Mounir B. Ibrahim

Cleveland State University, Cleveland, Ohio

Roy C. Tew, Jr.

Glenn Research Center, Cleveland, Ohio

Zhiguo Zhang

Cleveland State University, Cleveland, Ohio

David Gedeon

Gedeon Associates, Athens, Ohio

Terrence W. Simon

University of Minnesota, Minneapolis, Minnesota

CFD Modeling of Free-Piston Stirling Engines

NASA/TM—2001-211132

September 2001

IECEC2001–CT–38
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Prepared for the

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cosponsored by the ASME, IEEE, AIChE, ANS, SAE, and AIAA

Savannah, Georgia, July 29–August 2, 2001

Mounir B. Ibrahim

Cleveland State University, Cleveland, Ohio

Roy C. Tew, Jr.

Glenn Research Center, Cleveland, Ohio

Zhiguo Zhang

Cleveland State University, Cleveland, Ohio

David Gedeon

Gedeon Associates, Athens, Ohio

Terrence W. Simon

University of Minnesota, Minneapolis, Minnesota

CFD Modeling of Free-Piston Stirling Engines

NASA/TM—2001-211132 IECEC2001–CT–38
Acknowledgments

This work was conducted under a NASA Grant to Cleveland State University, NAG3–2482. Professor Ibrahim,

Cleveland State University, Professor Simon, University of Minnesota, and David Gedeon, Gedeon Associates,

are Co-Principle Investigators. The Thermo-Mechanical Branch, Power and On-Board Propulsion

Division, NASA Glenn Research Center, under the leadership of Mr. Richard Shaltens,

Branch Chief, is greatly acknowledged.

Available from

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ideas from a technical peer group.

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analysis proceeds.

Available electronically at http://gltrs.grc.nasa.gov/GLTRS
NASA/TM—2001-211132 1 

CFD MODELING OF FREE-PISTON STIRLING ENGINES

Mounir B. Ibrahim 

Cleveland State University 

Cleveland, Ohio 44115 

Roy C. Tew, Jr. 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Zhiguo Zhang 

Cleveland State University 

Cleveland, Ohio 44115 

David Gedeon 

Gedeon Associates 

Athens, Ohio 45701 

Terrence W. Simon 

University of Minnesota 

Minneapolis, Minnesota 55455 

ABSTRACT 

NASA Glenn Research Center (GRC) is funding Cleveland 

State University (CSU) to develop a reliable Computational 

Fluid Dynamics (CFD) code that can predict engine 

performance with the goal of significant improvements in 

accuracy when compared to one-dimensional (1-D) design 

code predictions. The funding also includes conducting code 

validation experiments at both the University of Minnesota 

(UMN) and CSU. In this paper a brief description of the work-inprogress is provided in the two areas (CFD & Experiments). 

Also, previous test results are compared with computational 

data obtained using: 1) a 2-D CFD code obtained from Dr. 

Georg Scheuerer and further developed at CSU and 2) a multidimensional commercial code CFD-ACE+. The test data and 

computational results are for: 1) a gas spring and 2) a single 

piston/cylinder with attached annular heat exchanger. The 

comparisons among the codes are discussed. The paper also 

discusses plans for conducting code validation experiments at 

CSU & UMN. 

INTRODUCTION 

The type of flows that are normally encountered in Stirling 

Engines are: 1) Unsteady (Oscillatory Flow/ Oscillatory 

Pressure), 2) 2-D/3-D Flows (Complex Geometry), 3) 

Compressible (low Mach Number), Laminar/Transition/ 

Turbulent, 4) Unsteady Conduction/Convection Heat Transfer, 

5) Sudden changes in cross-section, 6) Isothermal/adiabatic 

boundary conditions, and 7) Single phase with no chemical 

reaction. On the other hand the losses encountered in Stirling 

engines (Tew 1988 & Tew and Geng 1992) are: 1) Gas spring 

and working space hysteresis losses, 2) Gas-to-heat exchanger 

heat transfer inefficiencies, 3) Viscous (pressure drop) losses, 

4) Appendix gap losses (shuttle and pumping), 5) Mixing losses 

due to mixing of gases at different temperatures, 6) Conduction 

losses from the hot- to the cold-end of the engine. In addition, 

there are losses due to differences in flow distribution: a) from 

one heat exchanger flow passage to the other and/or b) across 

the regenerator (1-D codes assume uniform flow). 

