full-Multi-Grid (FMG) options now
becoming available in the industry along with the use of more advanced and robust
solvers based on the hybrid solver developed for this effort. Further work should also be
directed towards other combustor designs including ramp based injector modeling of full
scramjet engines using the system developed for this thesis for optimization efforts.
98
REFERENCES
1. Gage, P., and Kroo, I., ?A Role of Genetic Algorithms in a Preliminary Design
Environment,? AIAA Paper No. 93-3933.
2. Hartfield, Roy J., Jenkins, Rhonald M., Burkhalter, John E., ?Ramjet Powered
Missile Design Using a Genetic Algorithm,? AIAA 2004-0451, presented at the
forty-second AIAA Aerospace Sciences Meeting, Reno NV, January 5-8, 2004.
3. Burger, C. and Hartfield, R.J., ?Propeller Performance Optimization using Vortex
Lattice Theory and a Genetic Algorithm?, AIAA-2006-1067, presented at the
Forty-Fourth Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan 9-12,
2006.
4. Doyle, J., Hartfield, R.J., and Roy, C. ?Aerodynamic Optimization for Freight
Trucks using a Genetic Algorithm and CFD?, AIAA 2008-0323, presented at the
46th Aerospace Sciences Meeting and Exhibit, Reno, NV, January 2008.
5. J.E. Burkhalter, R.M. Jenkins, and R.J. Hartfield, M. B. Anderson, G.A. Sanders,
?Missile Systems Design Optimization Using Genetic Algorithms,? AIAA Paper
2002-5173, Classified Missile Systems Conference, Monterey, CA, November,
2002
6. Hartfield, R.J., Burkhalter, J. E., and Jenkins, R. M., ?Scramjet Missile Design
Using Genetic Algorithms?, Applied Mathematics and Computation, in press.
7. Hartfield, R. J., Jr., ?Planar Measurement of Flow Field Parameters in
Nonreacting Supersonic Flows with Laser-Induced Iodine Fluorescence?? PhD.
99
Dissertation, Department of Mechanical and Aerospace Engineering, University
of Virginia, May 1991.
8. David M. Peterson,_ Pramod K. Subbareddy_ and Graham V. Candler,
?Assessment of Synthetic Inflow Generation for Simulating Injection Into a
Supersonic Cross-flow?, University of Minnesota, Minneapolis, MN, 55455
9. J. McDaniel, D. Fletcher, R. Hartfield Jr and S. Hollo, ?Staged Transverse
Injection Into Mach-2 Flow Behind a Rearward-Facing step: a 3-D Compressible
Test Case for Hypersonic Combustor Code Validation?, AIAA Third International
Aerospace Planes Conference, 3-5 December 1991/ Orlando, FL
10. Eklund, D. R., Fletcher, D. G., Hartfield, R. J., Jr., Northam, G. B., and Dancey,
C. L., ?A Comparative Computational/Experimental Compressible Flow Field
Investigation. Mach 2 Flow Over a Rearward-Facing Step,? Computers in Fluids,
Vol. 24, No. 5, pp. 593-608.
11. Eklund, D. R., Fletcher, D. G., Hartfield, R. J., Jr., McDaniel, J. C., Northam, G.
B., Dancey, C. L., and Wang, J. A., ?A Comparative Computational/Experimental
Compressible Flow Field Investigation. Staged Normal Injection into a Mach 2
Flow Behind a Rearward-Facing Step,? AIAA Journal, Vol. 32, No. 5, May 1994,
pp. 907-916.
12. Dean R. Eklund, Douglas G. Fletcher, G. Burton Northam, J. C. McDaniel and
Roy J. Hartfield, Jr., ?Mach-2 Flow over a Rearward-Facing step: a comparison
between calculation and measurement?
13. K. Uenishi and R. C. Rogers, ?Three-Dimensional Computation of Mixing of
Transverse injector in a ducted supersonic airstream?, AIAA/SAE/ASME 22nd
Joint Propulsion Conference
14. Abbitt, J. D., Hartfield, R. J. and McDaniel, J. C., ?Mole Fraction Imaging of
Transverse Injection in a Ducted Supersonic Flow?, AIAA Journal, Vol. 29, No
3, March 1991, pp. 431-435.
100
15. R. Courant, K. Friedrichs and H. Lewy, ?On the partial difference equations of
mathematical physics?, IBM Journal, March 1967, pp. 215-234, English
translation of the 1928 German original
16. Murray B. Anderson, ?An Optimization Software Package based on Genetic
Algorithms?, Sverdrup Technology Inc./TEAS Group.
