Rodger W. Dyson

Glenn Research Center, Cleveland, Ohio

Scott D. Wilson

Sest, Inc., Middleburg Heights, Ohio

Roy C. Tew

Glenn Research Center, Cleveland, Ohio

Rikako Demko

Sest, Inc., Middleburg Heights, Ohio

Stirling Analysis Comparison of Commercial

Versus High-Order Methods

NASA/TM—2005-213976

October 2005

AIAA–2005–5580
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Rodger W. Dyson

Glenn Research Center, Cleveland, Ohio

Scott D. Wilson

Sest, Inc., Middleburg Heights, Ohio

Roy C. Tew

Glenn Research Center, Cleveland, Ohio

Rikako Demko

Sest, Inc., Middleburg Heights, Ohio

Stirling Analysis Comparison of Commercial

Versus High-Order Methods

NASA/TM—2005-213976

October 2005

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

Third International Energy Conversion Engineering Conference

sponsored by the American Institute of Aeronautics and Astronautics

San Francisco, California, August 15–18, 2005

AIAA–2005–5580
Acknowledgments

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paper, intended to solicit comments and

ideas from a technical peer group.

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This report contains preliminary

findings, subject to revision as

analysis proceeds.

Available electronically at http://gltrs.grc.nasa.gov
Recently, three-dimensional Stirling engine simulations have been accomplished utilizing commercial Computational Fluid Dynamics software. The validations reported can be

somewhat inconclusive due to the lack of precise time accurate experimental results from

engines, export control/proprietary concerns, and the lack of variation in the methods

utilized. The last issue may be addressed by solving the same flow problem with alternate methods. In this work, a comprehensive examination of the methods utilized in the

commercial codes is compared with more recently developed high-order methods. Specifically, Lele’s Compact scheme and Dyson’s Ultra Hi-Fi method will be compared with

the SIMPLE and PISO methods currently employed in CFD-ACE, FLUENT, CFX, and

STAR-CD (all commercial codes which can in theory solve a three-dimensional Stirling

model although sliding interfaces and their moving grids limit the effective time accuracy).

We will initially look at one-dimensional flows since the current standard practice is to

design and optimize Stirling engines with empirically corrected friction and heat transfer

coefficients in an overall one-dimensional model. This comparison provides an idea of the

range in which commercial CFD software for modeling Stirling engines may be expected

to provide accurate results. In addition, this work provides a framework for improving

current one-dimensional analysis codes.

Nomenclature

β Nondimensional Wavenumber

 Turbulent Dissipation Rate,(m2/s3)

η Kolmogorov Length Scale,(m)

μ Dynamic (Molecular) Viscosity,(Ns · m−2)

ν Kinematic Viscosity,(μ · ρ−1)

ρ Density,(kg · m−3)

τw Surface Shear Stress,(kg · m−1s−2)

c Constant Convective Velocity

c4o0 UHF Method - 4 point stencil - no derivatives

c4o1 UHF Method - 4 point stencil - one derivative

c4o2 UHF Method - 4 point stencil - two derivatives

c4o3 UHF Method - 4 point stencil - three derivatives

Cp Specific Heat Constant Pressure

Cv Specific Heat Constant Volume

G Complex Amplification Factor

k Wavenumber

L Length of Numerical Domain

r Courant Number

uτ Friction Velocity,(kg · m−3)

v Von Neuman Number

CFL Courant-Friedrichs-Lewy

NASA/TM—2005-213976 1

Rodger W. Dyson 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Scott D. Wilson 

Sest, Inc. 

Middleburg Heights, Ohio 44130 

Roy C. Tew 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Rikako Demko 

Sest, Inc. 

Middleburg Heights, Ohio 44130 

Stirling Analysis Comparison of Commercial Versus High-Order Methods 
I. Introduction

P

ower conversion with free-piston Stirling engines1 promises to deliver high efficiency, low mass solutions

for longer and more varied space missions.2

In addition to using advanced high-temperature materials to

increase the Carnot temperature ratio, it is anticipated that advanced computational fluid dynamics (CFD)

will help to identify the following losses4, 5, 6(also shown in figure 1):

1. Inefficient heat exchange and pressure loss in the heat exchangers (heater,regenerator, and cooler)

2. Gas spring and working space loss due to hysterisis and turbulence,

3. Appendix gap losses due to pumping and shuttle effects,

4. Mixing gas losses from unequal temperature distributions or losses from mixing of gas streams, or

elements of gas at different temperatures,

5. Conduction losses from the hot to cold regions

In addition, the following artificial numerical losses must be considered when computational simulations

are performed (also shown in figure 1):

1. Moving/deforming mesh losses from repeated low order flow field interpolations,

2. Transient/Unsteady heat transfer and flow loss from inconsistent and inaccurate time discretization,

3. Flow loss from low order approaches resulting in effectively adding artificial dissipation terms along

sliding interfaces, at structured/unstructured grid interfaces, and within interior.

Minimizing those artificial losses is best accomplished through higher order approaches.7 While this

approach is common in aeroacoustics, computational electromagnetics, and exterior flow problems, high

order techniques have not yet been applied to simulating a Stirling device. Moreover, the following difficulties

are often encountered when using high-order approaches:

1. Generation of high-order, smooth, body-fitted grids around complex configurations can be difficult.8

2. High-order formulations can lack nonlinear robustness.8

3. The general usefulness of high-order methods is limited by first order accurate shock capturing.9

Fortunately, with the exception of the possibly random geometry in the regenerator, the free-piston design

is essentially smooth and admits curvilinear structured grids (with some geometry simplification). The issue

of nonlinear robustness (i.e. maintaining design accuracy with nonlinear equations) is an open issue, but

preliminary results are encouraging. And finally, the working gas is subsonic and shockless throughout the

entire region10 (However, steep temperature gradients can exist at solid/fluid interfaces). For these reasons,

a high-order approach is investigated for ”whole engine” Stirling analysis and compared to commonly used

techniques.

II. Description of the Problem

The dual opposed configuration shown in figure 2

11, 12 is being developed for multimission uses.13

Many methods in general use stop at 4th order accuracy for time dependent problems since they use

Runge-Kutta methods. High-order Runge-Kutta methods become notoriously difficult to derive because the

number of nonlinear order conditions that need to be solved grows exponentially (i.e., a 12th order method

has 7813 nonlinear order conditions). The advantages of using Runge-Kutta methods at orders less than 6

are commonly cited as flexibility, large stability limits, and ease of programming.14 The practical limit on

their order has been an impediment to the analysis of their use in high order approaches for time dependent

applications.15

In this paper we use a series of explicit, local, high order methods which have the same order of accuracy

in space as in time16, 17 for inviscid flow (lower accuracy in time for viscous flows). These methods use

Hermite interpolation on stencils that are four points wide, and a Cauchy-Kowalewski recursive procedure18

NASA/TM—2005-213976 2
Figure 1. Schematic of Actual and Artificial Numerical Losses in a Free-Piston Convertor

for obtaining time derivatives from the space derivatives of the interpolant. The time derivatives are then

used to advance the primitive variables and their spatial derivatives in time with a Taylor series expansion.

This general approach is called the Modified Expansion Solution Approximation (MESA) method19 and the

new finite volume variation of this is called the Ultra HI-FI (UHF) method.20 This method can be used

to derive and implement algorithms with arbitrarily high orders of accuracy in multiple space dimensions if

their complexity is properly managed and the computer’s floating point precision is sufficiently high.21

First, some of the known exact solutions of the viscous Burger’s equation are provided and the linear

case is solved with both state-of-practice Compact schemes, new UHF methods, and various commercial

code solvers. The one-dimensional Navier-Stokes equations reduce to the linear viscous Burger’s equation

in certain circumstances and provide a means for testing both heat transfer effectiveness and turbulent

transition efficiency of each method.

Figure 2. Dual Opposed Stirling Convertors Reduce Vibration

NASA/TM—2005-213976 3
III. Exact Solutions For Method Comparison

Since the nonlinear Navier-Stokes equations generally do not have exact solutions nor known stability

limits, preliminary development and testing of new numerical methods is best accomplished by starting with

the viscous Burger’s equation. This equation describes flow behaviour in specialized circumstances, but more

importantly, its mathematical properties are very similar to the full Navier-Stokes equations and it admits

exact solutions.

For future reference and convenience, some of the known exact solutions are shown below (only the linear

viscous Burger’s equation will be required in this work).

A. Complete Nonlinear Viscous Burger’s Equation

The viscous Burger’s equation is written as:

∂u

∂t + u

∂u

∂x = μ

∂

2u

∂x2

(1)

where u is the convective velocity term and μ can be considered the dynamic viscosity.

Exact steady-state solution, limt→∞u(x, t) exists for the case with boundary conditions:

u(0, t) = u0 (2)

u(L, t) = 0 (3)

and it is given by:

u = u0u ̄



1 − exp[ ̄uReL(x/L − 1)]

1 + exp[ ̄uReL(x/L − 1)]

(4)

where ReL =

u0L

μ

(note this is a modified Reynold’s number)

u ̄ is a solution of the equation

u ̄ − 1

u ̄ + 1

= exp(−uRe  ̄ L) (5)

Other solutions for the nonlinear viscous burgers equation are:

1.

ut + uux − μuxx = 0, μ = 0.1, x ∈ (0, 1) (6)

u(x, 0) = 0, u(1, t) = −tanh 

1

2μ



, u(0, t) = 0 (7)

has exact solution

u(x, t) = −tanh 

x

2μ



(8)

2. Fully nonlinear equation:

ut + uux −

1

2

(uu ̇ x)x = 0, x ∈ (0, 1) (9)

with initial and boundary conditions as:

u(x, 0) = expx, u(0, t) = 1, u(1, t) = e (10)

has the following exact solution u(x, t) = expx.

NASA/TM—2005-213976 4
3. An exact solution of the nondimensional form of the Nonlinear Burger’s equationn22

∂u

∂t + u

∂u

∂x = μ

∂

2u

∂x2

(11)

Nondimensionalized by:

x

∗ =

x

L

, u∗ =

uL

μ

, t∗ =

μ

L2

t (12)

producing:

u

∗

t

∗

+ u

∗ ∂u∗

∂x∗

=

∂

2u∗

∂x∗2

(13)

One stationary solution is:

u

∗ = −

2 sinh x

∗

cosh x

∗ − exp−t

∗

(14)

B. Linear Viscous Burger’s Equation

For simplicity and more thorough stability analyses, the viscous Burger’s equation may be linearized with

constant convective velocity, c, and dynamic viscosity, μ:

∂u

∂t + c

∂u

∂x = μ

∂

2u

∂x2

(15)

The exact steady-state solution with the same boundary conditions as in Eqs.(2) and (3) is:

u = u0



1 − exp[RL(x/L − 1)]

1 − exp(−RL)



(16)

where RL =

cL

μ

(modified Reynolds number).

The exact solution for the linearized equation with initial condition, u(x, 0) = sin(kx), and periodic

boundary conditions is:

u(x, t) = exp(−k

2μt) sin k(x − ct) (17)

This is useful for evaluating the temporal accuracy of a method and this will be used in comparing Compact

and UHF techniques.23

In addition, this equation form also describes the time-accurate temperature distribution in a moving

solid or within a moving fluid in a channel in which case μ = α =

k

ρCv

is interpreted as the thermal diffusivity,

and T is the temperature:

∂T

∂t + u

∂T

∂x = α

∂

2T

∂x2

(18)

This will be used for comparing the heat transfer capabilities of Compact, UHF, and commercial code

solvers.

C. 2-D Nonlinear Viscous Burgers’ Equation

The 2-D Burger’s equation with nonlinear convection terms,24

∂u

∂t + u

∂u

∂x + v

∂u

∂y − v(

∂

2u

∂x2

+

∂

2u

∂y2

) − fx = 0 (19)

∂v

∂t + u

∂v

∂x + v

∂v

∂y − v(

∂

2v

∂x2

+

∂

2v

∂y2

) − fy = 0 (20)

with

fx = −

1

(1 + t)

2

+

x

2 + 2xy

(1 + t)

+ 3x

3

y

2 − 2vy (21)

NASA/TM—2005-213976 5
fy = −

1

(1 + t)

2

+

y

2 + 2xy

(1 + t)

+ 3y

3x2 − 2vx (22)

has exact solution:

u =

1

1 + t

+ x

2

y (23)

v =

1

1 + t

+ xy2

(24)

D. 2-D Linear Burger’s Equation

The 2D (Burger’s) linear convection and diffusion equation:25

ut + c(ux + uy) − μ(uxx + uyy) = 0,(x, y) ∈ (−1, 1) × (−1, 1); c = 1, μ = 0.01 (25)

with the initial condition u(x, y, 0) = sin(π(x+y)) and periodic boundary condition has the exact solution

is:

u(x, y, t) = exp−2πμt sin(π(x + y − 2ct)) (26)

Additional exact solutions may be found in the paper by Benton and Platzman.26

IV. Application of Compact Scheme

The currently accepted state-of-the-art approach to high fidelity numerical simulations is based on Lele’s

6

th order Compact scheme.27 This approach implicitly solves the spatial derivative terms and utilizes standard Runge-Kutta time advance. Each time step requires solving a tridiagonal matrix given by the equation:

α



∂u

∂x

i−1

+



∂u

∂x

i

+ α



∂u

∂x

i+1

= a

ui+1 − ui−1

2∆x

+ b

ui+2 − ui−2

4∆x

(27)

with α = 1/3, a = 14/9, b = 1/9.

Similarly, the second order derivatives are found by:

α



∂

2u

∂x2



i−1

+



∂

2u

∂x2



i

+ α



∂

2u

∂x2



i+1

= a

ui+1 − 2ui + ui−1

∆x

2

+ b

ui+2 − 2ui + ui−2

4∆x

2

(28)

with α = 2/11, a = 12/11, b = 3/11.

The 4th order Runge-Kutta method is described by:24

R(u) = −cux + μuxx (29)

u

(1) = un +

∆t

2

R

n

(30)

u

(2) = un +

∆t

2

R

1

(31)

u

(3) = un + ∆tR2

(32)

u

n+1 = un +

∆t

6



R

n + 2R(1) + 2R(2) + R(3)

(33)

(34)

with

R

(1) = R(u(1))

R

(2) = R(u(2))

R

(3) = R(u(3))

NASA/TM—2005-213976 6
The single step 6th order Compact scheme with 4th order Runge-Kutta on a domain [−4, 4] is given by:

u

n+1

i =



6296179542799261293r

4

2934584609881498112 +

98816474558753811vr3

18549839506204160 −

31068146188125r

3

42158726150464 +

55178557611v

2

r

2

19381094656 

u−4 +



−

287760797433vr2

133245025760 +

324338625r

2

2422636832 +

401096399245v

3

r

583491009024 

u−4 +



−

3588625v

2

r

3898048 +

573327vr

1093840 −

75r

8701 −

1230500161v

4

24196548096 −

268897v

3

1714608 +

16675v

2

127008 −

11v

180

u−4 +



−

76938698802699r

4

12527254840352 −

6211549394313vr3

860382166336 +

20491256151r

3

7918618736 −

186196891v

2

r

2

40043584 +

1236331119vr2

346090976 −

1463625r

2

2502724 

u−3 +



+

1778984341v

3

r

2210193216 +

77947v

2

r

44296 −

65875vr

69608 +

75r

1582 +

454075v

4

4148928 −

2032591v

3

24004512 −

1025v

2

3528 +

25v

252

u−3 +



12528089829r

4

1124897312 +

7466191437vr3

646417856 −

22282569r

3

3652264 +

441v

2

r

2

484 −

18974931vr2

2860256 +

103671r

2

47432 

u−2 +



−

111033673v

3

r

66975552 −

49v

2

r

66 +

15479vr

6328 −

75r

308 −

2401v

4

39366 +

267775v

3

889056 +

49v

2

162 −

25v

84 

u−2 +



−

211569041160693r

4

12527254840352 −

7860884553897vr3

860382166336 +

85890653865r

3

7918618736 +

186196891v

2

r

2

40043584 +

2047213695vr2

346090976 −

13246263r

2

2502724 

u−1 +



+

3480659477v

3

r

2210193216 −

77947v

2

r

44296 −

198763vr

69608 +

2637r

1582 −

454075v

4

4148928 −

9562127v

3

24004512 +

1025v

2

3528 +

221v

252 

u−1 +



28605737830234834755r

4

1467292304940749056 −

278097527037915r

3

21079363075232 −

72837778155v

2

r

2

9690547328 +

8619732963r

2

1211318416 

u0 +



+

19447891v

2

r

5847072 −

50889r

17402 +

2706289217v

4

12098274048 −

55091v

2

63504 + 1

u0 +



−

211569041160693r

4

12527254840352 +

7860884553897vr3

860382166336 +

85890653865r

3

7918618736 +

186196891v

2

r

2

40043584 −

2047213695vr2

346090976 −

13246263r

2

2502724 

u1 +



−

3480659477v

3

r

2210193216 −

77947v

2

r

44296 +

198763vr

69608 +

2637r

1582 −

454075v

4

4148928 +

9562127v

3

24004512 +

1025v

2

3528 −

221v

252 

u1 +



12528089829r

4

1124897312 −

7466191437vr3

646417856 −

22282569r

3

3652264 +

441v

2

r

2

484 +

18974931vr2

2860256 +

103671r

2

47432 

u2 +



+

111033673v

3

r

66975552 −

49v

2

r

66 −

15479vr

6328 −

75r

308 −

2401v

4

39366 −

267775v

3

889056 +

49v

2

162 +

25v

84 

u2 +



−

76938698802699r

4

12527254840352 +

6211549394313vr3

860382166336 +

20491256151r

3

7918618736 −

186196891v

2

r

2

40043584 −

1236331119vr2

346090976 −

1463625r

2

2502724 

u3 +



−

1778984341v

3

r

2210193216 +

77947v

2

r

44296 +

65875vr

69608 +

75r

1582 +

454075v

4

4148928 +

2032591v

3

24004512 −

1025v

2

3528 −

25v

252

u3 +



6296179542799261293r

4

2934584609881498112 −

98816474558753811vr3

18549839506204160 −

31068146188125r

3

42158726150464 +

55178557611v

2

r

2

19381094656 +

287760797433vr2

133245025760 +

324338625r

2

2422636832 

u4 +



−

401096399245v

3

r

583491009024 −

3588625v

2

r

3898048 −

573327vr

1093840 −

75r

8701 −

1230500161v

4

24196548096 +

268897v

3

1714608 +

16675v

2

127008 +

11v

180

u4 (35)

The linearized viscous Burger’s equation (Eq. 15) is solved with this Compact scheme with multiple

domain sizes to demonstrate the dependence of the implicitly derived spatial derivatives (and the stability

limit) on the size of the domain. A Fourier stability analysis is performed and the stability of the Compact

scheme as a function of Courant (r =

c∆t

∆x

), Von Neumann (v =

μ∆t

∆x2 ) numbers, and β = ∆xk is derived as:

<(G) =

28605737830234834755r

4

1467292304940749056 −

278097527037915r

3

21079363075232 −

72837778155v

2

r

2

9690547328 +

8619732963r

2

1211318416

+

19447891v

2

r

5847072 −

50889r

17402 +

2706289217v

4

12098274048 −

55091v

2

63504 +



−

211569041160693r

4

6263627420176 +

85890653865r

3

3959309368 +

186196891v

2

r

2

20021792 −

13246263r

2

1251362 

cos(β) +



−

77947v

2

r

22148 +

2637r

791 −

454075v

4

2074464 +

1025v

2

1764 

cos(β) +



12528089829r

4

562448656 −

22282569r

3

1826132 +

441v

2

r

2

242 +

103671r

2

23716 

cos(2β) +



−

49v

2

r

33 −

75r

154 −

2401v

4

19683 +

49v

2

81 

cos(2β) +

NASA/TM—2005-213976 7


−

76938698802699r

4

6263627420176 +

20491256151r

3

3959309368 −

186196891v

2

r

2

20021792 −

1463625r

2

1251362 

cos(3β) +



+

77947v

2

r

22148 +

75r

791 +

454075v

4

2074464 −

1025v

2

1764 

cos(3β) +



6296179542799261293r

4

1467292304940749056 −

31068146188125r

3

21079363075232 +

55178557611v

2

r

2

9690547328 +

324338625r

2

1211318416 

cos(4β)



−

3588625v

2

r

1949024 −

150r

8701 −

1230500161v

4

12098274048 +

16675v

2

63504 

cos(4β) + 1 (36)

The imaginary part of the amplification factor is:

=(G) =



7860884553897vr3

430191083168 −

2047213695vr2

173045488 −

3480659477v

3

r

1105096608 +

198763vr

34804 +

9562127v

3

12002256 −

221v

126 

sin(β)

+



−

7466191437vr3

323208928 +

18974931vr2

1430128 +

111033673v

3

r

33487776 −

15479vr

3164 −

267775v

3

444528 +

25v

42 

sin(2β)

+



6211549394313vr3

430191083168 −

1236331119vr2

173045488 −

1778984341v

3

r

1105096608 +

65875vr

34804 +

2032591v

3

12002256 −

25v

126

sin(3β)

+



−

98816474558753811vr3

9274919753102080 +

287760797433vr2

66622512880 −

401096399245v

3

r

291745504512 −

573327vr

546920 +

268897v

3

857304 +

11v

90 

sin(4β) (37)

The amplification factor is simply, |G| =

p

<(G)

2 + =(G)2, and the range of stability (green area) is

shown in Fig. 3. Notice how the stable region changes as the domain size, maxi (the number of grid points in

each direction), changes. Explicit spatial derivative operators do not exhibit this behavior since the stencil

size remains constant regardless of domain size.