A successful CFD code that simulates the engine 

adequately should capture the characteristics of the flow and 

the type of losses described above. 

Stirling One-Dimensional Codes 

During the late 70’s and early 80’s, NASA developed it’s 

own finite-difference Stirling engine performance code for use in 

monitoring the work of its Stirling engine contractors (Tew, et. 

al.1978 & Tew 1983). Later NASA gained unlimited rights to the 

MTI developed HFAST (Huang 1993) harmonic code, via a 

contract with MTI. NASA also purchased the GLIMPS Stirling 

engine code from David Gedeon. GLIMPS (now developed into 

Sage, Gedeon 1995) has been the primary design tool used by 

the Stirling Technology Co. (STC) in recent years. HFAST and 

GLIMPS (now Sage) were both more time-efficient and userfriendly than the NASA’s finite-difference code; and they were 

both being used to design real hardware. 

Tew and Geng,1992 showed comparisons of the losses 

calculated by GLIMPS and HFAST for the 12.5 kWe 

Component Test Power Converter (CTPC). Although the overall 

engine power and efficiency predictions were quite close, there 

were substantial differences in some of the loss calculations. 

For example GLIMPS calculations suggested that cylinder 

hysteresis power losses were about 10% of the indicated power 

while HFAST calculated a much smaller value (~3%) for this 

loss. HFAST predicted larger viscous and mixing losses than 

GLIMPS, and so the overall engine performance predictions 

were similar. Geng and Tew, 1992 showed additional 

HFAST/GLIMPS comparisons, both “calibrated” and 

“uncalibrated”, for the 1.2 kW indicated power RE-1000 engine 

and the 14 kW indicated power Space Power Demonstrator 

Engine (SPDE). GLIMPS also calculated that cylinder 

hysteresis losses were about 10% of the indicated power for 

these engines. 

The Sage commercial code (Gedeon 1995) was introduced 

about six years ago. The Sage Stirling-cycle modeling software 

is the latest in a line of commercial software developed by 

Gedeon Associates. It is a direct descendant of the GLIMPS 

software, which was used widely within the Stirling industry for 

nearly ten years. Sage introduced a drag-and-drop visual 

interface where a user could assemble complete machines from 

standard components, such as pistons, cylinders, heatexchangers, etc. Sage also introduced an interactive 

optimization capability built into the visual interface. 
NASA/TM—2001-211132 2 

CAST and Modified CAST Code

A CFD code, CAST, received by Dr. Mounir Ibrahim from 

Dr. Georg Scheuerer, was developed by Peric and Scheuerer 

1989. CAST, an acronym for Computer Aided Simulation of 

Turbulent Flows, is a computer program which uses the finite 

volume method for predicting two-dimensional flow and heat 

transfer phenomena. The CAST code is written in FORTRAN 

IV. CAST is similar in structure to existing fluid flow prediction 

procedures like TEACH (Gosman and Ideriah, 1976) and TEAM 

(Huang and Lewchziner, 1983); it differs from those codes in 

the following ways (1) it has a co-located variable arrangement, 

(2) it has a different discretization scheme, (3) the solution 

algorithms are different for the linear equation systems arising 

from the discretization, and (4) the pressure-velocity coupling is 

adapted to the co-located variable storage. CAST solves the 

Navier-Stokes equations. For turbulent flows, the Reynoldsaveraged Navier-Stokes equations are solved in connection 

with the k-ε, two-equation turbulence model of Launder and 

Spalding. 

The CAST code was used by several Cleveland State 

University (CSU) graduate students working under the direction 

of Dr. Mounir Ibrahim for two-dimensional modeling of “Stirling 

machine like” components. This work was conducted under 

grants from NASA Glenn Research Center (which was NASA 

Lewis at the time of the work) in the period from 1989 to1994. 

Below is a summary of the code development topics: 

1) Oscillating inlet & outlet velocity conditions, 2) Sudden 

change in channel cross section, i.e. sudden expansion in one 

half of the cycle and sudden contraction in the other half, 3) 

Different Low-Reynolds Number, k-ε, turbulent models, 4) A 

model for laminar-transition-turbulent in a pipe flow, 5) 

Incompressible-flow with moving boundaries and, finally, 6) 

Compressible-flow with moving boundaries. 