17. Anderson, G. Y., ?An examination of Injector/Combustor Design effects on
Scramjet Performance?, Proceedings of the 2nd International Symposium on Air-
Breathing Engines, Sheffield, England, March 1974.
18. William H. Heiser and David T. Pratt, with Daniel H. Daley and Unmeel B.
Mehta, ?Hypersonic Airbreathing Propulsion?, AIAA Education series
publication
19. Pai, S. I., ?The Fluid Dynamics of Jet?, D. Van Nostrand, New York, 1954
20. A. Abdelhafez, A. Gupta, R. Balar and K. Yu, ?Evaluation of Oblique and
Traverse Fuel Injection in a Supersonic Combustor?, AIAA Paper 5026, 43rd
AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 2007
21. C. Kim, E. Jeong, J. Kim and I. Jeung, Seoul National University, Seoul, South
Korea , ?Mixing and Penetration Studies of Transverse Jet into a Supersonic
Crossflow?, AIAA Paper 5420, 43rd AIAA/ASME/SAE/ASEE Joint Propulsion
Conference & Exhibit, 2007
22. Rogers, R. C., ?Mixing of Hydrogen injected from multiple injectors Normal to a
supersonic airstream?, NASA TN D-6476, September 1971.
23. Amjad Ali Sheikh, ?A study of overall mixing loss downstream of a blunt trailing
edge in a subsonic stream?, Tuskegee Institute, Alabama 36088, May 1982.
101
APPENDIX
MATHEMATICAL DERIVATION OF THE AVERAGE TOTAL PRESSURE
ACROSS A FLOW PLANE
The equations used in this analysis have been customized to handle rectangular
geometries and meshed surfaces. The mass-momentum averaging technique is explained
by the further development of the following fundamental equations as detailed in the
following sections. All notations remain as detailed in the main body of the thesis.
A. THE CONTINUITY EQUATION:
The continuity equation for a given grid plane in the three dimensional flow field can be
represented as shown in equation (1):
uni2211 g2025g3046g4666g1861g4667g1874g3051g4666g1861g4667g1827g4666g1861g4667 = g2025g3046g3015g3036g2880g2869 g1874g3051g1827 (1)
Equation (1) can be modified by dividing both terms with the termg4666g2025g3019g3006g3007g4593 g1853g3019g3006g3007g4667g4593 ) to get the
following result:
uni2211 g3096g3294g4666g3036g4667g3096?
Rg3137g3138g3028g3267g3254g3255g4594
g1874g3051g4666g1861g4667g1827g4666g1861g4667g3015g3036g2880g2869 = g3096g3294g3096
g3267g3254g3255?
g3049g3299
g3028g3267g3254g3255g4594 g1827 (2)
102
Now, before going any further, it should be noted that for the defined reference and total
conditions, using the ideal gas equation, assuming adiabatic conditions and using
isentropic relationships, the following results are obtained:
a) Adiabatic flow conditions:
g1846g4593 = g1846g3019g3006g3007g4593 (3)
b) Ideal gas law:
g3096g4594
g3096g3267g3254g3255g4594 =
g3017g4594
g3017g3267g3254g3255g4594 (4)
c) Definition for sonic velocity:
g1853g3019g3006g3007g4593 = g3493g2011g1844g1846g3019g3006g3007 = g3495g4672g2870g3082g3019g3021g3267g3254g3255g4594g3082g2878g2869 g4673 (5)
d) Isentropic Relationships:
g4672g3021g3294g3021?g4673 = g4678g883g3398g4672g3082g2879g2869g3082g2878g2869g4673g3436 g3049g3299g3118g3028
g3267g3254g3255g4594
g3118g3440g4679 (6)
These relationships will prove highly useful in the mathematical manipulations that
follow. Returning to equation (2), the right hand side of the equation can be modified in
the following manner:
103
uni2211 g3096g3294g4666g3036g4667g3096?
Rg3137g3138g3028g3267g3254g3255
g4594
g1874g3051g4666g1861g4667g1827g4666g1861g4667g3015g3036g2880g2869 = g3096g3294g3096? g4672 g3096g4594g3096
g3267g3254g3255?
g4673 g3049g3299g3028
g3267g3254g3255g4594
g1827 (7)
Substituting equation (4) in the right hand side of equation (7), yields:
uni2211 g3096g3294g4666g3036g4667g3096?
Rg3137g3138g3028g3267g3254g3255
g4594
g1874g3051g4666g1861g4667g1827g4666g1861g4667g3015g3036g2880g2869 = g3096g3294g3096? g4672 g3017g4594g3017
g3267g3254g3255?
g4673 g3049g3299g3028
g3267g3254g3255g4594
g1827 (8)
Let either side of equation (2) be represented by a function ?I?.
