The conclusions are: Low Reynold s number, wall bounded flow allows economical use of large eddy simulation for turbulent transition modeling. UHF and Compact comparable at conjugate heat transfer. UHF much better for turbulence modeling Modern methods much more efficient than those currently available commercially

Demko, Rikako

Dyson, Rodger W.

Tew, Roy C.

Wilson, Scott D.

NASA Technical Reports Server

1Glenn Research Center at Lewis Field
Stirling Analysis Comparison of 
Commercial VS. High-Order 
Methods
Rodger W. Dyson, Scott D. Wilson
Roy C. Tew, and Rikako Demko
Thermal Energy Conversion Branch
August 15, 2005
https://ntrs.nasa.gov/search.jsp?R=20060005155 2019-08-29T21:17:34+00:00Z
2Glenn Research Center at Lewis Field
Stirling Simulation Numerical Error
•Remeshing interpolation
•Layering interpolation
•Diffusive time advance
•Low grid quality/skewness
•Sliding interface interpolation
•Artificial entropy
•Turbulence transition
3Glenn Research Center at Lewis Field
Dual Opposed Convertors
• High Efficiency – Low Mass Space Power
• Free Piston Geometry is Essentially Smooth
4Glenn Research Center at Lewis Field
Whole Engine Simulation
•Bounded by Walls – No need for 
nonreflecting B.C.
•Kolmogorov scales fairly large
•Steep thermal gradients
•No shocks/subsonic/transitioning
•High-order friendly
5Glenn Research Center at Lewis Field
Curvilinear Features
6Glenn Research Center at Lewis Field
Stability Analysis
• Courant (CFL) number,  
r = c Δt / Δx
• Von Neumann number,  
v = μ Δt / Δx2
• Linear Viscous Burger’s 
Equation
2
2
x
u
x
uc
t
u
∂
∂=∂
∂+∂
∂ μ
7Glenn Research Center at Lewis Field
Compact Scheme 6th Order in Space
9/1,9/14,3/1
42
2211
11
===
Δ
−+Δ
−=⎟⎠
⎞⎜⎝
⎛
∂
∂+⎟⎠
⎞⎜⎝
⎛
∂
∂+⎟⎠
⎞⎜⎝
⎛
∂
∂ −+−+
+−
ba
x
uub
x
uua
x
u
x
u
x
u iiii
iii
α
αα
11/3,11/12,11/2
4
22
2
22
2
11
1
2
2
2
2
1
2
2
===
Δ
+−+Δ
+−=⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂+⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂+⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂ −+−+
+−
ba
x
uuub
x
uuua
x
u
x
u
x
u iiiiii
iii
α
αα
8Glenn Research Center at Lewis Field
Runge-Kutta 4th Order
( )
)(),(),(
22
6
2
2
)(
)3()3()2()2()1()1(
)3()2()1()1(
2)3(
1)2(
)1(
uRRuRRuRR
RRRRtuu
tRuu
Rtuu
Rtuu
ucuuR
nnn
n
n
nn
xxx
===
+++Δ+=
Δ+=
Δ+=
Δ+=
+−=
+
μ
9Glenn Research Center at Lewis Field
Current Practice
• Implicit 1st or 2nd order in time commercially
• 1st or 2nd order in space implicit
• Explicit/implicit 4th order in time academically
• Implicit 6th order compact scheme in space
10
Glenn Research Center at Lewis Field
Compact Scheme Stability Range
• Domain size affects stability since implicit
11
Glenn Research Center at Lewis Field
Basic UHF Technique
12
Glenn Research Center at Lewis Field
Derivatives of Cell Averages
13
Glenn Research Center at Lewis Field
Grid Singularity Resolution
14
Glenn Research Center at Lewis Field
C4o0 Linear Viscous Burger’s Equation
15
Glenn Research Center at Lewis Field
Efficiency Improves with Accuracy
16
Glenn Research Center at Lewis Field
Computational Domain Schematic
17
Glenn Research Center at Lewis Field
Amplification Factor Comparison
18
Glenn Research Center at Lewis Field
UHF Stability Range
19
Glenn Research Center at Lewis Field
Overview of Segregated Solution
20
Glenn Research Center at Lewis Field
Heat Transfer Test
0
3
4)( =⎟⎠
⎞⎜⎝
⎛ +−+∂
∂+∂
∂
xxv
t quuupTC
xt
E μρ
pC
k
x
T
x
Tu
t
T
ρα
α
=
∂
∂=∂
∂+∂
∂ ,2
2
Reduces to:
21
Glenn Research Center at Lewis Field
Comparison of Commercial & Advanced
22
Glenn Research Center at Lewis Field
Commercial Comparison
23
Glenn Research Center at Lewis Field
Turbulence Transition Efficiency
24
Glenn Research Center at Lewis Field
Points per Kolmogorov Wavelength
25
Glenn Research Center at Lewis Field
Wave Equation Amplification
26
Glenn Research Center at Lewis Field
Aliased Frequency Amplification
27
Glenn Research Center at Lewis Field
Conclusions
• Low Reynold’s number, wall bounded flow 
allows economical use of large eddy 
simulation for turbulent transition modeling
• UHF and Compact comparable at conjugate 
heat transfer
• UHF much better for turbulence modeling
• Modern methods much more efficient than 
those currently available commercially
28
Glenn Research Center at Lewis Field
29
Glenn Research Center at Lewis Field
30
Glenn Research Center at Lewis Field
31
Glenn Research Center at Lewis Field
32
Glenn Research Center at Lewis Field
33
Glenn Research Center at Lewis Field
34
Glenn Research Center at Lewis Field


Stirling Analysis Comparison of Commercial VS. High-Order Methods

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

Stirling Analysis Comparison of Commercial VS. High-Order Methods

Abstract

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