This work is summarized below: 

Ahn 1992, added a low-Reynolds-Number k-ε turbulence model 

to CAST for purposes of conducting studies of laminar to 

turbulent transition in pipe flows. This modified CAST code was 

used in cooperation with experimental studies at the University 

of Minnesota to help determine that upon relaminarization of 

turbulent pipe flow over a periodic cycle, the laminar flow at the 

beginning of the new cycle had attained the characteristics of a 

steady, uniform velocity distribution (in the radial direction). 

Hashim, 1992, configured the CAST code to simulate 

oscillating flow and heat transfer in parallel-plate channels with 

a sudden change in cross-section. The flow was assumed to 

be laminar and incompressible with the inflow velocity uniform 

over the channel cross-section but varying sinusoidally with 

time. Ibrahim et. al., 1992, using the computational results, 

were able to determine instantaneous friction factors and heat 

transfer coefficients for laminar oscillating flow between parallel 

plates with a sudden change in cross-section. It was found that 

instantaneous friction factors and heat transfer coefficients 

deviated substantially from the steady-state values for the same 

dimensionless flow parameters. 

Kannapareddy, 1993, used CAST to model laminar, 

incompressible, oscillatory flow in the heater, regenerator and 

cooler of the NASA Stirling Space Power Research Engine 

(SPRE). The heater and cooler tubes were modeled as circular 

pipes with isothermal walls; the regenerator was modeled as 

two-parallel plates. Although the flow was oscillatory, when the 

flow was into the heater or cooler, the input temperatures were 

assumed to be constant. The study was carried out for a wide 

range of Maximum Reynolds numbers (based on maximum flow 

velocity), Valensi numbers (non-dimensional frequencies), and 

relative amplitudes of fluid displacement in the component of 

interest; the ranges chosen were based on the operating 

ranges of the NASA SPRE. The instantaneous friction factor, 

wall heat flux and heat transfer coefficient were examined. It 

was concluded that the friction factor and heat transfer 

coefficients are larger under oscillatory flow conditions for larger 

Valensi numbers (i.e., larger non-dimensional frequencies). 

Also the thermal efficiency of the heat exchangers decreased 

for the lower fluid displacement values and the steady-state 

definition for the heat transfer coefficient did not appear valid for 

use with oscillatory-flow ranges studied. A concise presentation 

of this work was reported by Ibrahim and Kannapareddy, 1992. 

In work that was not documented, Kannapareddy modified 

CAST to incorporate a moving boundary model to allow 

simulation of piston motion. These modifications were based 

upon information in Ferziger and Peric, 1997. 

Bauer, 1993, used CAST with the low-Reynolds number 

k-ε, turbulence model option (Lam-Bremhorst form), to assist 

Ibrahim et al., 1994, in developing an empirical transition model 

that could be used to predict when in the University of 

Minnesota (UMN) periodic-pipe-flow experiments (Qiu and 

Simon, 1994) transition laminar to turbulent flow would occur. 

The empirical transition model was used to activate the 

turbulence model at the appropriate time within the cycle for a 

given axial location in the tube. This analytical technique for 

modeling unsteady flow and heat transfer in Stirling engine 

heater and cooler tubes (Simon, et al. 1992, and Ibrahim, et al., 

1994) is now an essential component of the 1-D system 

simulation code Sage (at that time called GLIMPS). The tests 

used for this development showed important features of 

oscillatory flow, which led to enhanced understanding of the 

mechanisms for losses in Stirling systems. Such features 

include transition from the laminar flow to turbulence in each 

cycle of oscillation and the rapid dissipation of the turbulence, 

which remains from one cycle as a result of the extremely 

strong axial straining of the fluid upon temporal acceleration at 

the beginning of the following cycle. This collaborative effort 

showed the value of combining computational results with 

experimental finding for better understanding of Stirling engine 

flow and heat transfer processes. 

In searching for a compressible flow simulation techniques 

suitable for simulation of Stirling machines, the literature was 

first searched for papers relating to “low Mach number 

compressible flow.” Many such papers were obtained. For 

example, papers by Pletcher and Chen, 1993, Chen and 

Pletcher, 1991, Sesterhenn, Muller et al.1993, Turkel, 1987, 

Weiss and Smith, 1993, and Horibata, 1992 were reviewed. 