B. MOMENTUM EQUATION:
Similar to the continuity equation, the momentum equation can be represented for a given
grid surface as follows:
uni2211 [g2025g3046g4666g1861g4667g1874g3051g2870g4666g1861g4667g1827g4666g1861g4667g3397 g1842g3046g4666g1861g4667g1827g4666g1861g4667]g3015g3036g2880g2869 = g2025g3046g1874g3051g2870g1827g3397g1842g3046g1827 (9)
Let equation (9) be represented as:
J = J1 + J2 = J3 + J4 (10)
Where ?J? signifies either of the two sides of the momentum equation whereas the other
terms are defined as:
g13g883 = uni2211 [g2025g3046g4666g1861g4667g1874g3051g2870g4666g1861g4667g1827g4666g1861g4667]g3015g3036g2880g2869 (11)
104
g13g884 = uni2211 [g1842g3046g4666g1861g4667g1827g4666g1861g4667]g3015g3036g2880g2869 (12)
g13g885 = g2025g3046g1874g3051g2870g1827 (13)
g13g886 = g1842g3046g1827 (14)
Dividing the entire expression by g2025g4593g2902g2889g2890g1853g3019g3006g3007? g2870 and simplifying the terms individually, the
following sets of results are obtained. From equation (11)
g13g883 = uni2211 [g2025g3046g4666g1861g4667g1874g3051
g2870g4666g1861g4667g1827g4666g1861g4667]g3015g3036g2880g2869
g2025g4593g2902g2889g2890g1853g3019g3006g3007? g2870
Applying the ideal gas law to equation (11):
g13g883 = g4678 g2902g2904g4594
g3017g4594Rg3137g3138g3028g3267g3254g3255? g3118
g4679uni2211 g4674g4672 g3017g3294g4666g3036g4667g3019g3021
g3268g4666g3036g4667
g4673g1874g3051g2870g4666g1861g4667g1827g4666g1861g4667g4675g3015g3036g2880g2869 (15)
But g1839g3051g4666g1861g4667 = g3049g3299g4666g3036g4667g3493g3082g3019g3021
g3268g4666g3010g4667
, so that a modification of equation (16) is:
g13g883 = g4678 g2963g2902g2904g4594
g3017g4594Rg3137g3138g3028g3267g3254g3255? g3118
g4679uni2211 g3427g1842g3046g4666g1861g4667g1839g3051g2870g4666g1861g4667g1827g4666g1861g4667g3431g3015g3036g2880g2869 (16)
105
From equation (12)
g13g884 = uni2211 [g1842g3046g4666g1861g4667g1827g4666g1861g4667]
g3015g3036g2880g2869
g2025g4593g2902g2889g2890g1853g3019g3006g3007? g2870
Substituting equation (5) in above equation:
g13g884 = uni2211 [g1842g3046g4666g1861g4667g1827g4666g1861g4667]
g3015g3036g2880g2869
g2025g4593g2902g2889g2890 g3436g884g2011g1844g1846g3019g3006g3007
g4593
g2011 g3397g883 g3440
This on further substitution with equation (4) yields:
g13g884 = g4672g2963g2878g2869g2870g2963 g4673uni2211 [g3017g3294g4666g3036g4667g3002g4666g3036g4667]g3263g3284g3128g3117g3017g4594
Rg3137g3138
(17)
From equation (13),
g13g885 = g3096g3294g3049g3299g3118g3002
g3096g4594Rg3137g3138g3028g3267g3254g3255? g3118
(18)
Applying some simple modifications to equation (18):
g13g885 = g3096g3294g3049g3299g3118g3002
g3096g4594g3028g3267g3254g3255? g3118
g3436 g3096g4594g3096g4594
Rg3137g3138
g3440 (19)
106
Substituting equation (4) in equation (19),
g13g885 = g3096g3294g3049g3299g3118g3002
g3096g4594g3028g3267g3254g3255? g3118
g4672 g3017g4594g3017g4594
Rg3137g3138
g4673 (20)
g13g886 = g3017g3294g3002
g3096g4594Rg3137g3138g3028g3267g3254g3255? g3118
(21)
Substituting equation (5) in equation (21):
g13g886 = g3017g3294g3002
g3096g4594Rg3137g3138g3118g3330g3267g3269g3267g3254g3255
g4594
g3330g3126g3117
(22)
Now substituting equation (4) in equation (22), we get:
g13g886 = g4672g2963g2878g2869g2870g2963 g4673g4672g3017g3294g3017?g4673g4672 g3017g4594g3017