V. Application of UHF Method

The MESA and Ultra Hi-Fi Methods are actually a procedure for designing ever more accurate numerical

methods in which additional information is stored at each cell or grid point. For this comparison, the solution

variable and up to it’s third spatial derivative will be stored at each grid. The notation, c4od, represents an

UHF method with a 4 point stencil and only the solution variable and up to d derivatives on the grid. The

basic procedure has been previously published for inviscid problems.20

The c4o0 UHF method will use a 4 point interpolation stencil to determine spatial derivatives as shown

in Fig 4. A simple 1st order Taylor series in time is used:

u

n+1

i = u

n + ut(∆t) (38)

in which the time derivative is found from the governing equation (Eq. 15) to be:

∂u

∂t = −cux + μuxx (39)

and the solution variables, u, ux, uxx, are interpolated to the center of the four-point stencil using:

u

n

i =

1

16



−ui− 3

2

+ 9ui− 1

2

+ 9ui+ 1

2

− ui+ 3

2



(40)

u

n

xi = −

−ui− 3

2

+ 27ui− 1

2

) − 27ui+ 1

2

+ ui+ 3

2

24∆x

(41)

u

n

xxi = −

−ui− 3

2

+ ui− 1

2

+ ui+ 1

2

− ui+ 3

2

2∆x

2

(42)

This results in a single time equation:

u

n+(1/2)

i =



μ∆t

2h

2

−

c∆t

24h

−

1

16

uui− 3

2

+



−

μ∆t

2h

2

+

9c∆t

8h

+

9

16

uui− 1

2

+



−

μ∆t

2h

2

−

9c∆t

8h

+

9

16

uui+ 1

2

+



μ∆t

2h

2

+

c∆t

24h

−

1

16

uui+ 3

2

(43)

NASA/TM—2005-213976 8
(a) MAXI=2 (b) MAXI=3

(c) MAXI=4

Figure 3. Lele Compact Linear Viscous Burgers Equation Stability Range

NASA/TM—2005-213976 9
n+1

n+1/2

n

4

4 4 44 4 4 4 4

-4 -3 -2 -1 0 1 2 3 4

4 4 4 4 4 4 444

Figure 4. UHF Staggered Grid Diagram

And after two time steps, the original ”unstaggered” grid has been updated with the following:

u

n+1

i = ui−3



μ∆t

2h

2

−

c∆t

24h

−

1

162

+

2



−

μ∆t

2h

2

+

9c∆t

8h

+

9

16

ui−2



μ∆t

2h

2

−

c∆t

24h

−

1

16

+

 

−

μ∆t

2h

2

+

9c∆t

8h

+

9

162

+ 2 −

μ∆t

2h

2

−

9c∆t

8h

+

9

16 μ∆t

2h

2

−

c∆t

24h

−

1

16!

ui−1

+



2



−

μ∆t

2h

2

−

9c∆t

8h

+

9

16 −

μ∆t

2h

2

+

9c∆t

8h

+

9

16

+ 2 

μ∆t

2h

2

−

c∆t

24h

−

1

16 μ∆t

2h

2

+

c∆t

24h

−

1

16 ui +

 

−

μ∆t

2h

2

−

9c∆t

8h

+

9

162

+ 2 −

μ∆t

2h

2

+

9c∆t

8h

+

9

16 μ∆t

2h

2

+

c∆t

24h

−

1

16!

ui+1 +

2



−

μ∆t

2h

2

−

9c∆t

8h

+

9

16 μ∆t

2h

2

+

c∆t

24h

−

1

16

ui+2

+



μ∆t

2h

2

+

c∆t

24h

−

1

162

ui+3 (44)

As was done with the Compact algorithm, a Fourier stability analysis is completed with the real part of

amplification factor:

<(G) = (1152)−1(−2920v

2 + 288r(4r − 5) + 9 

348v

2 + 16(1 − 4r)r + 63

cos(β)

−18 12v

2 + 16r(4r − 5) + 9

cos(2β) + 9(1 − 8r)

2 + 4v2



cos(3β) + 738) (45)

And the imaginary part of the amplification factor:

=(G) = 1

48

v(−8r + (8r − 1) cos(β) + 5)(sin(2β) − 26 sin(β)) (46)

The amplification factor, |G| =

p

<(G)

2 + =(G)2, is plotted as in Fig. 5 as a function of Courant and

Von Neumann numbers, and β = ∆xk. The left figure (a) unwraps the polar plot in right figure (b). As

β increases the amplification factor simply repeats. It has been traditional to use a polar plot due to this

property. However, as soon as derivative information is stored on the grid, the amplification factor continues

to vary as β > 2π. This is due to the ability of these techniques to carry ultra-short wave information.

Amplification factors larger than one for any wavenumber imply it is an unstable method.

Following this same procedure, but adding additional derivatives on the grid to produce methods, c4o1,

c4o2, and c4o3 results in the algorithms shown in the appendix.

NASA/TM—2005-213976 10
(a) Amplification - c4o0 (b) Polar Diagram

Figure 5. c4o0 - Linear Viscous Burgers Equation

The full equations shown in the appendix are required to perform the Fourier stability analysis to enable

true method comparisons. The exact amplification factor is also shown since we know the exact solution.

Notice in Fig. 6 that the various UHF methods more closely approach the exact ampification factor as the

number of solution derivatives on the grid increases. Since the UHF methods can resolve ultra-short waves,

the wavenumber range in the figure could be extended past k = 2π.

The stability region for all the UHF methods is shown in Fig. 7. This gives an indication of the allowable

time step as the dynamic viscosity and convective terms vary. Notice (see definition of Von Neuman number

given earlier) the role viscosity plays in reducing time step size. Fortunately, the viscosity, μ is a small term

generally compared to convection, c. The Compact scheme has larger allowable time steps as shown in Fig. 3

for a given grid. However, more grid is required for the Compact scheme as shown in the next section and

despite this apparent advantage, a coarser grid actually results in an effectively larger time step for the UHF

methods.

VI. Application of the SIMPLE/PISO Methods

The Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) method28 and Pressure Implicit

with Splitting of Operators (PISO) method29, 30 are the mainstay of commercial fluid dynamics solvers.

An overview of the SIMPLE method is:

• Start the iterative process by guessing the pressure field.

• Use those pressure values to determine the velocity from the momentum equations.

• Determine a pressure correction such that the continuity equation is satisfied.

• Find corresponding velocity corrections, and use new pressure and velocity.

• Repeat this until a velocity field is found that does satisfy continuity.

The governing equations are linearized to produce a system of linear equations with one equation for

each cell in the domain. A point implicit (Gauss-Seidel) linear equation solver is used in conjunction with

an algebraic multigrid method to solve the resultant scalar system of equations.

The time stepping is first order accurate implicit, and the spatial accuracy is second order.

NASA/TM—2005-213976 11
(a) r=v=.01 (b) r=v=.05

(c) r=v=.10 (d) r=v=.5

Figure 6. Ampification/Stability Factor Comparison

NASA/TM—2005-213976 12
(a) c4o0 (b) c4o1

(c) c4o2 (d) c4o3

Figure 7. UHF Linear Viscous Burgers Equation Stability Range

NASA/TM—2005-213976 13
(a) Segregated Process

Figure 8. Overview of Segregated Solution Technique

The PISO algorithm moves the repeated calculations required by SIMPLE inside the solution stage of

the pressure-correction equation to more closely satisfy the continuity and momentum equations. The PISO

method takes more time per iteration, but often requires fewer iterations, particularly for transient problems

as will be demonstrated next.

Finally, the stability limits of these two approaches are large (limited by the need for accuracy) compared

to the Compact and UHF methods due to implicit time-stepping.

NASA/TM—2005-213976 14
VII. Nonlinear Navier Stokes – 1D

We will now test these techniques on an example of the one-dimensional Navier-Stokes equations that

reduces to a one-dimensional heat transfer problem governed by the linear viscous Burger’s equation. This

is what is solved in Sage, HFAST, and GLIMPS to produce reasonable Stirling analysis solutions.

The one-dimensional Navier-Stokes equations can be written as:23

∂U

∂t +

∂E

∂x =

∂Ev

∂x (47)

U =





ρ

ρu

Et





E =





ρu

ρu2 + p

(Et + p)u





Ev =





0

τxx

−uτxx + qx



 (48)

Et = ρ



e +

u

2

2



τxx =

4

3

μux qx = −k

∂T

∂x T =

p

ρR

k =

μcp

P r

(49)

Sage/Glimps32 utilize a slightly different form:

∂U

∂t +

∂E

∂x =

∂Ev

∂x + Q (50)

U =





ρ

ρu

Et





E =





ρu

ρu2 + p

(Et + p)u





Ev =





0

0

qi



 Q =





0

Cf ρu

Q5



 (51)

Et = ρ



e +

u

2

2



τxx =

4

3

μux qi = − < k > ∂T

∂x

ρ|u|

< ρ|u| >

T =

p

ρR

k =

μcp

P r

(52)

Notice that Stoke’s stress tensor is replaced by source terms. This also simplifies the numerics since only

a single second derivative, ∂qi

∂x , must be calculated compared to three in Eq. 47. We will utilize the standard

Navier-Stokes form for comparison with commercial software. Extension of these ideas to the GLIMPS form

is direct.

A. Convection and Diffusion

Few exact solutions exist for the Navier-Stokes equations and therefore validating commercial codes is typically done with only approximate experimental information. One special case that both has an exact solution

and yet includes heat transfer physics relevant to oscillating Stirling engines will be shown.

First, commercial codes operate either in two or three dimensions. However, it is possible to cajole the

commercial code into solving a one-dimensional problem by solving the full Navier-Stokes equations with

an inherently one-dimensional problem such as flow through a pipe with an initial temperature ”shock” as

shown in Fig. 9.

u=1 m/s u=1 m/s

300K

✲ ✲

600K

-2 m 0 m 2 m

Figure 9. Heat Transfer Test

This problem can be solved with either two or three-dimensional solvers and it reduces to the onedimensional Navier-Stokes Eq. 47. Since the density and velocity are constant, both the continuity and

momentum equations are satisfied. Only the energy equation actually needs to be solved:

∂Et

∂t +

∂

∂x 

(ρCvT + p)u −

4

3

μuux + qx



= 0 (53)

NASA/TM—2005-213976 15
Using, Et = ρCvT, and dropping out terms that are zero, we have the following reduced form of the

energy equation:

ρCv

∂T

∂t = ρCvu

∂T

∂x = k

∂

2T

∂x2

(54)

Finally, dividing by ρCv and for essentially incompressible flows replacing Cv with Cp, we have the

following transport (linear viscous Burger’s equation):

∂T

∂t + u

∂T

∂x = α

∂

2T

∂x2

(55)

with the thermal diffusivity α =

k

ρCp

.

Furthermore, commercial solvers utilize the dimensional form of the Navier-Stokes, but by utlizing the

following non-dimensionalization (∗ = nondimensional quantity):

T

∗ =

T

T0

, x∗ =

x

L

, t∗ =

t

τ0

, u∗ = u

τ0

L

, α∗ =

τ0

L2

α (56)

and using the following nondimensional boundary conditions:

T

∗

(x∗, 0) = 2 or 1, x ∈ [−2, 2], t∗ = [0, 1], u∗ = 1.0, ∆x

∗ = .1, ∆t∗ = .01, α∗ = .03003 (57)

We have an exact solution, by the separation-of-variables,33 (as N is large):

T

∗

(x

∗

, t∗) = 1 + 0.5 −

2

π

X

N

k=1

sin (2k − 1)π(x

∗ − u∗

t

∗

)

L



exp[−α

∗

(2k − 1)2π

2

t

∗/L2

]

2k − 1

(58)

And finally, by choosing the characteristic constants as:

τ0 = 1s, L = 1m, T0 = 300K (59)

We have the following problem definition for the commercial solvers:

x ∈ [−2m, 2m], T ∈ [300K, 600K], t ∈ [0s, 1s], u = 1m/s, ∆x = .1m, ∆t = .01s,

α = .03003m2/s, ρ = 998.2kg/m2, Cp = 4182J/(kg · K), k = 125359J/(s · K), p = 101325P a (60)

This problem is solved and the results are compared as various techniques are applied to the problem.

In Fig. 10, the exact solution is shown along with four UHF techniques, a 6th order Compact scheme, a

segregated spatially and temporally implicit method (SIMPLE) used in Fluent, a coupled spatially and

temporally implicit method (PISO) used in Fluent, and a segregated central difference with a blended

(averaged) Euler and Crank-Nicolson technique used in CFD-ACE. Fluent only allows 1st order accuracy in

time when moving meshes are applied as when the Stirling engine is simulated. However, CFD-ACE allows

for up to 2nd order accuracy in time when the meshes are compressed with a spring analogy.

The best techniques in this test case are the Compact and c4o3 UHF schemes. The commercially used

solvers are noticeably less able to model time accurate heat transfer. However, the coupled solver should

be used when commercial codes are applied to oscillating Stirling simulations and when available, 2nd order

time accuracy should be used (as when using CFD-ACE).

VIII. Fidelity and Turbulence Transition

Modeling turbulence transition is a difficult problem due to the large disparity in both spatial and

temporal scales caused when velocity gradients are high. In the Stirling engine velocity gradients are high

near walls and regions of sheared flow due to oscillating/reversing flows. As the velocity gradients increase,

the flow becomes rotational, leading to a vigorous stretching of vortex lines, which cannot be supported in

two dimensions.24 For this reason, truly turbulent simulations cannot be done in one-dimension.

NASA/TM—2005-213976 16
(a) Bird’s View (b) Temperature Slope

(c) Close-Up (d) Commercial Comparison

(e) Lower Curve (f) Upper Curve

Figure 10. Heat Diffusion and Convection Comparison

NASA/TM—2005-213976 17
The Stirling engine exhibits turbulent and laminar behaviour simultaneously.34 It is desirable therefore

to avoid a Reynolds Averaged approach since the time averaged equations combined with some turbulence

model35, 36, 37 assumes turbulence everywhere. One promising approach is Large Eddy Simulation (LES) in

which the Navier-Stokes equations are solved in time, but with spatial filtering applied, leaving the small

eddies still unresolved. Since small eddies are essentially isotropic, the modeling is much easier compared

to Reynolds time averaging. Moreover, the entire flow is bounded by walls making boundary condition

specification much easier than the typical open domain problems encountered in modeling, for example, jet

flow turbulence.

The smallest scales of turbulence are the Kolmogorov scales of length, time and velocity:38

η = (ν

3

/)

1/4

, τ = (ν/)

1/2

, v = (ν)

1/4

(61)

where ν is the kinematic viscosity, and  is the dissipation rate. The Reynolds number in the Kolmogorov

region, Re = vη/ν = 1, shows the ratio of inertial and visous forces is unity because most of the energy is

dissipated in this wavenumber region.

In small Stirling engines we can estime the smallest scales as follows. With the average flow assumptions

for Helium in the engine:

μ = 350e − 7(Ns/m2), ν = 502e − 6(m2/s), k = 278e − 3W/(mK), α = 29.9341e − 6m2/s,

P r = 0.654, ω = 2πf = 502.655(rad/s), Cp = 5.19e3J/(kgK), T = 700K, p = 2.6e6P a, ρ = 1.78838(kg/m3) (62)

Then from West,39 for oscillating flow, the average thermal boundary layer thickness, p2α/ω = .345115mm,

and the average flow boundary layer thickness, p

2ν/ω = 1.41329mm.

The maximum surface shear stress may be approximated with this information by (assuming average

maximum flow speed of 10m/s and using the flow boundary layer thickness):

τw = μ

∆u

∆y

= .247649N/m2

(63)

The friction velocity, uτ =

qτw

ρ = .372125m/s, is then used in the following equation for dissipation in

channel flow:38

 ≈ 2u

2

τUm/.0253007m) = 54.7325m2

/s3(64)

Finally, we can estimate the Kolmogorv spatial wave length as:

η = (ν

3

/)

1/4 = 1.23301mm (65)

Assuming a representative engine length of (1in. = 25.3007mm), the domain consists of 20 to 30 Kolmogorov wavelengths, or roughly 8000 regions (Kolmogorov boxes) in 3D where isotropic turbulence modeling can be employed. This corresponds to a turbulent Reynolds number of Reτ = uτ (.0253007m)/(2ν) =

9.37751. This is low Reynolds number flow and appears ideal for applying Large Eddy Simulation.

One would like to model the smallest turbulent scales, which in the case of LES is the Kolmogorov

wavenumber range. The allowable time should also be of the same order as the Kolmogorov time scale.

τ = (ν/)

1/2 = .00302s (66)

This time scale is not prohibitive, since the time step size is already smaller than this in current commercial

simulations due to the numerical issues involving moving grids.

We would like to utilize the most efficient technique to minimize the computational cost. We know

the minimum wavelengths required to simulate turbulence in the Stirling engine and the wave convection/dissipation problem solution from Eq. 17 is used to determine the fewest grid points required for each

method. The results are shown in Fig. 11.

This solution represents a traveling (convecting) wave with a dissipating amplitude. Many methods will

convect at the wrong speed (dispersive error) or will excessively dissipate the amplitude (dissipative error).

A key measure of efficiency is how many grid points are required per wavelength to propagate this wave

within some predetermined error bound. As k, the wavenumber in Eq. 17 , is increased, the number of grid

points must increase to properly simulate the wave.

NASA/TM—2005-213976 18
(a) Fidelity Comparison

Method Spacing Error

c4o0 .1 2.54229 10−2

c4o0 .2 4.27563 10−2

c4o0 .4 4.68577 10−2

c4o1 .1 3.11163 10−6

c4o1 .2 2.96551 10−5

c4o1 .4 8.4702 10−4

c4o2 .1 1.0178 10−10

c4o2 .2 3.1935 10−9

c4o2 .4 6.41079 10−8

c4o3 .1 3.44169 10−15

c4o3 .2 2.27818 10−13

c4o3 .4 3.12925 10−11

Compact .1 1.0993 10−6

Compact .2 7.10929 10−5

Compact .4 5.31245 10−3

(b) Errors at t=1

Figure 11. Wave Propagation Fidelity

NASA/TM—2005-213976 19
Currently, Compact schemes are regarded as requiring approximately 6 grid points per wavelength for

reasonable solutions. A comparison of the Compact scheme and the UHF schemes with up to 3 spatial

derivatives stored on the grid, as shown in the table (Fig. 11(b), demonstrates the Compact scheme is

comparable to method c4o1 (one solution derivative per grid point). This is expected since the Compact

scheme also utilizes 1st derivative information. However, extending Compact schemes to include higher

derivatives involves complicated matrices which may be difficult or intractable to solve.

The c4o3 method can match the results of the Compact scheme using 16 times fewer grid points per

dimension. Specifically, one additional test of the c4o3 method with k = 8π and ∆x = .2, the error at t = 1

was 2.44291 ∗

− 5, which is more accurate than the Compact scheme with considerably fewer grid points per

wavelength. Note that the previous comparisons had k = π and here it was multiplied by eight and we use

half as many grid points for a net change of sixteen grid points per wavelength.

Clearly, the UHF schemes are more efficient and can be formulated explicitly for easier parallelization.

This implies smoothly transitioning turbulent flows can be more efficiently simulated with UHF techniques.

IX. Conclusion

One-dimensional Navier-Stokes equations are currently utilized for Stirling engine design and optimization

with reasonable success. Recent attempts at multi-dimensional simulations have relied upon commercial

solvers and this report examined that practice more closely.

This report has shown that the techniques used in commercial codes for simulating Stirling engines are

not as capable as more recently developed approaches available in the literature. Moreover, the unique

environment of the Stirling engine in which flow is simultaneously turbulent and laminar makes large eddy

simulation desirable, while the low Reynold’s number, wall bounded flow provides for modest grid requirements and well defined boundary conditions.

Despite the larger stability limit (4 times larger) of Compact schemes for a given grid spacing, the UHF

method results in an effective time-step that is 4 times larger than Compact schemes since the grid can be 16

times coarser per dimension. The 6th order Compact schemes performed well with the heat transfer test and

apparently would work well in regions of conjugate heat transfer. However, the Compact scheme is not as

efficient at predicting turbulent transition compared to UHF methods. The c4o3 method performs similarly

to the Compact scheme for steep temperature gradients (conjugate heat transfer) but is up to 163 = 4096

times more efficient when three-dimensional transitional flows need modeled.