These papers appeared to take the approach of starting with a 

fully compressible model (including acoustics) and making 

suitable modifications to the solution technique so that the 

calculations could be extended to low Mach number flow. If any 

of these approaches had been taken to modify the CAST code, 

it appeared that it would be almost equivalent to starting over 

and developing a new code. And, these approaches did not 

appear to offer the time benefits that could be achieved by 

complete elimination of acoustic phenomena. 

For all practical Stirling engines with which these authors 

have familiarity, flow occurs at very low Mach numbers (usually 

much less than 0.1). However, compressibility is very important 

because volume changes due to piston motion in an enclosed 

volume produce large changes in the engine pressure level and 

density over the cycle. Due to the compact nature of Stirling 
NASA/TM—2001-211132 3 

machines and the moderate frequencies of operation 

(< 125 Hz), spatial pressure variations at any given time are 

relatively small so that spatial density variations are due almost 

entirely to spatial temperature variations. Calculations have 

shown that it appears to be a good approximation to assume 

that the speed of sound is infinitely fast inside most Stirling 

machines. Some authors, such as Organ, 1992, appear to 

contest this assumption 

The literature search continued for a compressible flow 

approach based on the assumption that acoustics are 

negligible. An excellent paper explaining such an approach 

was found in the 1997 Journal of Fluid Mechanics entitled “A 

model of unsteady subsonic flow with acoustics excluded” by 

A.T. Fedorchenko, 1997. The derivations include a set of 

equations for simulating subsonic flow of a heat-conducting, 

viscous gas. The simplifications in the Navier-Stokes equations 

used to eliminate acoustics include: (1) Pressure at any time 

and spatial location is split into a mean pressure level that 

varies only with time and a delta-pressure relative to a 

reference location that varies with spatial position and time; (2) 

The pressure appearing in the equation of state (for example, 

the ideal gas equation of state) is the mean pressure that varies 

only with time. Therefore, from the equation of state, density is 

a function only of the mean pressure level and the temperature 

(which varies with spatial position and time); and (3) The 

pressure appearing in the momentum equations is just the 

delta-pressure relative to the reference location. 

Thus, Fedorchenko’s paper provided the basis for 

modifying the CAST code. A technique from an early finitedifference one-dimensional flow non-acoustic code (Tew, 

Jefferies and Miao, 1978, Tew, 1983) was incorporated to 

ensure an accurate mass balance. It was not necessary to 

modify the incompressible form of the SIMPLE algorithm used 

in CAST since density does not vary with the delta-pressure 

from the reference location. More details are available in (Tew & 

Ibrahim, 2001) 

The modified CAST can now be used to simulate: (1) fixedboundary, incompressible flow with an inlet and an outlet 

boundary, (2) incompressible flow with a moving boundary 

(piston, for example) with one inlet/outlet, or (3) compressible, 

non-acoustic, flow in a completely enclosed volume with a 

moving boundary. 

CFD-ACE+ Code

CFD-ACE+ a commercial CFD code product of CFD

Research Corporation. It iis a set of computer programs for 

multi-physics computational analysis. The programs provide an 

integrated geometry and grid generation module, a graphical 

user interface for preparation of the model, a computational 

solver for performing the simulation, and an interactive 

visualization program for examination and analysis of the 

simulation results. The standard CFD-ACE+ package includes 

the following applications: geometry and grid generation, 

advanced polyhedral solver and a post processor. 

CFD-ACE+ has been used for design and analysis of a 

wide variety of industrial applications. It can model 2-D or 3-D 

geometries and includes acoustics. 

 

ANALYSIS

For this paper analyses were performed using CFD-ACE+ 

and the Modified CAST code for gas spring and 

piston/cylinder/heat exchanger (or two-space) cases (See 

Appendix A for test rig dimensions originating from Kornhauser, 

1989): 

(1) Gas Spring case: The computational domain is 2-D and 

axisymmetric, with minimum dimensions of 0.0762 (x-direction) 

*0.0254(m) Piston stroke is 0.0762 m. Also 86*40 grids and 

200 time-steps/cycle were used. 

(2) Two-space case: The computational domain is 2-D and 

axisymmetric, with dimensions of ~0.445 (x-direction) 

*0.0254(m). The piston stroke was also 0.0762 m for this case. 

148*48 grids and 960 time-steps/cycle were used. 

Helium was used as the working fluid, with the standard k-ε, 

turbulence model and upwind finite differencing scheme. 