g3267g3254g3255g4594
g4673g1827 (23)
Substituting equations (15), (17), (20) and (23) in equation (10), the following expression
for the momentum equation is obtained:
g13 = g4678 g2963g2902g2904g4594
g3017g4594Rg3137g3138g3028g3267g3254g3255? g3118
g4679uni2211 g3427g1842g3046g4666g1861g4667g1839g3051g2870g4666g1861g4667g1827g4666g1861g4667g3431g3015g3036g2880g2869 g3397 g4672g2963g2878g2869g2870g2963 g4673uni2211 [g3017g3294g4666g3036g4667g3002g4666g3036g4667]
g3263g3284g3128g3117
g3017g4594Rg3137g3138 =
g3096g3294g3049g3299g3118g3002
g3096g4594g3028g3267g3254g3255? g3118
g4672 g3017g4594g3017g4594
Rg3137g3138
g4673g3397 g4672g2963g2878g2869g2870g2963 g4673g4672g3017g3294g3017?g4673g4672 g3017g4594g3017
g3267g3254g3255g4594
g4673g1827 (24)
107
Now, from the right hand side terms of equations (8) and (24):
g3011
g3010 =
g3344g3294g3297g3299g3118g3250
g3344g4594g3276g3267g3254g3255? g3118
g3436 g3265g4594g3265g4594
Rg3137g3138
g3440g2878 g4672g3211g3126g3117g3118g3211 g4673g4672g3265g3294g3265?g4673g4678 g3265g4594g3265
g3267g3254g3255g4594
g4679
g3344g3294
g3344?g4678
g3265g4594
g3265g3267g3254g3255? g4679
g3297g3299
g3276g3267g3254g3255g4594
(25)
Equation (25) on further simplification gives us the following expression:
g3011
g3010 =
g3344g3294g3297g3299g3118g3250
g3344g4594g3276g3267g3254g3255? g3118
g2878 g4672g3211g3126g3117g3118g3211 g4673g4672g3265g3294g3265?g4673
g3344g3294
g3344?
g3297g3299
g3276g3267g3254g3255g4594
(26)
Now defining a variable ?c? so that:
g1855 = g3011g3010 (27)
Substituting equation (27) in equation (26) and modifying the equation:
g3436 g3049g3299g3118g3028
g3267g3254g3255g4594
g3118 g3440 g514 g1855g4672
g3049g3299
g3028g3267g3254g3255g4594 g4673g3397 g4672
g2963g2878g2869
g2870g2963 g4673g4672
g3017g3294
g3017?g4673g4672
g3096g4594
g3096g3294g4673 = g882 (28)
Substituting equation (4) in equation (28):
108
g3436 g3049g3299g3118g3028
g3267g3254g3255g4594
g3118 g3440 g514 g1855g4672
g3049g3299
g3028g3267g3254g3255g4594 g4673g3397 g4672
g2963g2878g2869
g2870g2963 g4673g4672
g3021g3294
g3021?g4673 = g882 (28)
Substituting equation (6) in equation (28):
g3436 g3049g3299g3118g3028
g3267g3254g3255g4594
g3118 g3440 g514g4672
g2870g3030g3082
g3082g2878g2869g4673g4672
g3049g3299
g3028g3267g3254g3255g4594 g4673g3397 g883 = g882 (29)
Equation (29) is a quadratic equation for the variable g4672 g3049g3299g3028
g3267g3254g3255g4594
g4673 and can be solved as:
g3049g3299
g3028g3267g3254g3255g4594 = g4672
g3082g3030
g3082g2878g2869g4673g3398 g3495g4666
g3082g3030
g3082g2878g2869g4667
g2870 g3398 g883 (30)
Also, from isentropic relationships, the following relationship for the density ratio is
obtained:
g3096g3294
g3096? = [g883 g3398 g4672
g3082g2879g2869
g3082g2878g2869g4673g484
g3049g3299g3118
g3028g3267g3254g3255? g3118
]g4672
g3117
g3330g3127g3117g4673 (31)
Now, from equation (8):
g1835 = g3533 g2025g3046g4666g1861g4667g2025?
g2902g2889g2890g1853g3019g3006g3007
g4593
g1874g3051g4666g1861g4667g1827g4666g1861g4667
g3015
g3036g2880g2869
= g2025g3046g2025? g4678 g1842
g4593
g1842g3019g3006g3007? g4679
g1874g3051
g1853g3019g3006g3007g4593 g1827
109
Substituting equation (4) in the left hand side term of equation (8):
g1835 = uni2211 g4672g3096g3294g4666g3036g4667g3019g3021?g3017
g3267g3254g3255 ?