It would be desirable to compare c4o4 methods and higher in the future. Future work should examine utilizing UHF methods in a steady-harmonic formulation with Detached Eddy Simulation and more

complicated moving grid tests should be performed.

NASA/TM—2005-213976 20
X. Appendix

In what follows are the full equations used to time advance the solutions in this paper and their stability

amplification factors. These equations are lengthy, but are provided for completeness and to allow for

independent verification.

A. c4o1

The single step c4o1 is given by:

u

n+(1/2)

i =



−

15μ

3∆t3

h6 +

785cμ2∆t

3

36h5 +

45c

2μ∆t3

8h4 −

281c

3∆t3

1728h3 +

45μ

2∆t2

8h4 −

281cμ∆t

2

288h3 −

27c

2∆t2

128h2 −

27μ∆t

64h2 +

5c∆t

256h +

13

512

ui− 3

2

+



15μ

3∆t3

h6 +

345cμ2∆t

3

4h5 −

45c

2μ∆t3

8h4 −

171c

3∆t3

64h3 −

45μ

2∆t2

8h4 −

513cμ∆t

2

32h3 +

27c

2∆t2

128h2 +

27μ∆t

64h2 +

405c∆t

256h +

243

512

ui− 1

2

+



15μ

3∆t3

h6 −

345cμ2∆t

3

4h5 −

45c

2μ∆t3

8h4 +

171c

3∆t3

64h3 −

45μ

2∆t2

8h4 +

513cμ∆t

2

32h3 +

27c

2∆t2

128h2 +

27μ∆t

64h2 −

405c∆t

256h +

243

512

ui+ 1

2

+



−

15μ

3∆t3

h6 −

785cμ2∆t

3

36h5 +

45c

2μ∆t3

8h4 +

281c

3∆t3

1728h3 +

45μ

2∆t2

8h4 +

281cμ∆t

2

288h3 −

27c

2∆t2

128h2 −

27μ∆t

64h2 −

5c∆t

256h +

13

512

ui+ 3

2

+



−

5μ

3∆t3

h5 +

55cμ2∆t

3

12h4 +

11c

2μ∆t3

8h3 −

19c

3∆t3

576h2 +

11μ

2∆t2

8h3 −

19cμ∆t

2

96h2 −

19c

2∆t2

384h +

c∆t

256 −

19μ∆t

192h +

3h

512

uxi− 3

2

+



−

15μ

3∆t3

h5 +

285cμ2∆t

3

4h4 +

57c

2μ∆t3

8h3 −

99c

3∆t3

64h2 +

57μ

2∆t2

8h3 −

297cμ∆t

2

32h2 −

99c

2∆t2

128h +

81c∆t

256 −

99μ∆t

64h +

81h

512 

uxi− 1

2

+



15μ

3∆t3

h5 +

285cμ2∆t

3

4h4 −

57c

2μ∆t3

8h3 −

99c

3∆t3

64h2 −

57μ

2∆t2

8h3 −

297cμ∆t

2

32h2 +

99c

2∆t2

128h +

81c∆t

256 +

99μ∆t

64h −

81h

512 

uxi+ 1

2

+



5μ

3∆t3

h5 +

55cμ2∆t

3

12h4 −

11c

2μ∆t3

8h3 −

19c

3∆t3

576h2 −

11μ

2∆t2

8h3 −

19cμ∆t

2

96h2 +

19c

2∆t2

384h +

c∆t

256 +

19μ∆t

192h −

3h

512

uxi+ 3

2

(67)

(ux)

n+1

i =



770μ

3∆t3

9h7 +

45cμ2∆t

3

h6 −

785c

2μ∆t3

36h5 −

15c

3∆t3

8h4 −

785μ

2∆t2

36h5 −

45cμ∆t

2

4h4 +

281c

2∆t2

576h3 +

281μ∆t

288h3 +

27c∆t

64h2 −

5

256h



ui− 3

2

+



210μ

3∆t3

h7 −

45cμ2∆t

3

h6 −

345c

2μ∆t3

4h5 +

15c

3∆t3

8h4 −

345μ

2∆t2

4h5 +

45cμ∆t

2

4h4 +

513c

2∆t2

64h3 +

513μ∆t

32h3 −

27c∆t

64h2 −

405

256h



ui− 1

2

+



−

210μ

3∆t3

h7 −

45cμ2∆t

3

h6 +

345c

2μ∆t3

4h5 +

15c

3∆t3

8h4 +

345μ

2∆t2

4h5 +

45cμ∆t

2

4h4 −

513c

2∆t2

64h3 −

513μ∆t

32h3 −

27c∆t

64h2 +

405

256h



ui+ 1

2

+



−

770μ

3∆t3

9h7 +

45cμ2∆t

3

h6 +

785c

2μ∆t3

36h5 −

15c

3∆t3

8h4 +

785μ

2∆t2

36h5 −

45cμ∆t

2

4h4 −

281c

2∆t2

576h3 −

281μ∆t

288h3 +

27c∆t

64h2 +

5

256h



ui+ 3

2

+



70μ

3∆t3

3h6 +

15cμ2∆t

3

h5 −

55c

2μ∆t3

12h4 −

11c

3∆t3

24h3 −

55μ

2∆t2

12h4 −

11cμ∆t

2

4h3 +

19c

2∆t2

192h2 +

19μ∆t

96h2 +

19c∆t

192h −

1

256

uxi− 3

2

+



210μ

3∆t3

h6 +

45cμ2∆t

3

h5 −

285c

2μ∆t3

4h4 −

19c

3∆t3

8h3 −

285μ

2∆t2

4h4 −

57cμ∆t

2

4h3 +

297c

2∆t2

64h2 +

297μ∆t

32h2 +

99c∆t

64h −

81

256

uxi− 1

2

+



210μ

3∆t3

h6 −

45cμ2∆t

3

h5 −

285c

2μ∆t3

4h4 +

19c

3∆t3

8h3 −

285μ

2∆t2

4h4 +

57cμ∆t

2

4h3 +

297c

2∆t2

64h2 +

297μ∆t

32h2 −

99c∆t

64h −

81

256

uxi+ 1

2

+



70μ

3∆t3

3h6 −

15cμ2∆t

3

h5 −

55c

2μ∆t3

12h4 +

11c

3∆t3

24h3 −

55μ

2∆t2

12h4 +

11cμ∆t

2

4h3 +

19c

2∆t2

192h2 +

19μ∆t

96h2 −

19c∆t

192h −

1

256

uxi+ 3

2

(68)

The c4o1 full step (two staggered half-steps) is:

u

n+1

i = (69)



770μ

3∆t3

9h7 +

45cμ2∆t

3

h6 −

785c

2μ∆t3

36h5 −

15c

3∆t3

8h4 −

785μ

2∆t2

36h5 −

45cμ∆t

2

4h4 +

281c

2∆t2

576h3 +

281μ∆t

288h3 +

27c∆t

64h2 −

5

256h



ui− 3

2

+



210μ

3∆t3

h7 −

45cμ2∆t

3

h6 −

345c

2μ∆t3

4h5 +

15c

3∆t3

8h4 −

345μ

2∆t2

4h5 +

45cμ∆t

2

4h4 +

513c

2∆t2

64h3 +

513μ∆t

32h3 −

27c∆t

64h2 −

405

256h



ui− 1

2

+



−

210μ

3∆t3

h7 −

45cμ2∆t

3

h6 +

345c

2μ∆t3

4h5 +

15c

3∆t3

8h4 +

345μ

2∆t2

4h5 +

45cμ∆t

2

4h4 −

513c

2∆t2

64h3 −

513μ∆t

32h3 −

27c∆t

64h2 +

405

256h



ui+ 1

2

+



−

770μ

3∆t3

9h7 +

45cμ2∆t

3

h6 +

785c

2μ∆t3

36h5 −

15c

3∆t3

8h4 +

785μ

2∆t2

36h5 −

45cμ∆t

2

4h4 −

281c

2∆t2

576h3 −

281μ∆t

288h3 +

27c∆t

64h2 +

5

256h



ui+ 3

2

NASA/TM—2005-213976 21
+



70μ

3∆t3

3h6 +

15cμ2∆t

3

h5 −

55c

2μ∆t3

12h4 −

11c

3∆t3

24h3 −

55μ

2∆t2

12h4 −

11cμ∆t

2

4h3 +

19c

2∆t2

192h2 +

19μ∆t

96h2 +

19c∆t

192h −

1

256

uxi− 3

2

+



210μ

3∆t3

h6 +

45cμ2∆t

3

h5 −

285c

2μ∆t3

4h4 −

19c

3∆t3

8h3 −

285μ

2∆t2

4h4 −

57cμ∆t

2

4h3 +

297c

2∆t2

64h2 +

297μ∆t

32h2 +

99c∆t

64h −

81

256

uxi− 1

2

+



210μ

3∆t3

h6 −

45cμ2∆t

3

h5 −

285c

2μ∆t3

4h4 +

19c

3∆t3

8h3 −

285μ

2∆t2

4h4 +

57cμ∆t

2

4h3 +

297c

2∆t2

64h2 +

297μ∆t

32h2 −

99c∆t

64h −

81

256

uxi+ 1

2

+



70μ

3∆t3

3h6 −

15cμ2∆t

3

h5 −

55c

2μ∆t3

12h4 +

11c

3∆t3

24h3 −

55μ

2∆t2

12h4 +

11cμ∆t

2

4h3 +

19c

2∆t2

192h2 +

19μ∆t

96h2 −

19c∆t

192h −

1

256

uxi+ 3

2

(70)

The real part of c4o1 amplification factor:

<(G) = (13824−1)(32 1084v

3 − 3(8r(1160r − 271) + 225)v + 216

cos(β) +



2248v

3 + 972(80r − 3)v2 − 48r(6280r − 281)v − 270v − 648r(40r(8r − 3) + 9) + 351

cos(2β) −

48β



5760r

3 + 48(400v − 51)r2 − 6



408v

2 + 436v − 79

r + v((237 − 436v)v + 90)sin(β) +

81 2560r

3 + 320(46v − 3)r2 − 24(2v(20v + 57) − 3)r + 6v



−76v

2 + 6v + 45

+ 28β sin(β) + 81−

3β



23040r

3 + 2112(10v − 3)r2 − 24 

264v

2 + 38v − 19

r − (2v − 3) 76v

2 − 9

 sin(2β)) (71)

The imaginary part of c4o1 amplification factor:

=(G) = (13824−1)(−81β



2560r

3 − 1216(10v + 1)r2 − 8(2v(76v − 99) − 33)r + 3(2v + 1) 

44v

2 − 9

 +

6β



9(82v − 39) + 4 −910v

3 − 3(1104r − 139)v2 + 5460r(8r − 1)v + 6r(24r(40r − 23) + 139) cos(β) +

3β



23040r

3 + 2112(10v − 3)r2 − 24 

264v

2 + 38v − 19

r − (2v − 3) 76v

2 − 9

 cos(2β)

+2 19592v

3 − 972(80r − 3)v2 − 6(8r(15560r − 2449) + 1845)v + 27(24r(40r(8r − 3) + 9) + 115)

sin(β)

+



2248v

3 + 972(80r − 3)v2 − 48r(6280r − 281)v − 270v − 648r(40r(8r − 3) + 9) + 351

sin(2β)) (72)

B. c4o2

The single step c4o2 (with r = μ

∆t

∆x2 and v = c

∆t

∆x

is:

u

n+(1/2)

i =



2835r

5 −

29575vr4

4 −

7875r

4

8 −

7875v

2

r

3

4 +

113995vr3

144 

ui− 3

2

+



+

4425r

3

32 +

113995v

3

r

2

288 +

13275v

2

r

2

64 −

111115vr2

3456 −

2025r

2

256 

ui− 3

2

+



+

4425v

4

r

128 −

111115v

3

r

10368 −

2025v

2

r

256 +

13297vr

18432 +

1215r

4096 

ui− 3

2

+



−

22223v

5

41472 −

675v

4

1024 +

13297v

3

110592 +

1215v

2

8192 −

155v

16384 −

383

32768

ui− 3

2

+



−2835r

5 +

250425vr4

4 +

7875r

4

8 +

7875v

2

r

3

4 −

148365vr3

16 −

4425r

3

32 

ui− 1

2

+



−

148365v

3

r

2

32 −

13275v

2

r

2

64 +

98415vr2

128 

ui− 1

2

+



+

2025r

2

256 −

4425v

4

r

128 +

32805v

3

r

128 +

2025v

2

r

256 −

80919vr

2048 −

1215r

4096 

ui− 1

2

+



+

6561v

5

512 +

675v

4

1024 −

26973v

3

4096 −

1215v

2

8192 +

32805v

16384 +

16767

32768

ui− 1

2

+



−2835r

5 −

250425vr4

4 +

7875r

4

8 +

7875v

2

r

3

4 +

148365vr3

16 

ui+ 1

2

+



−

4425r

3

32 +

148365v

3

r

2

32 −

13275v

2

r

2

64 −

98415vr2

128 

ui+ 1

2

+



+

2025r

2

256 −

4425v

4

r

128 −

32805v

3

r

128 +

2025v

2

r

256 +

80919vr

2048 −

1215r

4096 

ui+ 1

2

+



−

6561v

5

512 +

675v

4

1024 +

26973v

3

4096 −

1215v

2

8192 −

32805v

16384 +

16767

32768

ui+ 1

2

+

NASA/TM—2005-213976 22


2835r

5 +

29575vr4

4 −

7875r

4

8 −

7875v

2

r

3

4 −

113995vr3

144 +

4425r

3

32 −

113995v

3

r

2

288 

ui+ 3

2

+



+

13275v

2

r

2

64 +

111115vr2

3456 −

2025r

2

256 +

4425v

4

r

128 +

111115v

3

r

10368 

ui+ 3

2

+



−

2025v

2

r

256 −

13297vr

18432 +

1215r

4096 +

22223v

5

41472 −

675v

4

1024 −

13297v

3

110592 

ui+ 3

2

+



+

1215v

2

8192 +

155v

16384 −

383

32768

ui+ 3

2

+



105h

2

r

5 −

245h

2

r

4

8 −

735

4

h

2vr4 +

115h

2

r

3

32 −

245

4

h

2v2

r

3 +

805

48 h

2vr3



uxxi− 3

2

+



+

805

96 h

2v3

r

2 −

149h

2

r

2

768 +

345

64 h

2v2

r

2 −

745h

2

vr2

1152 

uxxi− 3

2

+



+

115

128h

2v4

r −

745h

2

v

3

r

3456 +

29h

2

r

4096 −

149

768h

2v2

r +

29h

2

vr

2048 −

149h

2

v

5

13824 

uxxi− 3

2

+



−

149h

2

v

4

9216 +

29h

2

v

3

12288 −

9h

2

32768 +

29h

2

v

2

8192 −

3h

2

v

16384

uxxi− 3

2

+



−945h

2

r

5 +

3045h

2

r

4

8 +

27405

4

h

2vr4 −

2235h

2

r

3

32 +

3045

4

h

2v2

r

3



uxxi− 1

2

+



−

15645

16 h

2vr3 −

15645

32 h

2v3

r

2 +

1863h

2

r

2

256 −

6705

64 h

2v2

r

2



uxxi− 1

2

+



+

9315

128 h

2vr2 −

2235

128 h

2v4

r +

3105

128 h

2v3

r −

1701h

2

r

4096 +

1863

256 h

2v2

r −

5103h

2

vr

2048 +

621h

2

v

5

512 

uxxi− 1

2

+



+

621h

2

v

4

1024 −

1701h

2

v

3

4096 +

729h

2

32768 −

1701h

2

v

2

8192 +

729h

2

v

16384 

uxxi− 1

2

+



−945h

2

r

5 +

3045h

2

r

4

8 −

27405

4

h

2vr4 −

2235h

2

r

3

32 +

3045

4

h

2v2

r

3 +

15645

16 h

2vr3 +

15645

32 h

2v3

r

2



uxxi+ 1

2

+



+

1863h

2

r

2

256 −

6705

64 h

2v2

r

2 −

9315

128 h

2vr2 −

2235

128 h

2v4

r −

3105

128 h

2v3

r −

1701h

2

r

4096 +

1863

256 h

2v2

r +

5103h

2

vr

2048 

uxxi+ 1

2

+



−

621h

2

v

5

512 +

621h

2

v

4

1024 +

1701h

2

v

3

4096 +

729h

2

32768 −

1701h

2

v

2

8192 −

729h

2

v

16384 

uxxi+ 1

2

+



105h

2

r

5 −

245h

2

r

4

8 +

735

4

h

2vr4 +

115h

2

r

3

32 −

245

4

h

2v2

r

3 −

805

48 h

2vr3 −

805

96 h

2v3

r

2 −

149h

2

r

2

768 +

345

64 h

2v2

r

2+



uxxi+ 3

2

+



745h

2

vr2

1152 +

115

128h

2v4

r +

745h

2

v

3

r

3456 +

29h

2

r

4096 −

149

768h

2v2

r −

29h

2

vr

2048 

uxxi+ 3

2

+



+

149h

2

v

5

13824 −

149h

2

v

4

9216 −

29h

2

v

3

12288 −

9h

2

32768 +

29h

2

v

2

8192 +

3h

2

v

16384

uxxi+ 3

2

+



1015hr5 −

7945hr4

24 −

8925

4

hvr4 −

7945

12 hv2

r

3 +

4055hr3

96 +

10535

48 hvr3



uxi− 3

2

+



+

10535

96 hv3

r

2 +

4055

64 hv2

r

2 −

5401hr2

2304 −

9995hvr2

1152 +



uxi− 3

2

+



4055

384 hv4

r −

9995hv3r

3456 −

5401hv2r

2304 +

355hr

4096 +

1181hvr

6144 −

1999hv5

13824 −

5401hv4

27648 

uxi− 3

2

+



+

1181hv3

36864 +

355hv2

8192 −

111h

32768 −

41hv

16384

uxi− 3

2

+



945hr5 −

2205hr4

8 +

89775

4

hvr4 −

2205

4

hv2r

3 +

555hr3

32 −

58695

16 hvr3



uxi− 1

2

+



−

58695

32 hv3

r

2 +

1665

64 hv2

r

2 +

1377hr2

256 +

44145

128 hvr2 +

555

128hv4

r



uxi− 1

2

+



14715

128 hv3

r +

1377

256 hv2

r −

6075hr

4096 −

38637hvr

2048 +

2943hv5

512 +

459hv4

1024 −

12879hv3

4096 −

6075hv2

8192 +

5103h

32768 +

8019hv

16384 

uxi− 1

2

+



−945hr5 +

2205hr4

8 +

89775

4

hvr4 +

2205

4

hv2r

3 −

555hr3

32 −

58695

16 hvr3 −

58695

32 hv3

r

2 −

1665

64 hv2

r

2



uxi+ 1

2

+



−

1377hr2

256 +

44145

128 hvr2 −

555

128hv4

r



uxi+ 1

2

+



14715

128 hv3

r −

1377

256 hv2

r +

6075hr

4096 −

38637hvr

2048 +

2943hv5

512 

uxi+ 1

2

+



−

459hv4

1024 −

12879hv3

4096 +

6075hv2

8192 −

5103h

32768 +

8019hv

16384 

uxi+ 1

2

+

NASA/TM—2005-213976 23


−1015hr5 +

7945hr4

24 −

8925

4

hvr4 +

7945

12 hv2

r

3 −

4055hr3

96 +

10535

48 hvr3 +

10535

96 hv3

r

2 −

4055

64 hv2

r

2+



uxi+ 3

2 

5401hr2

2304 −

9995hvr2

1152 −

4055

384 hv4

r −

9995hv3r

3456 +

5401hv2r

2304 −

355hr

4096 +

1181hvr

6144 

uxi+ 3

2 

−

1999hv5

13824 +

5401hv4

27648 +

1181hv3

36864 −

355hv2

8192 +

111h

32768 −

41hv

16384

uxi+ 3

2

(73)

The double step from u

n

i

to u

n+1

i

can be immediately derived by using two single steps above. The

equations are ommitted due to length.

1. c4o2 Amplification

The equations for the amplification factor of this method are sufficiently large that their Fortran form is

shown instead.