RESULTS 

Gas Spring Hysteresis Losses

For a gas spring, the hysteresis loss is the work that is 

dissipated by the spring per cycle at steady operating 

conditions; it’s also equal to the heat generated in and 

transferred out of the spring. A good way to compare 

computational and measured hysteresis losses is via plots of 

dimensionless work as a function of oscillating flow Peclet 

number. Dimensionless work and oscillating flow Peclet 

numbers are defined, respectively, as follows: 

W ˆ 

loss = ∫ P dV

PoVo

Pa

Po

 

   

 

   

2

γ − 1

γ

 

     

 ( 1 ) 

 Peω =

ρocpω Dh

2

4λ

 ( 2 ) 

Where: 

c p specific heat at constant pressure, 

Dh hydraulic diameter (=4 x wetted area/wetted perimeter), 

P mean spatial pressure, 

Pa amplitude of mean spatial pressure 

Po arithmetic mean of max. and min. mean spatial pressures 

V volume 

Vo arithmetic mean of maximum and minimum volumes 

W ˆ 

loss non-dimensional work or hysteresis loss 

Greek letters 

γ ratio of specific heats of fluid 

λ molecular thermal conductivity 

ρ density, fluid mass per unit volume 

ω angular velocity (rad/sec) 

is
NASA/TM—2001-211132 4 

Figure 1: Comparison Among CFD Codes for 

Dimensionless Losses & Kornhauser's (1989) 

Experimentally Derived Losses. 

Figure 1 shows the dimensionless work plotted versus the 

Peclet number for the computations of CFD-ACE+, Modified 

CAST and Recktenwald (1989) together with Kornhauser’s 

(1989) data. Only four data points were chosen for CFD-ACE+ 

over a range of Pe from 0.97 to 843. The CFD-ACE+ 

predictions fit very well with all the other results presented in 

Figure 1. More detailed comparisons between the 

computational results and experimental data are given in a 

companion paper (Tew and Ibrahim 2001). 

Two-Space Test Rig Data and Calculation Comparisons

 Figures 2a and 2b show CFD results obtained from CFDACE+ and Modified CAST code respectively. The data are for 

the temperature difference between the wall and the center of 

the heat exchanger at the following different axial locations: 

entrance of the heat exchanger and 1/16, 1/8, 1/4 & 1/2 of the 

annulus total length, measured from the entrance. 

Figure2a: CFD-ACE+ Calculations of Temperature 

Difference from Heat Exchanger Center to Wall (148x48 

grids, 960 time steps/cycle). For Comparison with 

Kornhauser (1989) Exp. Data: Run #12071539, 201.7 RPM, 

1.008 MPa Mean Pressure. 

Figure 2b: Modified CAST Calculations of Temperature 

Difference from Heat Exchanger Center to Wall (34x20 

grids, 120 time steps/cycle). For Comparison with 

Kornhauser (1989) Exp. Data: Run #12071539, 201.7 RPM, 

1.008 MPa Mean Pressure.

Upon comparing Figures 2a & 2b, the two codes CFDACE+ & Modified CAST behave in a similar way to the 

experimental data (see Tew & Ibrahim 2001) with some 

differences in the following: 1) Modified CAST predicts 

maximum temperature differences (center/wall), except at the 

entrance, larger than CFD-ACE+ and closer to the experimental 

data, 2) Also Modified CAST shows effects of the axial location 

(up to 180 Degrees-Crank-Angle) more like the experimental 

data trend than CFD-ACE+, 3) Both codes show results 

independent of the axial location beyond 230 degrees-crankangle, unlike the experimental data. 

Figures 3a and 3b show CFD results obtained from CFDACE+ and the Modified CAST code, respectively. The results 

are for heat transfer per unit area at the following different 

locations in the heat exchanger: entrance of the heat 

exchanger and 1/16, 1/8, 1/4 & 1/2 of the annulus total length 

measured from the entrance. 

Figure 3a: CFD-ACE+ Calculations of Heat Transfer per 

Unit Area at Different Locations in the Heat Exchanger 

Relative to Entrance to Cylinder (148x48 grids, 960 time 

steps/cycle). For Comparison with Kornhauser (1989) Exp. 

Data: Run #12071539, 201.7 RPM, 1.008 MPa Mean 

Pressure. 
NASA/TM—2001-211132 5 

Figure 3b: Modified CAST Calculations of Heat Transfer 

per Unit Area at Different Locations in the Heat Exchanger 

Relative to Entrance to Cylinder (34x20 grids, 120-time 

steps/cycle). For Comparison with Kornhauser (1989) Exp. 