g4673g4672g3049g3299g4666g3036g4667g3028
g3267g3254g3255g4594
g4673g1827g4666g1861g4667g3015g3036g2880g2869 (32)
On further manipulations of equation (35):
g1835 = uni2211 g4672g3017g3294g4666g3036g4667g3082g3019g3021?g3017
g3267g3254g3255 ?
g4673g4672 g3014g3299g4666g3036g4667g3118g3028
g3267g3254g3255g4594 g3049g3299g4666g3036g4667
g4673g1827g4666g1861g4667g3015g3036g2880g2869 (33)
Or on simplifying the equation:
g1835 = g4672 g3082g3019g3021?g3017
g3267g3254g3255 ? g3028g3267g3254g3255g4594
g4673uni2211 g1842g3046g4666g1861g4667g4672g3014g3299g4666g3036g4667g3118 g3049
g3299g4666g3036g4667
g4673g1827g4666g1861g4667g3015g3036g2880g2869 (34)
Substituting equations (24) and (34) into equation (27):
g1855 =
g3437 g3211RTg4594
g3265g4594Rg3137g3138g3276g3267g3254g3255? g3118
g3441uni2211 g3427g3017g3294g4666g3036g4667g3014g3299g3118g4666g3036g4667g3002g4666g3036g4667g3431g3263g3284g3128g3117 g2878 g4672g3211g3126g3117g3118g3211 g4673uni2211 [g3265g3294g4666g3284g4667g3250g4666g3284g4667]
g3263g3284g3128g3117
g3265g4594Rg3137g3138
g4678 g3330g3267g3269?g3265
g3267g3254g3255 ? g3276g3267g3254g3255g4594
g4679uni2211 g3017g3294g4666g3036g4667g3436g3262g3299g4666g3284g4667g3118 g3297g3299g4666g3284g4667 g3440g3002g4666g3036g4667g3263g3284g3128g3117
(35)
110
As can be seen, the g1842g4593g2902g2889g2890 term cancels from both the numerator and the denominator of
equation (35), and on further simplification:
g1855 =
g3437 g3211RTg4594
g3276g3267g3254g3255? g3118
g3441uni2211 g3427g3017g3294g4666g3036g4667g3014g3299g3118g4666g3036g4667g3002g4666g3036g4667g3431g3263g3284g3128g3117 g2878 g4672g3211g3126g3117g3118g3211 g4673uni2211 [g3017g3294g4666g3036g4667g3002g4666g3036g4667]g3263g3284g3128g3117
g4678g3330g3267g3269?g3276
g3267g3254g3255g4594
g4679uni2211 g3017g3294g4666g3036g4667g3436g3262g3299g4666g3284g4667g3118 g3297g3299g4666g3284g4667 g3440g3002g4666g3036g4667g3263g3284g3128g3117
(36)
Equation (36) provides the value for ?c? in terms of the grid data. Finally, equating the
right hand term of equation (7) and (34):
g3096g3294
g3096? g4672
g3096g4594
g3096g3267g3254g3255? g4673
g3049g3299
g3028g3267g3254g3255g4594 g1827 = g4672
g3082g3019g3021?
g3017g3267g3254g3255 ? g3028g3267g3254g3255g4594 g4673uni2211 g1842g3046g4666g1861g4667g4672
g3014g3299g4666g3036g4667g3118
g3049g3299g4666g3036g4667 g4673g1827g4666g1861g4667
g3015g3036g2880g2869 (37)
Equation (40) when further simplified, gives:
g1842? =
g4678g3330g3267g3269?g3276
g3267g3254g3255g4594
g4679uni2211 g3017g3294g4666g3036g4667g3436g3262g3299g4666g3284g4667g3118 g3297g3299g4666g3284g4667 g3440g3002g4666g3036g4667g3263g3284g3128g3117
g3344g3294
g3344?
g3297g3299
g3276g3267g3254g3255g4594 g3002
(38)
Equation (38) provides the final form of the equation which can be solved for the average
total pressure g1842? across a given grid plane in the three dimensional flow field. The terms
in the denominator for equation (38) can be found from equations (30), (31) and (36)
111
which are themselves a result of flow field grid data. Note that g1842g1314 and g1874g3051, while average
values, simultaneously satisfy the continuity and momentum equations. They thus have
physical as well as statistical meaning.