\Im(G)= (beta*v*(-130029412509 - 135591927894717235200*r**8 +

- 235688089173884928000*r**9 + 2053416202920*v**2 -

- 12925760780304*v**4 + 37818812555520*v**6 -

- 41659973309440*v**8 -

- 59454259200*r**7*(-629819125 + 3450959988*v**2) +

- 6193152000*r**6*(-1050833123 + 10984215328*v**2) +

- 7372800*r**5*(105946873317 - 1571117362052*v**2 +

- 2866080408576*v**4) -

- 368640*r**4*(179452656305 - 3380823300140*v**2 +

- 11635255448992*v**4) -

- 122880*r**3*(-33255849510 + 726770729579*v**2 -

- 3540668800920*v**4 + 3497307329712*v**6) +

- 192*r**2*(-955513044297 + 21845158240800*v**2 -

- 132867547584800*v**4 + 234180801868800*v**6) +

- 160*r*(37334893005 - 789506398980*v**2 + 5296053262104*v**4 -

- 12801424523744*v**6 + 7884402819840*v**8)))/7.3383542784e10\

- + (beta*v*(692362647 - 5462028784867737600*r**8 +

- 9469522436161536000*r**9 + 54343258344*v**2 -

- 535399711056*v**4 + 1707900809472*v**6 - 1864301675520*v**8 -

- 138726604800*r**7*(-11074625 + 58913076*v**2) +

- 2890137600*r**6*(-95039991 + 964377760*v**2) +

- 491520*r**5*(69443249579 - 998375458220*v**2 +

- 1771442856960*v**4) -

- 40960*r**4*(72884695365 - 1333734152300*v**2 +

- 4436424946464*v**4) -

- 40960*r**3*(-4525470270 + 98467155981*v**2 -

- 464360171848*v**4 + 442587839568*v**6) +

- 64*r**2*(-118439206263 + 2973197651040*v**2 -

- 17955426117600*v**4 + 30755816238080*v**6) +

- 32*r*(4848891741 - 163397666340*v**2 + 1199037538872*v**4 -

- 2885901281760*v**6 + 1738612309760*v**8))*Cos(beta))/

- 3.623878656e9 - (beta*v*

- (8037225 - 74750888824012800*r**8 + 129148920004608000*r**9 -

- 34263000*v**2 - 575412336*v**4 + 5181034752*v**6 -

- 10549410816*v**8 - 69363302400*r**7*(-276055 + 1711536*v**2) +

- 82575360*r**6*(-34655721 + 424199720*v**2) +

- 245760*r**5*(1083499015 - 20285505016*v**2 +

- 43052870784*v**4) -

- 40960*r**4*(362625615 - 10128202520*v**2 + 45084965856*v**4) -

- 2048*r**3*(-224343621 + 9745619820*v**2 - 72360806480*v**4 +

- 92721195840*v**6) +

- 64*r**2*(-76222053 + 7530750144*v**2 - 92593649088*v**4 +

NASA/TM—2005-213976 24
- 236994878720*v**6) +

- 8*r*(-22830093 - 572822928*v**2 + 13259861136*v**4 -

- 59079642624*v**6 + 50087375360*v**8))*Cos(2*beta))/

- 2.26492416e8 - (beta*v*

- (-164355237 + 534389864162918400*r**8 -

- 2562002937446400000*r**9 + 3250233864*v**2 - 26360268048*v**4 +

- 94274868480*v**6 - 127264420864*v**8 +

- 59454259200*r**7*(607585 + 17586156*v**2) +

- 247726080*r**6*(-81048203 + 261110560*v**2) -

- 1843200*r**4*(130398037 - 2750451916*v**2 + 9383105312*v**4) +

- 1474560*r**5*(1943667693 - 29362137364*v**2 +

- 22507355136*v**4) -

- 24576*r**3*(-538843482 + 12803987455*v**2 - 71553575640*v**4 +

- 66154527600*v**6) +

- 192*r**2*(-2579408793 + 65572339872*v**2 - 457570639904*v**4 +

- 1041142379520*v**6) +

- 32*r*(375564033 - 8984298708*v**2 + 70710058008*v**4 -

- 228004837984*v**6 + 212775217920*v**8))*Cos(3*beta))/

- 2.93534171136e11 + (v*

- (beta**2*(261842949 - 2194527814366003200*r**8 +

- 3724642866954240000*r**9 + 6159355992*v**2 -

- 95377169520*v**4 + 415611350784*v**6 - 559729462272*v**8 -

- 6606028800*r**7*(-91437875 + 508551372*v**2) +

- 1513881600*r**6*(-67322291 + 726020640*v**2) -

- 40960*r**4*(21099636639 - 445669712500*v**2 +

- 1634611927648*v**4) +

- 163840*r**5*(70054445343 - 1103008857580*v**2 +

- 2070815523840*v**4) -

- 40960*r**3*(-1054627614 + 28532469783*v**2 -

- 156407713592*v**4 + 164686577776*v**6) +

- 64*r**2*(-19978382781 + 697506487200*v**2 -

- 5261467218336*v**4 + 10338925880320*v**6) +

- 32*r*(489426543 - 28430441100*v**2 + 286355835144*v**4 -

- 845957587872*v**6 + 574710288640*v**8)) -

- 16*(134555175 - 1115128841895936000*r**8 +

- 1927070689001472000*r**9 + 9672938919*v**2 -

- 77637693564*v**4 + 255245658912*v**6 - 304576780800*v**8 -

- 111476736000*r**7*(-2734475 + 15275008*v**2) +

- 1135411200*r**6*(-45562643 + 488318640*v**2) -

- 307200*r**4*(1552020979 - 30919444147*v**2 +

- 110645478832*v**4) +

- 61440*r**5*(97159680473 - 1493878846040*v**2 +

- 2789562424320*v**4) -

- 20480*r**3*(-1325701170 + 31436422945*v**2 -

- 162423477334*v**4 + 166958390760*v**6) +

- 80*r**2*(-13555495233 + 349150858992*v**2 -

- 2308931970240*v**4 + 4292918703616*v**6) +

- 4*r*(6887181573 - 188364602520*v**2 + 1419968235096*v**4 -

- 3707569366400*v**6 + 2391656357120*v**8)))*Sin(beta))/

- 3.623878656e9 - (v*(1741890648224563200*r**8 -

- 3029855637602304000*r**9 +

- 178362777600*r**7*(-2524059 + 15419236*v**2) -

- 82575360*r**6*(-835546259 + 9961097280*v**2) +

- 122880*r**4*(3233787985 - 85348974940*v**2 +

- 363067915168*v**4) -

NASA/TM—2005-213976 25
- 491520*r**5*(13630408051 - 245209154932*v**2 +

- 507285257472*v**4) +

- 40960*r**3*(-324619695 + 13089759659*v**2 - 91073759432*v**4 +

- 111515155248*v**6) -

- 64*r**2*(-2872614537 + 218229409440*v**2 - 2469996553632*v**4 +

- 5974919813120*v**6) +

- 3*(-65098971 + 157579992*v**2 + 6234027408*v**4 -

- 48443322624*v**6 + 92309828608*v**8) -

- 64*r*(-53262927 - 2462239134*v**2 + 47336581836*v**4 -

- 197637357392*v**6 + 159618062720*v**8) +

- 2*beta**2*(4153113 - 37031679885312000*r**8 +

- 73234329580339200*r**9 - 39990024*v**2 + 24132816*v**4 +

- 1091715840*v**6 - 3011607552*v**8 -

- 33030144000*r**7*(-249575 + 1831788*v**2) +

- 27525120*r**6*(-38982313 + 549851040*v**2) -

- 40960*r**4*(91581147 - 3105008708*v**2 + 16241654240*v**4) +

- 163840*r**5*(519280935 - 11052165404*v**2 +

- 27241385472*v**4) -

- 8192*r**3*(-7629228 + 607651455*v**2 - 5642488280*v**4 +

- 8549724400*v**6) +

- 64*r**2*(34845471 + 1103282208*v**2 - 23979806496*v**4 +

- 71777561600*v**6) +

- 32*r*(-5093523 + 25329348*v**2 + 600224904*v**4 -

- 3713708832*v**6 + 3396972800*v**8)))*Sin(2*beta))/

- 1.811939328e9 - (v*(-3*beta**2*

- (4516155 - 6737118835507200*r**8 + 57173118183014400*r**9 -

- 90405720*v**2 + 745088112*v**4 - 2739588864*v**6 +

- 3881749504*v**8 - 178362777600*r**7*(15625 + 90076*v**2) -

- 247726080*r**6*(-3042061 + 20646880*v**2) -

- 1474560*r**5*(62705127 - 1050002828*v**2 +

- 1304454144*v**4) +

- 368640*r**4*(19645223 - 434387796*v**2 + 1661380448*v**4) +

- 24576*r**3*(-15672258 + 381876625*v**2 - 2275528200*v**4 +

- 2407898640*v**6) -

- 192*r**2*(-73413567 + 1893294048*v**2 - 13708750304*v**4 +

- 32725847040*v**6) -

- 32*r*(10491039 - 254143116*v**2 + 2039834376*v**4 -

- 6769699744*v**6 + 6493098240*v**8)) +

- 4*(139405941 - 634572669085286400*r**8 +

- 2429857522778112000*r**9 - 2722769964*v**2 +

- 21720929760*v**4 - 75428706432*v**6 + 96443866112*v**8 +

- 1981808640*r**6*(5826243 + 14884490*v**2) -

- 29727129600*r**7*(-498163 + 39563832*v**2) -

- 15482880*r**5*

- (131848619 - 1741393344*v**2 + 166073344*v**4) +

- 2580480*r**4*(71794645 - 1435990910*v**2 +

- 4239126144*v**4) +

- 122880*r**3*(-86251770 + 1995280729*v**2 -

- 10367448084*v**4 + 8023615488*v**6) -

- 384*r**2*(-1055068551 + 26440604040*v**2 -

- 177290319536*v**4 + 382393509120*v**6) -

- 16*r*(626385717 - 14795770608*v**2 + 114091371864*v**4 -

- 356757419584*v**6 + 323542759680*v**8)))*Sin(3*beta))/

- 2.93534171136e11

NASA/TM—2005-213976 26
C. c4o3

The single step c4o3 is :

u

n+(1/2)