Data: Run #12071539, 201.7 RPM, 1.008 MPa Mean 

Pressure. 

Again, upon comparing Figures 3a & 3b, the two codes 

CFD-ACE+ & Modified CAST behave in a similar way to the 

experimental data (see Tew & Ibrahim 2001) with some 

differences in the following: 1) Modified CAST shows the effect 

of the axial location more like the experimental data trend, 

however, 2) except at the entrance, the CFD-ACE+ heat 

transfer/unit area is closer to the experimental data than 

Modified CAST. 

 The results obtained at this point emphasize the fact that in 

order to obtain a reliable CFD simulation for the Stirling engine, 

there is a need for accurate experimental data. The section 

below describes the test facilities at CSU & UMN and the plan 

for running code validation experiments. 

CODE VALIDATION EXPERIMENTS 

CSU-SLRE Test Rig The CSU Stirling-Laboratory-ResearchEngine (SLRE) derives from a two-piston Stirling-engine test rig 

developed by NASA Jet Propulsion Laboratory in 1982 (Hoehn, 

1982). A special test section is being designed to be bolted 

between the two existing piston-cylinder flanges on the rig. The 

pistons will drive atmospheric-pressure air or helium through 

the test section at Reynolds and Valensi numbers 

representative of typical Stirling engine practice. Within the test 

section will be various passages representative of regions of 

the Stirling engine where flow area changes abruptly or flow 

velocity changes direction. This may include the ports leading to 

the compression space or the transition between tubes or 

rectangular passages leading to the expansion space. In the 

test rig, the passages will be two dimensional, either axial (x, r) 

or rectangular (x, y), so that the flows can be compared directly 

with 2-D codes. 

Flow measurement will be either by LDV (Laser-Doppler 

Velocimetry) or PIV (Particle Imaging Velocimetry), through 

transparent walls in the test section. Both methods require 

tracer particles within the gas flow. LDV measures individual 

tracer particle velocities one after another as it encounters them 

within its measurement volume, whereas PIV measures 

trajectories of many particles simultaneously within a 

measurement plane. PIV is the preferred method because it 

provides flow field rather than single point information. For best 

compatibility with laser measurements, test sections with flat 

walls are best. A cylindrical test section requires optical 

correction for measurements of other than the axial component 

of velocity. 

The following illustration shows a cylindrical test section 

comprising a transparent acrylic tube with an insert that 

produces an annular flow passage leading to an abrupt 

expansion at the center of the section. It will probably not be 

built, due to concerns about laser measurements through the 

circular wall. A rectangular test section has yet to be designed. 

UMN Test Rig

The University of Minnesota test rig is an oscillatory flow 

facility, which is scaled up in size and down in frequency so the 

actual system is dynamically simulated. The drive section of 

the facility is a simple piston cylinder arrangement. The piston 

is driven by a scotch yoke arrangement so that its motion is a 

sinusoidally oscillating flow with a time mean bulk velocity of 

zero. The dimensionless amplitude and frequency match those 

of the engine through the Reynolds number based on peak 

velocity of the cycle and the Valensi number. This facility has 

been used for Stirling engine development under the SPRE 

Program (Qiu and Simon 1994). The facility is larger than the 

CSU-SLRE engine, which allows detailed measurements and 

clear visualization. It cannot match the engine frequency and 

must rely on dynamic similarity. The test section is a 

representation of a section of an engine. It consists of a 

cylindrical extension of the piston-cylinder region of the drive 

section and a radial channel between two disks, one of which is 

attached to that cylinder. It represents flow passages that could 

be found in either of the two ends of the engine, cold or hot 

section, in which the flow is driven
NASA/TM—2001-211132 6 

Figure 4 A Cylindrical Test Section Comprising a 

Transparent Acrylic Tube with an Insert, designed for the 

CSU-SLRE Test Rig. 

axially, then radially, as it is pumped from the compression or 

expansion space radially outward toward the regenerator, then 

reverses. Flow through geometries of this type seems to not be 

computed well and there seems to be a need to improve 

turbulence closure and flow transition modeling. Measurements 

will include unsteady velocities, resolved in space and time. 