i =



−1351350r

7 +

819043225vr6

162 +

1819125r

6

4 +

5457375v

2

r

5

4 −

272189225vr5

432 −

2163105r

5

32 −

1360946125v

3

r

4

2592 

ui− 3

2

+



−

10815525v

2

r

4

64 +

432659675vr4

10368 +

1414875r

4

256 −

3605175v

4

r

3

64 +

432659675v

3

r

3

15552 +

1414875v

2

r

3

128 

ui− 3

2

+



−

1097084975vr3

746496 −

515025r

3

2048 +

86531935v

5

r

2

20736 +

1414875v

4

r

2

512 −

1097084975v

3

r

2

1492992 −

1545075v

2

r

2

4096 

ui− 3

2

+



+

7170265vr2

221184 +

127575r

2

16384 +

94325v

6

r

512 −

219416995v

5

r

2985984 −

515025v

4

r

8192 +

7170265v

3

r

663552 

ui− 3

2

+



+

127575v

2

r

16384 −

286345vr

589824 −

25515r

131072 −

31345285v

7

17915904 −

34335v

6

16384 +

1434053v

5

2654208 

ui− 3

2

+



+

42525v

4

65536 −

286345v

3

3538944 −

25515v

2

262144 +

7435v

1572864 +

5969

1048576

ui− 3

2

+



1351350r

7 +

115540425vr6

2 −

1819125r

6

4 −

5457375v

2

r

5

4 −

144542475vr5

16 +

2163105r

5

32 −

240904125v

3

r

4

32 

ui+ 1

2

+



+

10815525v

2

r

4

64 +

108103275vr4

128 −

1414875r

4

256 +

3605175v

4

r

3

64 +

36034425v

3

r

3

64 −

1414875v

2

r

3

128 

ui+ 1

2

+



−

52367175vr3

1024 +

515025r

3

2048 +

21620655v

5

r

2

256 −

1414875v

4

r

2

512 −

52367175v

3

r

2

2048 +

1545075v

2

r

2

4096 

ui+ 1

2

+



+

16996635vr2

8192 −

127575r

2

16384 −

94325v

6

r

512 −

10473435v

5

r

4096 +

515025v

4

r

8192 +

5665545v

3

r

8192 

ui+ 1

2

+



−

127575v

2

r

16384 −

4275585vr

65536 +

25515r

131072 −

498735v

7

8192 +

34335v

6

16384 +

1133109v

5

32768 

ui+ 1

2

+



−

42525v

4

65536 −

1425195v

3

131072 +

25515v

2

262144 +

1213785v

524288 +

518319

1048576

ui− 1

2

+



1351350r

7 −

115540425vr6

2 −

1819125r

6

4 −

5457375v

2

r

5

4 +

144542475vr5

16 +

2163105r

5

32 +

240904125v

3

r

4

32 

ui+ 1

2

+



+

10815525v

2

r

4

64 −

108103275vr4

128 −

1414875r

4

256 +

3605175v

4

r

3

64 −

36034425v

3

r

3

64 −

1414875v

2

r

3

128 

ui+ 1

2

+



+

52367175vr3

1024 +

515025r

3

2048 −

21620655v

5

r

2

256 −

1414875v

4

r

2

512 +

52367175v

3

r

2

2048 +

1545075v

2

r

2

4096 

ui+ 1

2

+



−

16996635vr2

8192 −

127575r

2

16384 −

94325v

6

r

512 +

10473435v

5

r

4096 +

515025v

4

r

8192 −

5665545v

3

r

8192 

ui+ 1

2

+



−

127575v

2

r

16384 +

4275585vr

65536 +

25515r

131072 +

498735v

7

8192 +

34335v

6

16384 −

1133109v

5

32768 

ui+ 1

2

+



−

42525v

4

65536 +

1425195v

3

131072 +

25515v

2

262144 −

1213785v

524288 +

518319

1048576

ui+ 1

2

+



−1351350r

7 −

819043225vr6

162 +

1819125r

6

4 +

5457375v

2

r

5

4 +

272189225vr5

432 −

2163105r

5

32 +

1360946125v

3

r

4

2592 

ui+ 3

2

+



−

10815525v

2

r

4

64 −

432659675vr4

10368 +

1414875r

4

256 −

3605175v

4

r

3

64 −

432659675v

3

r

3

15552 +

1414875v

2

r

3

128 

ui+ 3

2

+



+

1097084975vr3

746496 −

515025r

3

2048 −

86531935v

5

r

2

20736 +

1414875v

4

r

2

512 +

1097084975v

3

r

2

1492992 −

1545075v

2

r

2

4096 

ui+ 3

2

+



−

7170265vr2

221184 +

127575r

2

16384 +

94325v

6

r

512 +

219416995v

5

r

2985984 −

515025v

4

r

8192 −

7170265v

3

r

663552 

ui+ 3

2

+



+

127575v

2

r

16384 +

286345vr

589824 −

25515r

131072 +

31345285v

7

17915904 −

34335v

6

16384 −

1434053v

5

2654208 

ui+ 3

2

+



+

42525v

4

65536 +

286345v

3

3538944 −

25515v

2

262144 −

7435v

1572864 +

5969

1048576

ui+ 3

2

+



−

200200

3

h

2

r

7 +

63140h

2

r

6

3 +

1796795

9

h

2vr6 −

23135h

2

r

5

8 + 63140h

2v2

r

5 −

554785

24 h

2vr5 −

2773925

144 h

2v3

r

4



uxxi− 3

2

+



+

188405h

2

r

4

864 −

115675

16 h

2v2

r

4 +

818965

576 h

2vr4 −

115675

48 h

2v4

r

3 +

818965

864 h

2v3

r

3 −

131695h

2

r

3

13824 

uxxi− 3

2

+



+

188405

432 h

2v2

r

3 −

2000635h

2

vr3

41472 +

163793h

2

v

5

r

2

1152 +

188405h

2

v

4

r

2

1728 −

2000635h

2

v

3

r

2

82944 +

443h

2

r

2

1536 

uxxi− 3

2

+

NASA/TM—2005-213976 27


−

131695h

2

v

2

r

2

9216 +

38425h

2

vr2

36864 +

37681h

2

v

6

r

5184 −

400127h

2

v

5

r

165888 −

131695h

2

v

4

r

55296 +

38425h

2

v

3

r

110592 

uxxi− 3

2

+



−

233h

2

r

32768 +

443h

2

v

2

r

1536 −

505h

2

vr

32768 −

57161h

2

v

7

995328 −

26339h

2

v

6

331776 +

7685h

2

v

5

442368 

uxxi− 3

2

+



+

443h

2

v

4

18432 −

505h

2

v

3

196608 +

27h

2

131072 −

233h

2

v

2

65536 +

39h

2

v

262144

uxxi− 3

2

+



−

24651h

2

v

7

4096 −

981h

2

v

6

1024 +

3913

128 h

2

rv6 +

54675h

2

v

5

16384 +

1058589

128 h

2

r

2v5 −

517671h

2

rv5

2048 

uxxi− 1

2

+



−

29925

8

h

2

r

3v4 +

12879h

2

v

4

16384 +

58695

128 h

2

r

2v4 −

14715

512 h

2

rv4 −

11659725

16 h

2

r

4v3 +

1764315

32 h

2

r

3v3



uxxi− 1

2

+



−

54675h

2

v

3

65536 −

2588355h

2

r

2

v

3

1024 +

273375h

2

rv3

4096 + 31185h

2

r

5v2 −

89775

8

h

2

r

4v2 +

58695

32 h

2

r

3v2 −

8019h

2

v

2

32768 

uxxi− 1

2

+



−

44145

256 h

2

r

2v2 +

38637h

2

rv2

4096 + 5540535h

2

r

6v −

6995835

8

h

2

r

5v +

5292945

64 h

2

r

4v −

2588355

512 h

2

r

3v +

19683h

2

v

262144 

uxxi− 1

2

+



+

820125h

2

r

2

v

4096 −

164025h

2

rv

32768 + 10395h

2

r

6 −

17955h

2

r

5

4 +

58695h

2

r

4

64 −

14715h

2

r

3

128 +

6561h

2

262144 

uxxi− 1

2

+



+

38637h

2

r

2

4096 −

8019h

2

r

16384 

uxxi− 1

2

+



24651h

2

v

7

4096 −

981h

2

v

6

1024 +

3913

128 h

2

rv6 −

54675h

2

v

5

16384 −

1058589

128 h

2

r

2v5 +

517671h

2

rv5

2048 

uxxi+ 1

2

+



−

29925

8

h

2

r

3v4 +

12879h

2

v

4

16384 +

58695

128 h

2

r

2v4 −

14715

512 h

2

rv4 +

11659725

16 h

2

r

4v3 −

1764315

32 h

2

r

3v3



uxxi+ 1

2

+



+

54675h

2

v

3

65536 +

2588355h

2

r

2

v

3

1024 −

273375h

2

rv3

4096 + 31185h

2

r

5v2 −

89775

8

h

2

r

4v2 +

58695

32 h

2

r

3v2 −

8019h

2

v

2

32768 

uxxi+ 1

2

+



−

44145

256 h

2

r

2v2 +

38637h

2

rv2

4096 − 5540535h

2

r

6v +

6995835

8

h

2

r

5v −

5292945

64 h

2

r

4v +

2588355

512 h

2

r

3v −

19683h

2

v

262144 

uxxi+ 1

2

+



−

820125h

2

r

2

v

4096 +

164025h

2

rv

32768 + 10395h

2

r

6 −

17955h

2

r

5

4 +

58695h

2

r

4

64 −

14715h

2

r

3

128 

uxxi+ 1

2

+



+

6561h

2

262144 +

38637h

2

r

2

4096 −

8019h

2

r

16384 

uxxi+ 1

2

+



−

200200

3

h

2

r

7 +

63140h

2

r

6

3 −

1796795

9

h

2vr6 −

23135h

2

r

5

8 + 63140h

2v2

r

5 +

554785

24 h

2vr5 +

2773925

144 h

2v3

r

4



uxxi+ 3

2

+



+

188405h

2

r

4

864 −

115675

16 h

2v2

r

4 −

818965

576 h

2vr4 −

115675

48 h

2v4

r

3 −

818965

864 h

2v3

r

3 −

131695h

2

r

3

13824 

uxxi+ 3

2

+



+

188405

432 h

2v2

r

3 +

2000635h

2

vr3

41472 −

163793h

2

v

5

r

2

1152 +

188405h

2

v

4

r

2

1728 +

2000635h

2

v

3

r

2

82944 +

443h

2

r

2

1536 

uxxi+ 3

2

+



−

131695h

2

v

2

r

2

9216 −

38425h

2

vr2

36864 +

37681h

2

v

6

r

5184 +

400127h

2

v

5

r

165888 −

131695h

2

v

4

r

55296 −

38425h

2

v

3

r

110592 

uxxi+ 3

2

+



−

233h

2

r

32768 +

443h

2

v

2

r

1536 +

505h

2

vr

32768 +

57161h

2

v

7

995328 −

26339h

2

v

6

331776 −

7685h

2

v

5

442368 

uxxi+ 3

2

+



+

443h

2

v

4

18432 +

505h

2

v

3

196608 +

27h

2

131072 −

233h

2

v

2

65536 −

39h

2

v

262144

uxxi+ 3

2

+



−

10010

3

h

3

r

7 +

11935h

3

r

6

12 +

155155

18 h

3vr6 −

4165h

3

r

5

32 +

11935

4

h

3v2

r

5 −

45815

48 h

3vr5



uxxxi− 3

2

+



+

65345h

3

r

4

6912 −

229075

288 h

3v3

r

4 −

20825

64 h

3v2

r

4 +

65345h

3

vr4

1152 −

20825

192 h

3v4

r

3 −

22415h

3

r

3

55296 

uxxxi− 3

2

+



+

65345h

3

v

3

r

3

1728 +

65345h

3

v

2

r

3

3456 −

156905h

3

vr3

82944 +

13069h

3

v

5

r

2

2304 +

65345h

3

v

4

r

2

13824 +

199h

3

r

2

16384 

uxxxi− 3

2

+



−

156905h

3

v

3

r

2

165888 −

22415h

3

v

2

r

2

36864 +

995h

3

vr2

24576 +

13069h

3

v

6

r

41472 −

31381h

3

v

5

r

331776 −

22415h

3

v

4

r

221184 

uxxxi− 3

2

+



−

39h

3

r

131072 +

995h

3

v

3

r

73728 +

199h

3

v

2

r

16384 −

39h

3

vr

65536 −

4483h

3

v

7

1990656 −

4483h

3

v

6

1327104 

uxxxi− 3

2

+



+

199h

3

v

5

294912 +

199h

3

v

4

196608 +

9h

3

1048576 −

13h

3

v

3

131072 −

39h

3

v

2

262144 +

3h

3

v

524288

uxxxi− 3

2

+



−90090h

3

r

7 +

135135h

3

r

6

4 +

1756755

2

h

3vr6 −

188055h

3

r

5

32 +

405405

4

h

3v2

r

5 −

2068605

16 h

3vr5 +

156905h

3

r

4

256 

uxxxi− 1

2

+



−

3447675

32 h

3v3

r

4 −

940275

64 h

3v2

r

4 +

1412145

128 h

3vr4 −

313425

64 h

3v4

r

3 −

84015h

3

r

3

2048 +

470715

64 h

3v3

r

3



uxxxi− 1

2

+



+

156905

128 h

3v2

r

3 −

588105h

3

vr3

1024 +

282429

256 h

3v5

r

2 +

156905

512 h

3v4

r

2 +

28917h

3

r

2

16384 −

588105h

3

v

3

r

2

2048 

uxxxi− 1

2

+

NASA/TM—2005-213976 28


−

252045h

3

v

2

r

2

4096 +

144585h

3

vr2

8192 +

31381h

3

v

6

r

1536 −

117621h

3

v

5

r

4096 −

84015h

3

v

4

r

8192 −

7533h

3

r

131072 

uxxxi− 1

2

+



+

48195h

3

v

3

r

8192 +

28917h

3

v

2

r

16384 −

22599h

3

vr

65536 −

5601h

3

v

7

8192 −

5601h

3

v

6

16384 +

9639h

3

v

5

32768 

uxxxi− 1

2

+



+

9639h

3

v

4

65536 +

2187h

3

1048576 −

7533h

3

v

3

131072 −

7533h

3

v

2

262144 +

2187h

3

v

524288 

uxxxi− 1

2

+



90090h

3

r

7 −

135135h

3

r

6

4 +

1756755

2

h

3vr6 +

188055h

3

r

5

32 −

405405

4

h

3v2

r

5 −

2068605

16 h

3vr5 −

156905h

3

r

4

256 

uxxxi+ 1

2

+



−

3447675

32 h

3v3

r

4 +

940275

64 h

3v2

r

4 +

1412145

128 h

3vr4 +

313425

64 h

3v4

r

3 +

84015h

3

r

3

2048 +

470715

64 h

3v3

r

3



uxxxi+ 1

2

+



−

156905

128 h

3v2

r

3 −

588105h

3

vr3

1024 +

282429

256 h

3v5

r

2 −

156905

512 h

3v4

r

2 −

28917h

3

r

2

16384 −

588105h

3

v

3

r

2

2048 

uxxxi+ 1

2

+



+

252045h

3

v

2

r

2

4096 +

144585h

3

vr2

8192 −

31381h

3

v

6

r

1536 −

117621h

3

v

5

r

4096 +

84015h

3

v

4

r

8192 +

7533h

3

r

131072 

uxxxi+ 1

2

+



+

48195h

3

v

3

r

8192 −

28917h

3

v

2

r

16384 −

22599h

3

vr

65536 −

5601h

3

v

7

8192 +

5601h

3

v

6

16384 +

9639h

3

v

5

32768 

uxxxi+ 1

2

+



−

9639h

3

v

4

65536 −

2187h

3

1048576 −

7533h

3

v

3

131072 +

7533h

3

v

2

262144 +

2187h

3

v

524288 

uxxxi+ 1

2

+



10010h

3

r

7

3 −

11935h

3

r

6

12 +

155155

18 h

3vr6 +

4165h

3

r

5

32 −

11935

4

h

3v2

r

5 −

45815

48 h

3vr5



uxxxi+ 3

2

+



−

65345h

3

r

4

6912 −

229075

288 h

3v3

r

4 +

20825

64 h

3v2

r

4 +

65345h

3

vr4

1152 +

20825

192 h

3v4

r

3 +

22415h

3

r

3

55296 

uxxxi+ 3

2

+



+

65345h

3

v

3

r

3

1728 −

65345h

3

v

2

r

3

3456 −

156905h

3

vr3

82944 +

13069h

3

v

5

r

2

2304 −

65345h

3

v

4

r

2

13824 −

199h

3

r

2

16384 

uxxxi+ 3

2

+



−

156905h

3

v

3

r

2

165888 +

22415h

3

v

2

r

2

36864 +

995h

3

vr2

24576 −

13069h

3

v

6

r

41472 −

31381h

3

v

5

r

331776 +

22415h

3

v

4

r

221184 

uxxxi+ 3

2

+



+

39h

3

r

131072 +

995h

3

v

3

r

73728 −

199h

3

v

2

r

16384 −

39h

3

vr

65536 −

4483h

3

v

7

1990656 +

4483h

3

v

6

1327104 

uxxxi+ 3

2

+



+

199h

3

v

5

294912 −

199h

3

v

4

196608 −

9h

3

1048576 −

13h

3

v

3

131072 +

39h

3

v

2

262144 +

3h

3

v

524288

uxxxi+ 3

2

+



−

4454450hr7

9 +

5837755hr6

36 +

90265175

54 hvr6 +

5837755

12 hv2

r

5 −

2234785hr5

96 −

29020915

144 hvr5



uxi− 3

2

+



−

145104575

864 hv3

r

4 −

11173925

192 hv2

r

4 +

37870805hr4

20736 +

44422525hvr4

3456 −

11173925

576 hv4

r

3 +

44422525hv3r

3

5184 

uxi− 3

2

+



+

37870805hv2r

3

10368 −

13501235hr3

165888 −

110513725hvr3

248832 +

8884505hv5r

2

6912 +

37870805hv4r

2

41472 −

110513725hv3r

2

497664 

uxi− 3

2

+



−

13501235hv2r

2

110592 +

367337hr2

147456 +

714695hvr2

73728 +

7574161hv6r

124416 −

22102745hv5r

995328 −

13501235hv4r

663552 

uxi− 3

2

+



+

714695hv3r

221184 +

367337hv2r

147456 −

8105hr

131072 −

28355hvr

196608 −

3157535hv7

5971968 −

2700247hv6

3981312 

uxi− 3

2

+



+

142939hv5

884736 +

367337hv4

1769472 −

28355hv3

1179648 −

8105hv2

262144 +

1887h

1048576 +

733hv

524288

uxi− 3

2

+



−1351350hr7 +

1943865hr6

4 +

69594525

2

hvr6 +

5831595

4

hv2r

5 −

2563785hr5

32 −

84168315

16 hvr5 −

140280525

32 hv3

r

4



uxi− 1

2

+



−

12818925

64 hv2

r

4 +

2008335hr4

256 +

60418575

128 hvr4 −

4272975

64 hv4

r

3 +

20139525

64 hv3

r

3 +

2008335

128 hv2

r

3



uxi− 1

2

+



−

1039905hr3

2048 −

27986175hvr3

1024 +

12083715

256 hv5

r

2 +

2008335

512 hv4

r

2 −

27986175hv3r

2

2048 −

3119715hv2r

2

4096 

uxi− 1

2

+



+

451251hr2

16384 +

8817255hvr2

8192 +

133889

512 hv6

r −

5597235hv5r

4096 −

1039905hv4r

8192 +

2939085hv3r

8192 

uxi− 1

2

+



+

451251hv2r

16384 −

287955hr

131072 −

2176065hvr

65536 −

266535hv7

8192 −

69327hv6

16384 +

587817hv5

32768 

uxi− 1

2

+



+

150417hv4

65536 −

725355hv3

131072 −

287955hv2

262144 +

190269h

1048576 +

347733hv

524288 

uxi− 1

2

+



1351350hr7 −

1943865hr6

4 +

69594525

2

hvr6 −

5831595

4

hv2r

5 +

2563785hr5

32 −

84168315

16 hvr5 −

140280525

32 hv3

r

4



uxi+ 1

2

+



+

12818925

64 hv2

r

4 −

2008335hr4

256 +

60418575

128 hvr4 +

4272975

64 hv4

r

3 +

20139525

64 hv3

r

3 −

2008335

128 hv2

r

3



uxi+ 1

2

+



+

1039905hr3

2048 −

27986175hvr3

1024 +

12083715

256 hv5

r

2 −

2008335

512 hv4

r

2 −

27986175hv3r

2

2048 +

3119715hv2r

2

4096 

uxi+ 1

2

+

NASA/TM—2005-213976 29


−

451251hr2

16384 +

8817255hvr2

8192 −

133889

512 hv6

r −

5597235hv5r

4096 +

1039905hv4r

8192 +

2939085hv3r

8192 

uxi+ 1

2

+



−

451251hv2r

16384 +

287955hr

131072 −

2176065hvr

65536 −

266535hv7

8192 +

69327hv6

16384 +

587817hv5

32768 

uxi+ 1

2

+



−

150417hv4

65536 −

725355hv3

131072 +

287955hv2

262144 −

190269h

1048576 +

347733hv

524288 

uxi+ 1

2

+



4454450hr7

9 −

5837755hr6

36 +

90265175

54 hvr6 −

5837755

12 hv2

r

5 +

2234785hr5

96 −

29020915

144 hvr5



uxi+ 3

2

+



−

145104575

864 hv3

r

4 +

11173925

192 hv2

r

4 −

37870805hr4

20736 +

44422525hvr4

3456 +

11173925

576 hv4

r

3 +

44422525hv3r

3

5184 

uxi+ 3

2

+



−

37870805hv2r

3

10368 +

13501235hr3

165888 −

110513725hvr3

248832 +

8884505hv5r

2

6912 −

37870805hv4r

2

41472 −

110513725hv3r

2

497664 

uxi+ 3

2

+



+

13501235hv2r

2

110592 −

367337hr2

147456 +

714695hvr2

73728 −

7574161hv6r

124416 −

22102745hv5r

995328 +

13501235hv4r

663552 

uxi+ 3

2

+



+

714695hv3r

221184 −

367337hv2r

147456 +

8105hr

131072 −

28355hvr

196608 −

3157535hv7

5971968 +

2700247hv6

3981312 

uxi+ 3

2

+



+

142939hv5

884736 −

367337hv4

1769472 −

28355hv3

1179648 +

8105hv2

262144 −

1887h

1048576 +

733hv

524288

uxi+ 3

2

(74)

1. c4o3 Amplification Factor

The real part of the c4o3 technique’s amplification factor is shown in Fortran form due to its large size:

\Re(G)=

(v*(-588807047879148744*beta + 5029688315970720*beta**3 +

- 37059670203005168640*beta*r - 328971137267483136*beta**3*r -

- 1534749197369015130624*beta*r**2 +

- 11801835930422937600*beta**3*r**2 +

- 40832065370654591385600*beta*r**3 -

- 56904951116416352256*beta**3*r**3 -

- 622294838643747115008000*beta*r**4 -

- 12851851697346192998400*beta**3*r**4 +

- 992192170426717962240000*beta*r**5 +

- 643479477553473666416640*beta**3*r**5 +

- 201463802968439566903541760*beta*r**6 -

- 17461527698797546910515200*beta**3*r**6 -

- 5286721179261333240859852800*beta*r**7 +

- 313381129818851095672258560*beta**3*r**7 +

- 71155114020377497763315712000*beta*r**8 -

- 3822300621955502779347763200*beta**3*r**8 -

- 447028423148769786829209600000*beta*r**9 +

- 28368526140181720582638796800*beta**3*r**9 -

- 1601877496352558815064044339200*beta*r**10 -

- 59847788836847349695250432000*beta**3*r**10 +

- 58804023178508604265092612096000*beta*r**11 -

- 1145586894161661438329880576000*beta**3*r**11 -

- 501654460948288553120465682432000*beta*r**12 +

- 12480187407814541778213666816000*beta**3*r**12 +

- 1674235923789035490017311457280000*beta*r**13 -

- 44428501533548568301629603840000*beta**3*r**13 +

- 10856641400139384000*beta*v**2 -

- 61032243152201472*beta**3*v**2 -

- 868027203027200001024*beta*r*v**2 +

- 2811635262800179200*beta**3*r*v**2 +

- 35316960283429877145600*beta*r**2*v**2 +

- 127489092611209003008*beta**3*r**2*v**2 -

- 760525851586382093352960*beta*r**3*v**2 -

- 19737100631473481318400*beta**3*r**3*v**2 +

NASA/TM—2005-213976 30
- 3084002023769167527936000*beta*r**4*v**2 +

- 1039016222710842465976320*beta**3*r**4*v**2 +

- 314967737626958823476428800*beta*r**5*v**2 -

- 32380786633438227175833600*beta**3*r**5*v**2 -

- 10107460755831691793557094400*beta*r**6*v**2 +

- 661442776702898338434908160*beta**3*r**6*v**2 +

- 145254712110721815333253939200*beta*r**7*v**2 -

- 8735486181850552701498163200*beta**3*r**7*v**2 -

- 786248901199484763726938112000*beta*r**8*v**2 +

- 65791903274387010222450278400*beta**3*r**8*v**2 -

- 7318887101806147288826904576000*beta*r**9*v**2 -

- 97419093993513730330066944000*beta**3*r**9*v**2 +

- 157738390755256383450315227136000*beta*r**10*v**2 -

- 2984392362118233250213134336000*beta**3*r**10*v**2 -

- 959035275737495023494484721664000*beta*r**11*v**2 +

- 23052633926807309847814471680000*beta**3*r**11*v**2 -

- 79250271825534920064*beta*v**4 +

- 30583554037562880*beta**3*v**4 +

- 6421630897117767352320*beta*r*v**4 +

- 47492496337006411776*beta**3*r*v**4 -

- 203859543829363430473728*beta*r**2*v**4 -

- 6802413221178936852480*beta**3*r**2*v**4 +

- 486202955408031891456000*beta*r**3*v**4 +

- 442357687991309237747712*beta**3*r**3*v**4 +

- 179634315623296769686241280*beta*r**4*v**4 -

- 16792980732045908980531200*beta**3*r**4*v**4 -

- 6422236975344074427570585600*beta*r**5*v**4 +

- 400971483120935712923320320*beta**3*r**5*v**4 +

- 101773594843604206525266001920*beta*r**6*v**4 -

- 5885633663530072923753676800*beta**3*r**6*v**4 -

- 554704432089795085806128332800*beta*r**7*v**4 +

- 45308606820467398505366814720*beta**3*r**7*v**4 -

- 6059100162146023672974095155200*beta*r**8*v**4 -

- 26477965133207704812965068800*beta**3*r**8*v**4 +

- 97208121972520878672118284288000*beta*r**9*v**4 -

- 2004030344796760868503368499200*beta**3*r**9*v**4 +

- 280563375107936286720*beta*v**6 +

- 3124168881320337408*beta**3*v**6 -

- 16539252149112612175872*beta*r*v**6 -

- 741601712426909368320*beta**3*r*v**6 -

- 101728757391537173299200*beta*r**2*v**6 +

- 67557036781240920834048*beta**3*r**2*v**6 +

- 37767943066486311089602560*beta*r**3*v**6 -

- 3207640886027135012044800*beta**3*r**3*v**6 -

- 1396197232635062331113472000*beta*r**4*v**6 +

- 86683666972156365688012800*beta**3*r**4*v**6 +

- 22881291722928390218239180800*beta*r**5*v**6 -

- 1358716656503716645227724800*beta**3*r**5*v**6 -

- 125760631653986542616051712000*beta*r**6*v**6 +

- 10643200834575207229795860480*beta**3*r**6*v**6 -

- 996010134491996472481520025600*beta*r**7*v**6 -

- 11246255265294721400124211200*beta**3*r**7*v**6 -

- 358922944231258576896*beta*v**8 -

- 23015585822443315200*beta**3*v**8 -

- 11941623566508633292800*beta*r*v**8 +

NASA/TM—2005-213976 31
- 3743409734999328817152*beta**3*r*v**8 +

- 3210170415656980278804480*beta*r**2*v**8 -

- 250908047549192562278400*beta**3*r**2*v**8 -

- 140105518368166183698432000*beta*r**3*v**8 +

- 8288905329514443807129600*beta**3*r**3*v**8 +

- 2267320234111262990637465600*beta*r**4*v**8 -

- 134105137056035364667392000*beta**3*r**4*v**8 -

- 7259350157559910527034982400*beta*r**5*v**8 +

- 805428580219484723887472640*beta**3*r**5*v**8 -

- 495787807221085716480*beta*v**10 +

- 69672616691407847424*beta**3*v**10 +

- 88601438767640178327552*beta*r*v**10 -

- 7191691599933383639040*beta**3*r*v**10 -

- 5321169629212015853568000*beta*r**2*v**10 +

- 316068470842954088448000*beta**3*r**2*v**10 +

- 98710173002797443585146880*beta*r**3*v**10 -

- 5423944737688441690521600*beta**3*r**3*v**10 +

- 1283927763562788519936*beta*v**12 -

- 79564408586669260800*beta**3*v**12 -

- 46335501936495034368000*beta*r*v**12 +

- 3408981391065816760320*beta**3*r*v**12 -

- 243*beta*(beta**2*(-57683669631863209448177664000*r**12 +

- 173015870211123787302174720000*r**13 -

- 421906448646144000*r**11*

- (-21617280715 + 273889751564*v**2) +

- 2343924714700800*r**10*

- (-399594918611 + 10694645915880*v**2) -

- 951268147200*r**8*

- (7496356089035 - 273113326736812*v**2 +

- 2137216758218080*v**4) +

- 1902536294400*r**9*

- (41595043870467 - 1551530871852500*v**2 +

- 7494132128690592*v**4) +

- 82575360*r**6*

- (-717986639667879 + 21486595174684016*v**2 -

- 161240285877566592*v**4 + 512543000161882368*v**6) -

- 660602880*r**7*

- (-1054801734874837 + 32563398611203140*v**2 -

- 264101846109232464*v**4 + 583098278454795840*v**6) +

- 41287680*r**5*

- (94795433102277 - 3047017849889958*v**2 +

- 24033401976772592*v**4 - 78895185920097600*v**6 +

- 94232269568938368*v**8) -

- 61440*r**4*(3104552759774103 - 108755814337086168*v**2 +

- 992484062573122224*v**4 - 3457844668905410560*v**6 +

- 5184867573569725440*v**8) -

- 24576*r**3*(-271207080146889 + 10250945050778100*v**2 -

- 107797789683950562*v**4 + 436899751570237320*v**6 -

- 749392863017170400*v**8 + 438028314410897280*v**10) +

- 192*r**2*(-829348380467703 + 33079948668167184*v**2 -

- 389320831684735200*v**4 + 1889296626965601024*v**6 -

- 4024984550910324480*v**8 + 3693650363298488320*v**10) -

- 27*(297709173819 - 23981152303080*v**2 +

- 342299730399120*v**4 - 2140251513313536*v**6 +

- 6829065061044480*v**8 - 10372466907949056*v**10 +

NASA/TM—2005-213976 32
- 6410253269020672*v**12) +

- 32*r*(69296470633935 - 3020407834350708*v**2 +

- 38725919708165280*v**4 - 217803616412366016*v**6 +

- 580822312401040896*v**8 - 721684033577183232*v**10 +

- 339501476090900480*v**12)) -

- 3*(-775772574024358258148376576000*r**12 +

- 2438132793167793018151895040000*r**13 -

- 421906448646144000*r**11*

- (-257254367127 + 3614160898348*v**2) +

- 2343924714700800*r**10*

- (-3402185718667 + 126271369928760*v**2) +

- 66588770304000*r**9*

- (3620684948021 - 401361376717532*v**2 +

- 2603560564774752*v**4) -

- 951268147200*r**8*

- (-3376274173575 - 1342889495673180*v**2 +

- 20112535030790048*v**4) -

- 3303014400*r**7*

- (-218271568715125 + 12397919089960596*v**2 -

- 254809926086013648*v**4 + 1067011446179418432*v**6) +

- 82575360*r**6*

- (-2915872934335767 + 52293334784431120*v**2 -

- 203709865093847424*v**4 + 2553069348088170240*v**6) +

- 41287680*r**5*

- (651620035353411 - 14395064916992790*v**2 +

- 44726295355144816*v**4 - 135736772059360320*v**6 +

- 543435526019068800*v**8) -

- 61440*r**4*(29153895283741815 - 763599752521182360*v**2 +

- 4358868307579671984*v**4 - 6185181328054722560*v**6 +

- 8606299462809646080*v**8) -

- 245760*r**3*(-326013752062737 + 9434326912362198*v**2 -

- 73410801524614449*v**4 + 173104319746778292*v**6 -

- 76702873012875760*v**8 + 7731509141205696*v**10) +

- 192*r**2*(-12623593377224871 + 384637417525282320*v**2 -

- 3498380150575610592*v**4 + 12206778765661843200*v**6 -

- 14025471592493326080*v**8 + 4061847949873602560*v**10)\

- - 27*(6169818240603 - 461835791461800*v**2 +

- 5001715839214224*v**4 - 23824855185096960*v**6 +

- 57403294649274624*v**8 - 55426485040900096*v**10 +

- 6685742743433216*v**12) +

- 32*r*(1331862912087885 - 44860039946650836*v**2 +

- 439976175253469280*v**4 - 1899540445326107328*v**6 +

- 3591794938498652160*v**8 - 2682471909002545152*v**10 +

- 1185920327022161920*v**12)))*Cos(beta) -

- 972*beta*(beta**2*(-922892772507352674022195200*r**12 +

- 2152486478012855315595264000*r**13 -

- 84381289729228800*r**11*(-2365554469 + 24378299660*v**2) +

- 2343924714700800*r**10*

- (-12158409919 + 243221791632*v**2) -

- 761014517760*r**8*

- (306846388688 - 11037490968085*v**2 + 73879224732544*v**4)

- + 1902536294400*r**9*

- (1555698089817 - 44461683026068*v**2 +

- 157343066562528*v**4) +

- 82575360*r**6*

NASA/TM—2005-213976 33
- (-8390260233783 + 400592317432384*v**2 -

- 4796853064401408*v**4 + 16887687205248000*v**6) -

- 660602880*r**7*

- (-21735341420783 + 918912972202836*v**2 -

- 8740846641965616*v**4 + 16939526831464512*v**6) +

- 82575360*r**5*

- (319009475466 - 16669199445327*v**2 +

- 234442873415960*v**4 - 1144188407488224*v**6 +

- 1448247275003712*v**8) -

- 61440*r**4*(12813557189103 - 703861986264696*v**2 +

- 11174990937599664*v**4 - 67997567769399296*v**6 +

- 152053146144721920*v**8) -

- 12288*r**3*(-1500438468915 + 82527016827600*v**2 -

- 1422945532585836*v**4 + 10185289876987920*v**6 -

- 31139167116948800*v**8 + 28552272358337280*v**10) +

- 192*r**2*(-1650077479143 + 89009052247440*v**2 -

- 1602547661864160*v**4 + 12947818722667776*v**6 -

- 48796496561053440*v**8 + 76203903203262464*v**10) -

- 81*(-59934735 - 9592864344*v**2 + 218226663216*v**4 -

- 2010084284160*v**6 + 10031018607360*v**8 -

- 26368470194176*v**10 + 30903649701888*v**12) +

- 64*r*(46120044303 - 2849042564514*v**2 +

- 53345116022976*v**4 - 459076826220768*v**6 +

- 2011462884761472*v**8 - 4254001425607680*v**10 +

- 2909977010382848*v**12)) -

- 3*(-20133583894660502098280448000*r**12 +

- 51423299015953439407472640000*r**13 -

- 421906448646144000*r**11*

- (-9679278267 + 101865765716*v**2) +

- 2343924714700800*r**10*

- (-245765287109 + 4905031931328*v**2) -

- 1902536294400*r**8*

- (2959552682415 - 96219577428774*v**2 +

- 604433839198112*v**4) +

- 1902536294400*r**9*

- (33223356231995 - 906542266954940*v**2 +

- 3243439678639776*v**4) -

- 660602880*r**7*

- (-620539208944357 + 22640836761551100*v**2 -

- 188809621496456976*v**4 + 341796645237347520*v**6) +

- 82575360*r**6*

- (-288543121430805 + 11671815463754336*v**2 -

- 116857257267352320*v**4 + 364001677713148416*v**6) +

- 82575360*r**5*

- (13104388796445 - 584971692371685*v**2 +

- 6766961878348840*v**4 - 27939797535271200*v**6 +

- 31561275189465024*v**8) -

- 61440*r**4*(612130122784485 - 29493473801105160*v**2 +

- 389745499167568656*v**4 - 1958865473739317248*v**6 +

- 3697607015636997120*v**8) -

- 12288*r**3*(-80726104925487 + 4027934222870400*v**2 -

- 59360722913632644*v**4 + 352703853912613680*v**6 -

- 887189420132478400*v**8 + 681034036144062720*v**10) +

- 192*r**2*(-98476779983445 + 4922372676072816*v**2 -

- 77715621186994080*v**4 + 534442122526535424*v**6 -

NASA/TM—2005-213976 34
- 1675966626530069760*v**8 + 2177600498496274432*v**10) -

- 27*(14928433569 - 2207820187512*v**2 +

- 40585421655216*v**4 - 328743522743040*v**6 +

- 1427223068762880*v**8 - 3169036058834944*v**10 +

- 3055632709816320*v**12) +

- 128*r*(1651783345704 - 88594168691835*v**2 +

- 1464721895747088*v**4 - 11067394312967760*v**6 +

- 41628975096003264*v**8 - 73850069684021760*v**10 +

- 43397862838787072*v**12)))*Cos(2*beta) -

- 3*beta*(-6565413079503 + 398408459938326331671969792000*r**12 -

- 906088081101032576086179840000*r**13 + 110821561318728*v**2 -

- 401211159351120*v**4 - 4508563972688640*v**6 +

- 58656504616268544*v**8 - 257715918703392768*v**10 +

- 442123540826525696*v**12 +

- 421906448646144000*r**11*

- (-199054845767 + 2182021247692*v**2) -

- 2343924714700800*r**10*

- (-4729245686083 + 103656419886264*v**2) +

- 951268147200*r**8*

- (69844313562865 - 2972871503160924*v**2 +

- 22946621498684576*v**4) -

- 1902536294400*r**9*

- (531609461763455 - 17239248316624180*v**2 +

- 66134244014058528*v**4) -

- 82575360*r**6*(-1385166227803471 + 87325784946246928*v**2 -

- 1351565780861611392*v**4 + 5758348552837801728*v**6) +

- 660602880*r**7*(-4865642730162041 + 256056788770319940*v**2 -

- 2971496980680499728*v**4 + 6653161396905291840*v**6) -

- 41287680*r**5*(70947893367747 - 5326699377817030*v**2 +

- 103227072160039792*v**4 - 641970137232605760*v**6 +

- 979355319246328704*v**8) +

- 430080*r**4*(115071039657225 - 10705019299468200*v**2 +

- 257430127321849680*v**4 - 2138251218314289152*v**6 +

- 6115711802512051200*v**8) +

- 49152*r**3*(-7713649675629 + 1227142504767870*v**2 -

- 38765752221783573*v**4 + 414931844050079140*v**6 -

- 1732608805218292400*v**8 + 2050361230337542080*v**10) -

- 192*r**2*(36136825502337 + 1550210230422288*v**2 -

- 101250428823753696*v**4 + 1447856822418607872*v**6 -

- 8012778473780811520*v**8 + 16755880172081340416*v**10) -

- 32*r*(-11208197155347 + 166109466520332*v**2 +

- 2472521938752096*v**4 - 61285385287929024*v**6 +

- 446619625746905088*v**8 - 1360377446820857856*v**10 +

- 1186546227591983104*v**12) +

- 3*beta**2*(-659108509091251205072486400*r**12 +

- 1523320518370749755424768000*r**13 -

- 84381289729228800*r**11*(-1612681903 + 18059678780*v**2) +

- 2343924714700800*r**10*(-7484112451 + 168099152424*v**2) +

- 1902536294400*r**9*

- (821151171723 - 27290457944596*v**2 +

- 107281230487200*v**4) -

- 190253629440*r**8*

- (526084952279 - 22934452411900*v**2 +

- 181725049039072*v**4) +

- 82575360*r**6*

NASA/TM—2005-213976 35
- (-1956547557303 + 127748838546160*v**2 -

- 2032176036711552*v**4 + 8885105774274816*v**6) -

- 660602880*r**7*

- (-7130914417565 + 385061401604292*v**2 -

- 4585461138962640*v**4 + 10544008313745984*v**6) +

- 41287680*r**5*

- (93887635293 - 7526238878022*v**2 +

- 151021271377264*v**4 - 963450698392896*v**6 +

- 1510243631355264*v**8) -

- 61440*r**4*(892544804055 - 99481400362008*v**2 +

- 2549241380372400*v**4 - 21869195086548992*v**6 +

- 64186134339855360*v**8) -

- 24576*r**3*(-1149062193 + 2780159930100*v**2 -

- 104119336285650*v**4 + 1175524010390280*v**6 -

- 5058029287141600*v**8 + 6153750656565120*v**10) +

- 192*r**2*(110404472985 + 77801877648*v**2 -

- 119366959569120*v**4 + 1954509147636480*v**6 -

- 11342305865137920*v**8 + 24358626486591488*v**10) +

- 3*(3711406797 - 67873440024*v**2 + 358447305456*v**4 +

- 1117039880448*v**6 - 24498416113920*v**8 +

- 117600703752192*v**10 - 209771938164736*v**12) +

- 32*r*(-21509011593 + 420604058700*v**2 +

- 1291629672864*v**4 - 74872746083520*v**6 +

- 605030639961600*v**8 - 1917925378615296*v**10 +

- 1704777524703232*v**12)))*Cos(3*beta) -

- 729*(beta**2*(105741172866477573305008128000*r**12 -

- 345588754177326433582448640000*r**13 +

- 4640970935107584000*r**11*

- (-2850552659 + 44081501412*v**2) -

- 16407473002905600*r**10*

- (-37497573689 + 2176321827240*v**2) +

- 951268147200*r**8*

- (-8243164839537 + 23669349438860*v**2 +

- 1861865235800416*v**4) -

- 9512681472000*r**9*

- (-4347617889887 - 244038396953428*v**2 +

- 2266640436651360*v**4) +

- 82575360*r**6*

- (201824348917275 + 5393607906635120*v**2 -

- 117193940856745344*v**4 + 40426634189395200*v**6) +

- 3303014400*r**7*

- (114111407047925 - 3788820566188332*v**2 +

- 5949837097675920*v**4 + 93817478273572416*v**6) -

- 41287680*r**5*

- (94093784713071 - 1250846573090382*v**2 -

- 8969352529977392*v**4 + 49587992868653760*v**6 +

- 9853891643771520*v**8) -

- 61440*r**4*(-4931360757517671 + 111162807540305640*v**2 -

- 341488009675121904*v**4 - 1349570909297766400*v**6 +

- 3339682557973601280*v**8) -

- 122880*r**3*(115311651991713 - 3205582644247668*v**2 +

- 21186805831200198*v**4 - 22636216114826184*v**6 -

- 82502104862326240*v**8 + 85388234245087872*v**10) +

- 192*r**2*(2172688760787939 - 68755345498995120*v**2 +

- 594437324079273696*v**4 - 1732248770187859200*v**6 +

NASA/TM—2005-213976 36
- 657607468274369280*v**8 + 1979673708999024640*v**10) -

- 27*(-1009995530787 + 74309862642840*v**2 -

- 875005671090384*v**4 + 4293338196259584*v**6 -

- 9746141498802432*v**8 + 7380223941482496*v**10 +

- 1971581659811840*v**12) +

- 32*r*(-213827634433485 + 7809056613423684*v**2 -

- 78990312626485344*v**4 + 322917377690563776*v**6 -

- 504350861926325760*v**8 + 150948224433948672*v**10 +

- 13505346093854720*v**12)) +

- 2*(-579164488159548445053419520000*r**12 +

- 1961191715319373579576934400000*r**13 -

- 4219064486461440000*r**11*

- (-15359554547 + 260703519934*v**2) +

- 11719623573504000*r**10*

- (-83020302737 + 14746647361080*v**2) -

- 9512681472000*r**8*

- (-9761508015143 + 129985977196304*v**2 +

- 572297552058848*v**4) +

- 38050725888000*r**9*

- (-16875641979912 - 157709025385835*v**2 +

- 2841462403951248*v**4) -

- 6606028800*r**7*

- (992837565004693 - 29231979456070830*v**2 +

- 131191705522068768*v**4 + 125738952524375520*v**6) -

- 412876800*r**6*

- (-538711671355491 + 30466191272258128*v**2 -

- 325903046690708736*v**4 + 482733945503753472*v**6) -

- 41287680*r**5*

- (-103170194733375 - 8146923188365390*v**2 +

- 193505373764547392*v**4 - 736520632101652800*v**6 +

- 310399731783221760*v**8) +

- 307200*r**4*(-3133463798505879 + 29163429763252848*v**2 +

- 637275042406935408*v**4 - 5656716299520763904*v**6 +

- 9808111688628126720*v**8) +

- 61440*r**3*(948011830571583 - 19617646470228720*v**2 +

- 43577030101694544*v**4 + 681912210566310480*v**6 -

- 2834559992969425280*v**8 + 2110158314000513280*v**10) -

- 960*r**2*(2158755861170673 - 53898431669866944*v**2 +

- 346419096157008864*v**4 - 173053798755784704*v**6 -

- 3763428220024482560*v**8 + 6893215787690860544*v**10) +

- 9*(-20129692202145 + 1449671858802000*v**2 -

- 12296086881752592*v**4 + 47340492487649280*v**6 -

- 72658653626085120*v**8 - 40528416929239040*v**10 +

- 165266499427307520*v**12) -

- 160*r*(-278158668457275 + 7481171537805564*v**2 -

- 59885380704290976*v**4 + 176103003953732544*v**6 -

- 4250652217341696*v**8 - 619379876900419584*v**10 +

- 363292515836551168*v**12)))*Sin(beta) +

- 1458*(93149179290552303737634816000*r**12 -

- 246877865688655000523243520000*r**13 +

- 421906448646144000*r**11*(-43287796421 + 462914161996*v**2) -

- 11719623573504000*r**10*

- (-216435349655 + 4359492372864*v**2) -

- 104639496192000*r**9*

- (2698620485907 - 72933563676988*v**2 + 264833023123872*v**4)\

NASA/TM—2005-213976 37
- + 1902536294400*r**8*

- (13805511831569 - 434288957577530*v**2 +

- 2680609668773344*v**4) +

- 3303014400*r**7*

- (-613864546602751 + 21323646693005844*v**2 -

- 169854088665737520*v**4 + 301270771112410176*v**6) -

- 412876800*r**6*(-302987318486055 + 11629586218331552*v**2 -

- 109574401523831040*v**4 + 327268567196702208*v**6) -

- 82575360*r**5*(72551512578735 - 3088646615010355*v**2 +

- 33597781165796696*v**4 - 131158083360499680*v**6 +

- 142648823651019840*v**8) +

- 61440*r**4*(3532323834269955 - 163994687251347960*v**2 +

- 2053511071558037616*v**4 - 9719048608637419520*v**6 +

- 17333519957424491520*v**8) +

- 61440*r**3*(-95983707035637 + 4666317548673600*v**2 -

- 65918071771476876*v**4 + 370758980823898896*v**6 -

- 876913488411714880*v**8 + 633480120919505664*v**10) -

- 960*r**2*(-119780938536327 + 5879215360055376*v**2 -

- 89877487849922016*v**4 + 591656152031882496*v**6 -

- 1756770906870959872*v**8 + 2154121759149359104*v**10) -

- 640*r*(2071112379030 - 108494532089289*v**2 +

- 1746481645544688*v**4 - 12777458758845552*v**6 +

- 46084793355740736*v**8 - 77624201846410752*v**10 +

- 43548588811277312*v**12) +

- 9*(350489177685 - 41944107421080*v**2 +

- 745001267346288*v**4 - 5869666090817280*v**6 +

- 24618475207745280*v**8 - 52183701319014400*v**10 +

- 47443699365826560*v**12) +

- beta**2*(-5851842503476545885044736000*r**12 +

- 14584512691569717902573568000*r**13 -

- 84381289729228800*r**11*

- (-14478492031 + 149040099780*v**2) +

- 2343924714700800*r**10*

- (-75253922725 + 1470778685712*v**2) -

- 3805072588800*r**8*

- (448677732615 - 14739871368323*v**2 + 92102778883808*v**4)

- + 1902536294400*r**9*

- (10238612744539 - 276767674825180*v**2 +

- 967663477977504*v**4) -

- 660602880*r**7*

- (-181466996077885 + 6823946603640540*v**2 -

- 57932914257676944*v**4 + 104446520276894400*v**6) +

- 82575360*r**6*

- (-80535955555197 + 3395324450176640*v**2 -

- 35318602303764480*v**4 + 111698694999201792*v**6) +

- 82575360*r**5*

- (3488621369112 - 162446990875029*v**2 +

- 1973806767302536*v**4 - 8439247978353312*v**6 +

- 9655698751530432*v**8) -

- 430080*r**4*(22370894453907 - 1116354114342936*v**2 +

- 15490999132628976*v**4 - 81679616621400064*v**6 +

- 159694719017303040*v**8) -

- 12288*r**3*(-20043453482109 + 1025042979865680*v**2 -

- 15744856565515140*v**4 + 98344407722704560*v**6 -

- 259752971261963200*v**8 + 206870355639978240*v**10) +

NASA/TM—2005-213976 38
- 192*r**2*(-23881268931597 + 1213649688770352*v**2 -

- 19805244850217376*v**4 + 142185814593488640*v**6 -

- 468574408032318720*v**8 + 637535391567757312*v**10) -

- 27*(2583570897 - 510424962264*v**2 + 9739448713008*v**4 -

- 81264819767040*v**6 + 365709873537792*v**8 -

- 849889633572864*v**10 + 863173582524416*v**12) +

- 64*r*(775381807791 - 42530703638166*v**2 +

- 723105192345984*v**4 - 5648368335055776*v**6 +

- 22146137322791040*v**8 - 41206726443156480*v**10 +

- 25169881674348544*v**12)))*Sin(2*beta) +

- (9*beta**2*(258734708055 - 15549235181138349398163456000*r**12 +

- 35623071438746881345191936000*r**13 - 4550593234104*v**2 +

- 20409795982224*v**4 + 127896243169536*v**6 -

- 2012012824953600*v**8 + 9192060007925760*v**10 -

- 16060874463268864*v**12 -

- 928194187021516800*r**11*

- (-3497258887 + 38725811540*v**2) +

- 2343924714700800*r**10*

- (-180757714199 + 4008278601624*v**2) +

- 13317754060800*r**9*

- (2868920831899 - 94143364395740*v**2 +

- 365360529862944*v**4) -

- 951268147200*r**8*

- (2606643331919 - 112245625081044*v**2 +

- 877406250294112*v**4) +

- 82575360*r**6*

- (-50147232969059 + 3213399676226960*v**2 -

- 50398337480389248*v**4 + 217442907736898304*v**6) -

- 660602880*r**7*

- (-179206790012575 + 9548202735783300*v**2 -

- 112209081625691568*v**4 + 254474776093496640*v**6) +

- 41287680*r**5*

- (2494786113855 - 192837447947102*v**2 +

- 3798718875072080*v**4 - 23918871981926208*v**6 +

- 36971243669071488*v**8) -

- 61440*r**4*(26320552610031 - 2637942453019944*v**2 +

- 65270894920980336*v**4 - 550491961326555136*v**6 +

- 1594432797576299520*v**8) -

- 24576*r**3*(-319858286049 + 80848429244100*v**2 -

- 2742568464175470*v**4 + 30075460967614120*v**6 -

- 127397218441488800*v**8 + 152803760846808960*v**10) +

- 192*r**2*(1994151317589 + 31633066726704*v**2 -

- 3388133391429600*v**4 + 51329238364212480*v**6 -

- 290300163578612480*v**8 + 614902888963170304*v**10) +

- 32*r*(-470940516531 + 8160191550684*v**2 +

- 63861953714784*v**4 - 2080336736786112*v**6 +

- 15855413162964480*v**8 - 49194214246434816*v**10 +

- 43308340658917376*v**12)) -

- 2*(30166464489525 - 1842098813469762639567519744000*r**12 +

- 4164992816402544456923873280000*r**13 -

- 487728092905920*v**2 + 1308808254098448*v**4 +

- 26450188396715520*v**6 - 303662123843800320*v**8 +

- 1292370351419105280*v**10 - 2180785886053765120*v**12 -

- 5062877383753728000*r**11*

- (-77330254127 + 840198168667*v**2) +

NASA/TM—2005-213976 39
- 35158870720512000*r**10*

- (-1484489830757 + 32201219462232*v**2) -

- 7610145177600*r**8*

- (42023175324339 - 1769435026490695*v**2 +

- 13500411156203664*v**4) +

- 19025362944000*r**9*

- (253006881664231 - 8116017220136826*v**2 +

- 30818687380278336*v**4) -

- 118908518400*r**7*

- (-131705920213382 + 6852511340086811*v**2 -

- 78586799399187768*v**4 + 173909698940190224*v**6) +

- 412876800*r**6*

- (-1370490234992227 + 85171108096236432*v**2 -

- 1301996090061407232*v**4 + 5482204198604107008*v**6) +

- 41287680*r**5*

- (359545292830305 - 26358525922696670*v**2 +

- 503356757273686944*v**4 - 3094050621272811840*v**6 +

- 4663191899411308800*v**8) -

- 552960*r**4*(479017085154345 - 42174611799274080*v**2 +

- 990371724210410416*v**4 - 8113035789389404160*v**6 +

- 22930291222525137920*v**8) -

- 61440*r**3*(-41456131042059 + 5228930118173760*v**2 -

- 156526840186493112*v**4 + 1641186719654641744*v**6 -

- 6763031918331279360*v**8 + 7902816330307804416*v**10) +

- 960*r**2*(20048562602793 + 2098046110342752*v**2 -

- 106703534850323616*v**4 + 1459078409555891712*v**6 -

- 7925337652125707008*v**8 + 16375351377067941888*v**10)\

- + 1440*r*(-1068273665775 + 12808707849756*v**2 +

- 338133889156320*v**4 - 7104681800882496*v**6 +

- 49965166436758272*v**8 - 149745393445836800*v**10 +

- 129453832195702784*v**12)))*Sin(3*beta)))/

- 2.1912208461633946e17

The imaginary part of the amplification factor for the c4o3 method is shown here in Fortran form due to

its length:

\Im(G)=

(v*(-588807047879148744*beta + 5029688315970720*beta**3 +

- 37059670203005168640*beta*r - 328971137267483136*beta**3*r -

- 1534749197369015130624*beta*r**2 +

- 11801835930422937600*beta**3*r**2 +

- 40832065370654591385600*beta*r**3 -

- 56904951116416352256*beta**3*r**3 -

- 622294838643747115008000*beta*r**4 -

- 12851851697346192998400*beta**3*r**4 +

- 992192170426717962240000*beta*r**5 +

- 643479477553473666416640*beta**3*r**5 +

- 201463802968439566903541760*beta*r**6 -

- 17461527698797546910515200*beta**3*r**6 -

- 5286721179261333240859852800*beta*r**7 +

- 313381129818851095672258560*beta**3*r**7 +

- 71155114020377497763315712000*beta*r**8 -

- 3822300621955502779347763200*beta**3*r**8 -

NASA/TM—2005-213976 40
- 447028423148769786829209600000*beta*r**9 +

- 28368526140181720582638796800*beta**3*r**9 -

- 1601877496352558815064044339200*beta*r**10 -

- 59847788836847349695250432000*beta**3*r**10 +

- 58804023178508604265092612096000*beta*r**11 -

- 1145586894161661438329880576000*beta**3*r**11 -

- 501654460948288553120465682432000*beta*r**12 +

- 12480187407814541778213666816000*beta**3*r**12 +

- 1674235923789035490017311457280000*beta*r**13 -

- 44428501533548568301629603840000*beta**3*r**13 +

- 10856641400139384000*beta*v**2 -

- 61032243152201472*beta**3*v**2 -

- 868027203027200001024*beta*r*v**2 +

- 2811635262800179200*beta**3*r*v**2 +

- 35316960283429877145600*beta*r**2*v**2 +

- 127489092611209003008*beta**3*r**2*v**2 -

- 760525851586382093352960*beta*r**3*v**2 -

- 19737100631473481318400*beta**3*r**3*v**2 +

- 3084002023769167527936000*beta*r**4*v**2 +

- 1039016222710842465976320*beta**3*r**4*v**2 +

- 314967737626958823476428800*beta*r**5*v**2 -

- 32380786633438227175833600*beta**3*r**5*v**2 -

- 10107460755831691793557094400*beta*r**6*v**2 +

- 661442776702898338434908160*beta**3*r**6*v**2 +

- 145254712110721815333253939200*beta*r**7*v**2 -

- 8735486181850552701498163200*beta**3*r**7*v**2 -

- 786248901199484763726938112000*beta*r**8*v**2 +

- 65791903274387010222450278400*beta**3*r**8*v**2 -

- 7318887101806147288826904576000*beta*r**9*v**2 -

- 97419093993513730330066944000*beta**3*r**9*v**2 +

- 157738390755256383450315227136000*beta*r**10*v**2 -

- 2984392362118233250213134336000*beta**3*r**10*v**2 -

- 959035275737495023494484721664000*beta*r**11*v**2 +

- 23052633926807309847814471680000*beta**3*r**11*v**2 -

- 79250271825534920064*beta*v**4 +

- 30583554037562880*beta**3*v**4 +

- 6421630897117767352320*beta*r*v**4 +

- 47492496337006411776*beta**3*r*v**4 -

- 203859543829363430473728*beta*r**2*v**4 -

- 6802413221178936852480*beta**3*r**2*v**4 +

- 486202955408031891456000*beta*r**3*v**4 +

- 442357687991309237747712*beta**3*r**3*v**4 +

- 179634315623296769686241280*beta*r**4*v**4 -

- 16792980732045908980531200*beta**3*r**4*v**4 -

- 6422236975344074427570585600*beta*r**5*v**4 +

- 400971483120935712923320320*beta**3*r**5*v**4 +

- 101773594843604206525266001920*beta*r**6*v**4 -

- 5885633663530072923753676800*beta**3*r**6*v**4 -

- 554704432089795085806128332800*beta*r**7*v**4 +

- 45308606820467398505366814720*beta**3*r**7*v**4 -

- 6059100162146023672974095155200*beta*r**8*v**4 -

- 26477965133207704812965068800*beta**3*r**8*v**4 +

- 97208121972520878672118284288000*beta*r**9*v**4 -

- 2004030344796760868503368499200*beta**3*r**9*v**4 +

- 280563375107936286720*beta*v**6 +

NASA/TM—2005-213976 41
- 3124168881320337408*beta**3*v**6 -

- 16539252149112612175872*beta*r*v**6 -

- 741601712426909368320*beta**3*r*v**6 -

- 101728757391537173299200*beta*r**2*v**6 +

- 67557036781240920834048*beta**3*r**2*v**6 +

- 37767943066486311089602560*beta*r**3*v**6 -

- 3207640886027135012044800*beta**3*r**3*v**6 -

- 1396197232635062331113472000*beta*r**4*v**6 +

- 86683666972156365688012800*beta**3*r**4*v**6 +

- 22881291722928390218239180800*beta*r**5*v**6 -

- 1358716656503716645227724800*beta**3*r**5*v**6 -

- 125760631653986542616051712000*beta*r**6*v**6 +

- 10643200834575207229795860480*beta**3*r**6*v**6 -

- 996010134491996472481520025600*beta*r**7*v**6 -

- 11246255265294721400124211200*beta**3*r**7*v**6 -

- 358922944231258576896*beta*v**8 -

- 23015585822443315200*beta**3*v**8 -

- 11941623566508633292800*beta*r*v**8 +

- 3743409734999328817152*beta**3*r*v**8 +

- 3210170415656980278804480*beta*r**2*v**8 -

- 250908047549192562278400*beta**3*r**2*v**8 -

- 140105518368166183698432000*beta*r**3*v**8 +

- 8288905329514443807129600*beta**3*r**3*v**8 +

- 2267320234111262990637465600*beta*r**4*v**8 -

- 134105137056035364667392000*beta**3*r**4*v**8 -

- 7259350157559910527034982400*beta*r**5*v**8 +

- 805428580219484723887472640*beta**3*r**5*v**8 -

- 495787807221085716480*beta*v**10 +

- 69672616691407847424*beta**3*v**10 +

- 88601438767640178327552*beta*r*v**10 -

- 7191691599933383639040*beta**3*r*v**10 -

- 5321169629212015853568000*beta*r**2*v**10 +

- 316068470842954088448000*beta**3*r**2*v**10 +

- 98710173002797443585146880*beta*r**3*v**10 -

- 5423944737688441690521600*beta**3*r**3*v**10 +

- 1283927763562788519936*beta*v**12 -

- 79564408586669260800*beta**3*v**12 -

- 46335501936495034368000*beta*r*v**12 +

- 3408981391065816760320*beta**3*r*v**12 -

- 243*beta*(beta**2*(-57683669631863209448177664000*r**12 +

- 173015870211123787302174720000*r**13 -

- 421906448646144000*r**11*

- (-21617280715 + 273889751564*v**2) +

- 2343924714700800*r**10*

- (-399594918611 + 10694645915880*v**2) -

- 951268147200*r**8*

- (7496356089035 - 273113326736812*v**2 +

- 2137216758218080*v**4) +

- 1902536294400*r**9*

- (41595043870467 - 1551530871852500*v**2 +

- 7494132128690592*v**4) +

- 82575360*r**6*

- (-717986639667879 + 21486595174684016*v**2 -

- 161240285877566592*v**4 + 512543000161882368*v**6) -

- 660602880*r**7*

NASA/TM—2005-213976 42
- (-1054801734874837 + 32563398611203140*v**2 -

- 264101846109232464*v**4 + 583098278454795840*v**6) +

- 41287680*r**5*

- (94795433102277 - 3047017849889958*v**2 +

- 24033401976772592*v**4 - 78895185920097600*v**6 +

- 94232269568938368*v**8) -

- 61440*r**4*(3104552759774103 - 108755814337086168*v**2 +

- 992484062573122224*v**4 - 3457844668905410560*v**6 +

- 5184867573569725440*v**8) -

- 24576*r**3*(-271207080146889 + 10250945050778100*v**2 -

- 107797789683950562*v**4 + 436899751570237320*v**6 -

- 749392863017170400*v**8 + 438028314410897280*v**10) +

- 192*r**2*(-829348380467703 + 33079948668167184*v**2 -

- 389320831684735200*v**4 + 1889296626965601024*v**6 -

- 4024984550910324480*v**8 + 3693650363298488320*v**10) -

- 27*(297709173819 - 23981152303080*v**2 +

- 342299730399120*v**4 - 2140251513313536*v**6 +

- 6829065061044480*v**8 - 10372466907949056*v**10 +

- 6410253269020672*v**12) +

- 32*r*(69296470633935 - 3020407834350708*v**2 +

- 38725919708165280*v**4 - 217803616412366016*v**6 +

- 580822312401040896*v**8 - 721684033577183232*v**10 +

- 339501476090900480*v**12)) -

- 3*(-775772574024358258148376576000*r**12 +

- 2438132793167793018151895040000*r**13 -

- 421906448646144000*r**11*

- (-257254367127 + 3614160898348*v**2) +

- 2343924714700800*r**10*

- (-3402185718667 + 126271369928760*v**2) +

- 66588770304000*r**9*

- (3620684948021 - 401361376717532*v**2 +

- 2603560564774752*v**4) -

- 951268147200*r**8*

- (-3376274173575 - 1342889495673180*v**2 +

- 20112535030790048*v**4) -

- 3303014400*r**7*

- (-218271568715125 + 12397919089960596*v**2 -

- 254809926086013648*v**4 + 1067011446179418432*v**6) +

- 82575360*r**6*

- (-2915872934335767 + 52293334784431120*v**2 -

- 203709865093847424*v**4 + 2553069348088170240*v**6) +

- 41287680*r**5*

- (651620035353411 - 14395064916992790*v**2 +

- 44726295355144816*v**4 - 135736772059360320*v**6 +

- 543435526019068800*v**8) -

- 61440*r**4*(29153895283741815 - 763599752521182360*v**2 +

- 4358868307579671984*v**4 - 6185181328054722560*v**6 +

- 8606299462809646080*v**8) -

- 245760*r**3*(-326013752062737 + 9434326912362198*v**2 -

- 73410801524614449*v**4 + 173104319746778292*v**6 -

- 76702873012875760*v**8 + 7731509141205696*v**10) +

- 192*r**2*(-12623593377224871 + 384637417525282320*v**2 -

- 3498380150575610592*v**4 + 12206778765661843200*v**6 -

- 14025471592493326080*v**8 + 4061847949873602560*v**10)\

- - 27*(6169818240603 - 461835791461800*v**2 +

NASA/TM—2005-213976 43
- 5001715839214224*v**4 - 23824855185096960*v**6 +

- 57403294649274624*v**8 - 55426485040900096*v**10 +

- 6685742743433216*v**12) +

- 32*r*(1331862912087885 - 44860039946650836*v**2 +

- 439976175253469280*v**4 - 1899540445326107328*v**6 +

- 3591794938498652160*v**8 - 2682471909002545152*v**10 +

- 1185920327022161920*v**12)))*Cos(beta) -

- 972*beta*(beta**2*(-922892772507352674022195200*r**12 +

- 2152486478012855315595264000*r**13 -

- 84381289729228800*r**11*(-2365554469 + 24378299660*v**2) +

- 2343924714700800*r**10*

- (-12158409919 + 243221791632*v**2) -

- 761014517760*r**8*

- (306846388688 - 11037490968085*v**2 + 73879224732544*v**4)

- + 1902536294400*r**9*

- (1555698089817 - 44461683026068*v**2 +

- 157343066562528*v**4) +

- 82575360*r**6*

- (-8390260233783 + 400592317432384*v**2 -

- 4796853064401408*v**4 + 16887687205248000*v**6) -

- 660602880*r**7*

- (-21735341420783 + 918912972202836*v**2 -

- 8740846641965616*v**4 + 16939526831464512*v**6) +

- 82575360*r**5*

- (319009475466 - 16669199445327*v**2 +

- 234442873415960*v**4 - 1144188407488224*v**6 +

- 1448247275003712*v**8) -

- 61440*r**4*(12813557189103 - 703861986264696*v**2 +

- 11174990937599664*v**4 - 67997567769399296*v**6 +

- 152053146144721920*v**8) -

- 12288*r**3*(-1500438468915 + 82527016827600*v**2 -

- 1422945532585836*v**4 + 10185289876987920*v**6 -

- 31139167116948800*v**8 + 28552272358337280*v**10) +

- 192*r**2*(-1650077479143 + 89009052247440*v**2 -

- 1602547661864160*v**4 + 12947818722667776*v**6 -

- 48796496561053440*v**8 + 76203903203262464*v**10) -

- 81*(-59934735 - 9592864344*v**2 + 218226663216*v**4 -

- 2010084284160*v**6 + 10031018607360*v**8 -

- 26368470194176*v**10 + 30903649701888*v**12) +

- 64*r*(46120044303 - 2849042564514*v**2 +

- 53345116022976*v**4 - 459076826220768*v**6 +

- 2011462884761472*v**8 - 4254001425607680*v**10 +

- 2909977010382848*v**12)) -

- 3*(-20133583894660502098280448000*r**12 +

- 51423299015953439407472640000*r**13 -

- 421906448646144000*r**11*

- (-9679278267 + 101865765716*v**2) +

- 2343924714700800*r**10*

- (-245765287109 + 4905031931328*v**2) -

- 1902536294400*r**8*

- (2959552682415 - 96219577428774*v**2 +

- 604433839198112*v**4) +

- 1902536294400*r**9*

- (33223356231995 - 906542266954940*v**2 +

- 3243439678639776*v**4) -

NASA/TM—2005-213976 44
- 660602880*r**7*

- (-620539208944357 + 22640836761551100*v**2 -

- 188809621496456976*v**4 + 341796645237347520*v**6) +

- 82575360*r**6*

- (-288543121430805 + 11671815463754336*v**2 -

- 116857257267352320*v**4 + 364001677713148416*v**6) +

- 82575360*r**5*

- (13104388796445 - 584971692371685*v**2 +

- 6766961878348840*v**4 - 27939797535271200*v**6 +

- 31561275189465024*v**8) -

- 61440*r**4*(612130122784485 - 29493473801105160*v**2 +

- 389745499167568656*v**4 - 1958865473739317248*v**6 +

- 3697607015636997120*v**8) -

- 12288*r**3*(-80726104925487 + 4027934222870400*v**2 -

- 59360722913632644*v**4 + 352703853912613680*v**6 -

- 887189420132478400*v**8 + 681034036144062720*v**10) +

- 192*r**2*(-98476779983445 + 4922372676072816*v**2 -

- 77715621186994080*v**4 + 534442122526535424*v**6 -

- 1675966626530069760*v**8 + 2177600498496274432*v**10) -

- 27*(14928433569 - 2207820187512*v**2 +

- 40585421655216*v**4 - 328743522743040*v**6 +

- 1427223068762880*v**8 - 3169036058834944*v**10 +

- 3055632709816320*v**12) +

- 128*r*(1651783345704 - 88594168691835*v**2 +

- 1464721895747088*v**4 - 11067394312967760*v**6 +

- 41628975096003264*v**8 - 73850069684021760*v**10 +

- 43397862838787072*v**12)))*Cos(2*beta) -

- 3*beta*(-6565413079503 + 398408459938326331671969792000*r**12 -

- 906088081101032576086179840000*r**13 + 110821561318728*v**2 -

- 401211159351120*v**4 - 4508563972688640*v**6 +

- 58656504616268544*v**8 - 257715918703392768*v**10 +

- 442123540826525696*v**12 +

- 421906448646144000*r**11*

- (-199054845767 + 2182021247692*v**2) -

- 2343924714700800*r**10*

- (-4729245686083 + 103656419886264*v**2) +

- 951268147200*r**8*

- (69844313562865 - 2972871503160924*v**2 +

- 22946621498684576*v**4) -

- 1902536294400*r**9*

- (531609461763455 - 17239248316624180*v**2 +

- 66134244014058528*v**4) -

- 82575360*r**6*(-1385166227803471 + 87325784946246928*v**2 -

- 1351565780861611392*v**4 + 5758348552837801728*v**6) +

- 660602880*r**7*(-4865642730162041 + 256056788770319940*v**2 -

- 2971496980680499728*v**4 + 6653161396905291840*v**6) -

- 41287680*r**5*(70947893367747 - 5326699377817030*v**2 +

- 103227072160039792*v**4 - 641970137232605760*v**6 +

- 979355319246328704*v**8) +

- 430080*r**4*(115071039657225 - 10705019299468200*v**2 +

- 257430127321849680*v**4 - 2138251218314289152*v**6 +

- 6115711802512051200*v**8) +

- 49152*r**3*(-7713649675629 + 1227142504767870*v**2 -

- 38765752221783573*v**4 + 414931844050079140*v**6 -

- 1732608805218292400*v**8 + 2050361230337542080*v**10) -

NASA/TM—2005-213976 45
- 192*r**2*(36136825502337 + 1550210230422288*v**2 -

- 101250428823753696*v**4 + 1447856822418607872*v**6 -

- 8012778473780811520*v**8 + 16755880172081340416*v**10) -

- 32*r*(-11208197155347 + 166109466520332*v**2 +

- 2472521938752096*v**4 - 61285385287929024*v**6 +

- 446619625746905088*v**8 - 1360377446820857856*v**10 +

- 1186546227591983104*v**12) +

- 3*beta**2*(-659108509091251205072486400*r**12 +

- 1523320518370749755424768000*r**13 -

- 84381289729228800*r**11*(-1612681903 + 18059678780*v**2) +

- 2343924714700800*r**10*(-7484112451 + 168099152424*v**2) +

- 1902536294400*r**9*

- (821151171723 - 27290457944596*v**2 +

- 107281230487200*v**4) -

- 190253629440*r**8*

- (526084952279 - 22934452411900*v**2 +

- 181725049039072*v**4) +

- 82575360*r**6*

- (-1956547557303 + 127748838546160*v**2 -

- 2032176036711552*v**4 + 8885105774274816*v**6) -

- 660602880*r**7*

- (-7130914417565 + 385061401604292*v**2 -

- 4585461138962640*v**4 + 10544008313745984*v**6) +

- 41287680*r**5*

- (93887635293 - 7526238878022*v**2 +

- 151021271377264*v**4 - 963450698392896*v**6 +

- 1510243631355264*v**8) -

- 61440*r**4*(892544804055 - 99481400362008*v**2 +

- 2549241380372400*v**4 - 21869195086548992*v**6 +

- 64186134339855360*v**8) -

- 24576*r**3*(-1149062193 + 2780159930100*v**2 -

- 104119336285650*v**4 + 1175524010390280*v**6 -

- 5058029287141600*v**8 + 6153750656565120*v**10) +

- 192*r**2*(110404472985 + 77801877648*v**2 -

- 119366959569120*v**4 + 1954509147636480*v**6 -

- 11342305865137920*v**8 + 24358626486591488*v**10) +

- 3*(3711406797 - 67873440024*v**2 + 358447305456*v**4 +

- 1117039880448*v**6 - 24498416113920*v**8 +

- 117600703752192*v**10 - 209771938164736*v**12) +

- 32*r*(-21509011593 + 420604058700*v**2 +

- 1291629672864*v**4 - 74872746083520*v**6 +

- 605030639961600*v**8 - 1917925378615296*v**10 +

- 1704777524703232*v**12)))*Cos(3*beta) -

- 729*(beta**2*(105741172866477573305008128000*r**12 -

- 345588754177326433582448640000*r**13 +

- 4640970935107584000*r**11*

- (-2850552659 + 44081501412*v**2) -

- 16407473002905600*r**10*

- (-37497573689 + 2176321827240*v**2) +

- 951268147200*r**8*

- (-8243164839537 + 23669349438860*v**2 +

- 1861865235800416*v**4) -

- 9512681472000*r**9*

- (-4347617889887 - 244038396953428*v**2 +

- 2266640436651360*v**4) +

NASA/TM—2005-213976 46
- 82575360*r**6*

- (201824348917275 + 5393607906635120*v**2 -

- 117193940856745344*v**4 + 40426634189395200*v**6) +

- 3303014400*r**7*

- (114111407047925 - 3788820566188332*v**2 +

- 5949837097675920*v**4 + 93817478273572416*v**6) -

- 41287680*r**5*

- (94093784713071 - 1250846573090382*v**2 -

- 8969352529977392*v**4 + 49587992868653760*v**6 +

- 9853891643771520*v**8) -

- 61440*r**4*(-4931360757517671 + 111162807540305640*v**2 -

- 341488009675121904*v**4 - 1349570909297766400*v**6 +

- 3339682557973601280*v**8) -

- 122880*r**3*(115311651991713 - 3205582644247668*v**2 +

- 21186805831200198*v**4 - 22636216114826184*v**6 -

- 82502104862326240*v**8 + 85388234245087872*v**10) +

- 192*r**2*(2172688760787939 - 68755345498995120*v**2 +

- 594437324079273696*v**4 - 1732248770187859200*v**6 +

- 657607468274369280*v**8 + 1979673708999024640*v**10) -

- 27*(-1009995530787 + 74309862642840*v**2 -

- 875005671090384*v**4 + 4293338196259584*v**6 -

- 9746141498802432*v**8 + 7380223941482496*v**10 +

- 1971581659811840*v**12) +

- 32*r*(-213827634433485 + 7809056613423684*v**2 -

- 78990312626485344*v**4 + 322917377690563776*v**6 -

- 504350861926325760*v**8 + 150948224433948672*v**10 +

- 13505346093854720*v**12)) +

- 2*(-579164488159548445053419520000*r**12 +

- 1961191715319373579576934400000*r**13 -

- 4219064486461440000*r**11*

- (-15359554547 + 260703519934*v**2) +

- 11719623573504000*r**10*

- (-83020302737 + 14746647361080*v**2) -

- 9512681472000*r**8*

- (-9761508015143 + 129985977196304*v**2 +

- 572297552058848*v**4) +

- 38050725888000*r**9*

- (-16875641979912 - 157709025385835*v**2 +

- 2841462403951248*v**4) -

- 6606028800*r**7*

- (992837565004693 - 29231979456070830*v**2 +

- 131191705522068768*v**4 + 125738952524375520*v**6) -

- 412876800*r**6*

- (-538711671355491 + 30466191272258128*v**2 -

- 325903046690708736*v**4 + 482733945503753472*v**6) -

- 41287680*r**5*

- (-103170194733375 - 8146923188365390*v**2 +

- 193505373764547392*v**4 - 736520632101652800*v**6 +

- 310399731783221760*v**8) +

- 307200*r**4*(-3133463798505879 + 29163429763252848*v**2 +

- 637275042406935408*v**4 - 5656716299520763904*v**6 +

- 9808111688628126720*v**8) +

- 61440*r**3*(948011830571583 - 19617646470228720*v**2 +

- 43577030101694544*v**4 + 681912210566310480*v**6 -

- 2834559992969425280*v**8 + 2110158314000513280*v**10) -

NASA/TM—2005-213976 47
- 960*r**2*(2158755861170673 - 53898431669866944*v**2 +

- 346419096157008864*v**4 - 173053798755784704*v**6 -

- 3763428220024482560*v**8 + 6893215787690860544*v**10) +

- 9*(-20129692202145 + 1449671858802000*v**2 -

- 12296086881752592*v**4 + 47340492487649280*v**6 -

- 72658653626085120*v**8 - 40528416929239040*v**10 +

- 165266499427307520*v**12) -

- 160*r*(-278158668457275 + 7481171537805564*v**2 -

- 59885380704290976*v**4 + 176103003953732544*v**6 -

- 4250652217341696*v**8 - 619379876900419584*v**10 +

- 363292515836551168*v**12)))*Sin(beta) +

- 1458*(93149179290552303737634816000*r**12 -

- 246877865688655000523243520000*r**13 +

- 421906448646144000*r**11*(-43287796421 + 462914161996*v**2) -

- 11719623573504000*r**10*

- (-216435349655 + 4359492372864*v**2) -

- 104639496192000*r**9*

- (2698620485907 - 72933563676988*v**2 + 264833023123872*v**4)\

- + 1902536294400*r**8*

- (13805511831569 - 434288957577530*v**2 +

- 2680609668773344*v**4) +

- 3303014400*r**7*

- (-613864546602751 + 21323646693005844*v**2 -

- 169854088665737520*v**4 + 301270771112410176*v**6) -

- 412876800*r**6*(-302987318486055 + 11629586218331552*v**2 -

- 109574401523831040*v**4 + 327268567196702208*v**6) -

- 82575360*r**5*(72551512578735 - 3088646615010355*v**2 +

- 33597781165796696*v**4 - 131158083360499680*v**6 +

- 142648823651019840*v**8) +

- 61440*r**4*(3532323834269955 - 163994687251347960*v**2 +

- 2053511071558037616*v**4 - 9719048608637419520*v**6 +

- 17333519957424491520*v**8) +

- 61440*r**3*(-95983707035637 + 4666317548673600*v**2 -

- 65918071771476876*v**4 + 370758980823898896*v**6 -

- 876913488411714880*v**8 + 633480120919505664*v**10) -

- 960*r**2*(-119780938536327 + 5879215360055376*v**2 -

- 89877487849922016*v**4 + 591656152031882496*v**6 -

- 1756770906870959872*v**8 + 2154121759149359104*v**10) -

- 640*r*(2071112379030 - 108494532089289*v**2 +

- 1746481645544688*v**4 - 12777458758845552*v**6 +

- 46084793355740736*v**8 - 77624201846410752*v**10 +

- 43548588811277312*v**12) +

- 9*(350489177685 - 41944107421080*v**2 +

- 745001267346288*v**4 - 5869666090817280*v**6 +

- 24618475207745280*v**8 - 52183701319014400*v**10 +

- 47443699365826560*v**12) +

- beta**2*(-5851842503476545885044736000*r**12 +

- 14584512691569717902573568000*r**13 -

- 84381289729228800*r**11*

- (-14478492031 + 149040099780*v**2) +

- 2343924714700800*r**10*

- (-75253922725 + 1470778685712*v**2) -

- 3805072588800*r**8*

- (448677732615 - 14739871368323*v**2 + 92102778883808*v**4)

- + 1902536294400*r**9*

NASA/TM—2005-213976 48
- (10238612744539 - 276767674825180*v**2 +

- 967663477977504*v**4) -

- 660602880*r**7*

- (-181466996077885 + 6823946603640540*v**2 -

- 57932914257676944*v**4 + 104446520276894400*v**6) +

- 82575360*r**6*

- (-80535955555197 + 3395324450176640*v**2 -

- 35318602303764480*v**4 + 111698694999201792*v**6) +

- 82575360*r**5*

- (3488621369112 - 162446990875029*v**2 +

- 1973806767302536*v**4 - 8439247978353312*v**6 +

- 9655698751530432*v**8) -

- 430080*r**4*(22370894453907 - 1116354114342936*v**2 +

- 15490999132628976*v**4 - 81679616621400064*v**6 +

- 159694719017303040*v**8) -

- 12288*r**3*(-20043453482109 + 1025042979865680*v**2 -

- 15744856565515140*v**4 + 98344407722704560*v**6 -

- 259752971261963200*v**8 + 206870355639978240*v**10) +

- 192*r**2*(-23881268931597 + 1213649688770352*v**2 -

- 19805244850217376*v**4 + 142185814593488640*v**6 -

- 468574408032318720*v**8 + 637535391567757312*v**10) -

- 27*(2583570897 - 510424962264*v**2 + 9739448713008*v**4 -

- 81264819767040*v**6 + 365709873537792*v**8 -

- 849889633572864*v**10 + 863173582524416*v**12) +

- 64*r*(775381807791 - 42530703638166*v**2 +

- 723105192345984*v**4 - 5648368335055776*v**6 +

- 22146137322791040*v**8 - 41206726443156480*v**10 +

- 25169881674348544*v**12)))*Sin(2*beta) +

- (9*beta**2*(258734708055 - 15549235181138349398163456000*r**12 +

- 35623071438746881345191936000*r**13 - 4550593234104*v**2 +

- 20409795982224*v**4 + 127896243169536*v**6 -

- 2012012824953600*v**8 + 9192060007925760*v**10 -

- 16060874463268864*v**12 -

- 928194187021516800*r**11*

- (-3497258887 + 38725811540*v**2) +

- 2343924714700800*r**10*

- (-180757714199 + 4008278601624*v**2) +

- 13317754060800*r**9*

- (2868920831899 - 94143364395740*v**2 +

- 365360529862944*v**4) -

- 951268147200*r**8*

- (2606643331919 - 112245625081044*v**2 +

- 877406250294112*v**4) +

- 82575360*r**6*

- (-50147232969059 + 3213399676226960*v**2 -

- 50398337480389248*v**4 + 217442907736898304*v**6) -

- 660602880*r**7*

- (-179206790012575 + 9548202735783300*v**2 -

- 112209081625691568*v**4 + 254474776093496640*v**6) +

- 41287680*r**5*

- (2494786113855 - 192837447947102*v**2 +

- 3798718875072080*v**4 - 23918871981926208*v**6 +

- 36971243669071488*v**8) -

- 61440*r**4*(26320552610031 - 2637942453019944*v**2 +

- 65270894920980336*v**4 - 550491961326555136*v**6 +

NASA/TM—2005-213976 49
- 1594432797576299520*v**8) -

- 24576*r**3*(-319858286049 + 80848429244100*v**2 -

- 2742568464175470*v**4 + 30075460967614120*v**6 -

- 127397218441488800*v**8 + 152803760846808960*v**10) +

- 192*r**2*(1994151317589 + 31633066726704*v**2 -

- 3388133391429600*v**4 + 51329238364212480*v**6 -

- 290300163578612480*v**8 + 614902888963170304*v**10) +

- 32*r*(-470940516531 + 8160191550684*v**2 +

- 63861953714784*v**4 - 2080336736786112*v**6 +

- 15855413162964480*v**8 - 49194214246434816*v**10 +

- 43308340658917376*v**12)) -

- 2*(30166464489525 - 1842098813469762639567519744000*r**12 +

- 4164992816402544456923873280000*r**13 -

- 487728092905920*v**2 + 1308808254098448*v**4 +

- 26450188396715520*v**6 - 303662123843800320*v**8 +

- 1292370351419105280*v**10 - 2180785886053765120*v**12 -

- 5062877383753728000*r**11*

- (-77330254127 + 840198168667*v**2) +

- 35158870720512000*r**10*

- (-1484489830757 + 32201219462232*v**2) -

- 7610145177600*r**8*

- (42023175324339 - 1769435026490695*v**2 +

- 13500411156203664*v**4) +

- 19025362944000*r**9*

- (253006881664231 - 8116017220136826*v**2 +

- 30818687380278336*v**4) -

- 118908518400*r**7*

- (-131705920213382 + 6852511340086811*v**2 -

- 78586799399187768*v**4 + 173909698940190224*v**6) +

- 412876800*r**6*

- (-1370490234992227 + 85171108096236432*v**2 -

- 1301996090061407232*v**4 + 5482204198604107008*v**6) +

- 41287680*r**5*

- (359545292830305 - 26358525922696670*v**2 +

- 503356757273686944*v**4 - 3094050621272811840*v**6 +

- 4663191899411308800*v**8) -

- 552960*r**4*(479017085154345 - 42174611799274080*v**2 +

- 990371724210410416*v**4 - 8113035789389404160*v**6 +

- 22930291222525137920*v**8) -

- 61440*r**3*(-41456131042059 + 5228930118173760*v**2 -

- 156526840186493112*v**4 + 1641186719654641744*v**6 -

- 6763031918331279360*v**8 + 7902816330307804416*v**10) +

- 960*r**2*(20048562602793 + 2098046110342752*v**2 -

- 106703534850323616*v**4 + 1459078409555891712*v**6 -

- 7925337652125707008*v**8 + 16375351377067941888*v**10)\

- + 1440*r*(-1068273665775 + 12808707849756*v**2 +

- 338133889156320*v**4 - 7104681800882496*v**6 +

- 49965166436758272*v**8 - 149745393445836800*v**10 +

- 129453832195702784*v**12)))*Sin(3*beta)))/

- 2.1912208461633946e17

NASA/TM—2005-213976 50
References

1Beale, W.T.:”Free Piston Stirling Engines - Some Model Tests and Simulations”, [Preliminary investigation of free-piston

Stirling engines.], SAE International Automtive Engineering Conference, Jan. 1969 2Thieme, L.G.; Schreiber, J.G. : ”NASA GRC Stirling Technology Development Overview”, NASA/TM-2003-212454,

Aug. 2003 3Cengel, Y.A.; Boles, M.A. : ”Thermodynamics: An Engineering Approach”, McGraw-Hill, 1989

4Tew, R.C., Thieme, L.G., Dudenhoefer, J.E. : ”Recent Stirling Engine Loss-Understanding Results”, 25th IECEC,

NASA TM 103122, 1990 5Hirata, K., Kagawa, N., Takeuchi, M., Yamashita, I., Isshiki, N., Hamaguchi, K. : ”Test Results of Applicative 100 W

Stirling Engine”, 96204. 6Ibrahim, M.B., Tew, R.C., Zhang, Z., Gedeon, D. Simon, T. : ”CFD Modeling of Free-Piston Stirling Engines”,

NASA/TM-2001-211132, IECEC2001-CT-38, Sept. 2001. 7Dyson, R.W., ”Technique For Very High Order Nonlinear Simulation and Validation”, J. Comp. Acoustics, Vol. 10, No.

2, pp. 211-229, 2002. 8Carpenter, M.H., Nordstrom, J., Gottlieb, D. ”A Stable and Conservative Interface Treatment of Arbitrary Spatial

Accuracy”, Institute for Computer Applications in Science and Engineering (ICASE) Report No. 98-12, 1998. 9Carpeneter, M.H., Casper, J.H. ”The Accuracy of Shock Capturing in Two Spatial Dimensions”, AIAA 97-2107, July

1997.10Tew, R.C., Ibrahim, M.B. : ”Two-dimensional Compressible Non-Acoustic Modeling of Stirling Machine-Type Components”, J. of Prop. and Power, Vol. 19, No. 5, Sept.-Oct. 2003 11Thieme, L.G., Schreiber, J.G., Mason, L.S. : ”Stirling Technology Development at NASA GRC”, NASA/TM-2001-

211315/REV1, Jan. 2002 12Schreiber, J. : ”Tutorial on Free-Piston Stirling Power Conversion Technology (How Does it Work?), Space Technology

and Applications International Forum, Feb. 2001 13Furlong, R., Shaltens, R. : ”Technology Assessment of DOE’s 55-We Stirling Technology Demonstrator Convertor

(TDC)”, NASA/TM-2000-210509, Oct. 2000 14Stanescu, D., Habashi, W.G., ”2N-Storage Low Dissipation and Dispersion Runge-Kutta Schemes for Computational

Acoustics”, Journal of Computational Physics, Vol 143, no. 2, 1998, pp. 674. 15Goodrich, J.W., ”A Comparison of Numerical Methods for Computational Aeroacoustics”, AIAA 99-1943, May 1999. 16Goodrich, J.W., ”Application of a New High Order Finite Difference Scheme to Acoustic Propagation With the Linearized Euler Equation,” NASA/TM-1993-106454, June 1993. 17Goodrich, J.W. ”Accurate Finite Difference Algorithms for Computational Aeroacoustics”, Computational Fluid Dynamics Review 1998, edited by M.M Hafez and K. Oshima, World Scientific, Singapore, 1998. 18Garabedian, P.R., ”The Method of Power Series:The Cauchy-Kowalewski Theorem,” Partial Differential Equations,

2nd ed., Chelsea, New York, 1986, pp. 6-16. 19Goodrich, J.W., ”An Approach to the Development of Numerical Algorithms for First Order Linear Hyperbolic Systems

in Multiple Space Dimensions:The Constant Coefficient Case,” NASA/TM-1995-106928, September 1995. 20Dyson, R.W., ”Demonstration of Ultra HI-FI Methods”, Int. J. of Computational Fluid Dynamics”, accepted for

publication. 21Dyson, R.W., Goodrich, J.W. ”A Very High Order, Adaptable MESA Implementation for Aeroacoustic Computations”,

NASA/TM-2000-209944, April 2000. 22Hoffman, K.A., Chiang, S.T., ”Computational Fluid Dynamics For Engineers – Volume I and II”, Engineering Education System, Wichita, Kansas, 1993 23Anderson, D.; Tannehill, J.C. ; Pletcher, R.H. : ”Computational Fluid Mechanics and Heat Transfer”, McGraw-Hill,

1984.24Chung, T.J. : ”Computational Fluid Dynamics”, Cambridge University Press, 2002, pp. 394. 25Sun, Y. ; Wang, Z.J. : ”High-Order Spectral Volume Method for the Navier-Stokes Equations on Unstructured Grids”,

AIAA-2004-2133, 2004 26Benton, E.R. ; Platzman, G.W. : ”A Table of Solutions of the One-dimensional Burgers Equation, Q. Appl. Math.,

vol. 30, pp. 195-212, 1972. 27Lele, S.K.: ”Compact finite difference schemes with spectral-like resolution”, J. Comp. Phys., 183, 16-42., 1992 28Patankar, S.V. and Spalding, D.B., ”A Calculation procedure for heat, mass and momentum transfer in threedimensional parabolic flows.”, Int. J. Heat Mass Transfer, 15, 1787-1806. 29Issa, R. I., ”Solution of the implicitly discretized fluid flow equations by operator splitting.”, 62, 40-65 of compressible

and incompressible recirculating flows by a non-iterative scheme”, J. Comp. Phys., 62, 66-82. 30Issa, R. I., Gosman, A.D., and Watkins, A.P., ”The computation of compressible and incompressible recirculating

flows by a non-iterative scheme”, J. Comp. Phys., 62, 66-82. 31Anderson, J.D., ”Computational Fluid Dynamics: the Basics with Applications”, McGraw-Hill, 1995

NASA/TM—2005-213976 51
32Gedeon, D. : ”GLIMPS Version 4: User’s Manual”, Gedeon Associates, July 1992.

33Fletcher, C.A.J. : ”Computational Techniques for Fluid Dynamics - Volume I”, Springer-Verlag, 1988.

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Mar. 1988.

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346, Dec. 1992

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Geometries”, NASA CR 198416, Oct. 1995

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NASA/TM—2005-213976 52
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AIAA–2005–5580

E–15295

WBS–22–972–30–01

58

Stirling Analysis Comparison of Commercial Versus High-Order Methods

Rodger W. Dyson, Scott D. Wilson, Roy C. Tew, and Rikako Demko

CFD-ACE; FLUENT

Unclassified -Unlimited

Subject Category: 20

Prepared for the Third International Energy Conversion Engineering Conference sponsored by the American Institute of

Aeronautics and Astronautics, San Francisco, California, August 15–18, 2005. Rodger W. Dyson and Roy C. Tew,

NASA Glenn Research Center; and Scott D. Wilson and Rikako Demko, Sest, Inc., 18000 Jefferson Park, Suite 104,

Middleburg Heights, Ohio 44130. Responsible person, Rodger W. Dyson, organization code RPT, 216–433–9083.

Recently, three-dimensional Stirling engine simulations have been accomplished utilizing commercial Computational

Fluid Dynamics software. The validations reported can be somewhat inconclusive due to the lack of precise time accurate

experimental results from engines, export control/proprietary concerns, and the lack of variation in the methods utilized.

The last issue may be addressed by solving the same flow problem with alternate methods. In this work, a comprehensive

examination of the methods utilized in the commercial codes is compared with more recently developed high-order

methods. Specifically, Lele's compact scheme and Dyson's Ultra Hi-Fi method will be compared with the SIMPLE and

PISO methods currently employed in CFD-ACE, FLUENT, CFX, and STAR-CD (all commercial codes which can in

theory solve a three-dimensional Stirling model with sliding interfaces and their moving grids limit the effective time

accuracy). We will initially look at one-dimensional flows since the current standard practice is to design and optimize

Stirling engines with empirically corrected friction and heat transfer coefficients in an overall one-dimensional model.

This comparison provides an idea of the range in which commercial CFD software for modeling Stirling engines may be

expected to provide accurate results. In addition, this work provides a framework for improving current one-dimensional

analysis codes.