Further documentation includes visualization and measurement 

of unsteady flow separation and reattachment through the 

abrupt flow geometry change regions. Instruments which may 

be brought to bear on the problem are hot-wire anemometry, 

laser-Doppler anemometry and flow visualization with smoke, 

tufts and neutrally buoyant helium bubbles. Time records of the 

measurements will be made then decomposed according to 

time within the cycle and ensemble-averaged. 

CONCLUDING REMARKS 

Comparisons were made between the commercial CFDACE+ code and compressible non-acoustic calculations using 

Modified CAST, developed at CSU. The comparison was made 

for gas spring as well as two-space rig experimental data. The 

two codes predicted similar trends to the experimental data for, 

Work-Loss (gas spring), temperature differences (wall-center in 

the heat exchanger annulus) and wall heat flux per unit area. 

However, the two codes showed significant differences under 

certain conditions for the two-space test rig data. The reasons 

will be explored further. This work emphasizes the need for 

conducting accurate code validation experiments for oscillatory 

flow as it occurs in current Stirling engines. 

REFERENCES 

Ahn, K. and Ibrahim, M.B. (1992), “Laminar/Turbulent 

Oscillating Flow in Circular Pipes”, Int. J. of Heat and Fluid 

Flow, Vol. 13, No. 4, pp 340-346. 

Bauer, C. (1993), “Low-Reynolds Number k-epsilon and 

Empirical Transition Models for Oscillating Pipe Flow and Heat 

Transfer”, M. S. Thesis, Cleveland State University. 

Chen, K.-H. and R. H. Pletcher (1991), “Primitive Variable, 

Strongly Implicit Calculation Procedure for Viscous Flows at All 

Speeds”, AIAA Journal, Vol. 29, No. 8. 

Fedorchenko, A. T. (1997), “A Model of Unsteady Subsonic 

Flow with Acoustics Excluded”, J. Fluid Mechanics, Vol, 334, 

pp. 135-155. 

Ferziger, J. H. and Peric, M. (1997), “Computational 

Methods for Fluid Dynamics”, Springer-Verlag. 

Gedeon, D. (1995), Sage: Object-Oriented Software for 

Cryocooler Design, in: Cryocoolers 8, Edited by R.G. Ross, Jr., 

Plenum Press, pp. 281—292. 

Geng, S. M. and Tew, R. C. (1992), “Comparison of 

GLIMPS and HFAST Stirling Engine Code Predictions with 

Experimental Data, NASA TM-105549. 

Hashim, W. (1992), “Oscillating Flow and Heat Transfer in 

Channels with Sudden Change in Cross Section”, Master’s 

Thesis, Cleveland State University. 

Hoehn, F.W. (1982), Stirling Laboratory Research Engine, 

JPL Publication 82-13. 

Huang, S.C. (1993), “HFAST Version 2.0 Analysis Manual”, 

Prepared under NASA Contract Number NAS3-25330. 

Ibrahim, M., Hashim, W., Tew, R. C. and Dudenhoefer, J. E. 

(1992), “Heat Transfer in Oscillating Flows with Sudden Change 

in Cross Section”, NASA TM-105692. 

Ibrahim, M.B. and Kannapareddy, M. (1994)"Computational 

Heat Transfer Analysis for Oscillatory Channel Flows" First 

ISHMT-ASME Heat and Mass Transfer Conference, Bombay, 

India. 

Ibrahim, M., Bauer, C., Simon, T. and Qiu, S. (1994), 

“Modeling Oscillatory Laminar, Transitional, and Turbulent 

Channel Flows and Heat Transfer”, International Heat Transfer 

Conference, London, England. 

Kannapareddy, M. R. (1993), “Numerical Thermal Analysis 

of Heat Exchangers for the Stirling Engine Applications”, M. S. 

Thesis, Cleveland State University. 

Kornhauser, A. A. (1989), “Gas-Wall Heat Transfer During 

Compression and Expansion”, Massachusetts Institute of 

Technology S. D. Thesis. 

Peric, M. and Scheuerer, G.(1989), “CAST - A Finite 

Volume Method for Predicting Two-Dimensional Flow and Heat 

Transfer Phenomena”, GRS - Technische Notiz SRR-89-01. 

Pletcher, R.H. and Chen, K.-H. (1993), “On Solving the 

Compressible Navier-Stokes Equations for Unsteady Flows at 

Very Low Mach Numbers”, AIAA 93-3368 and NASA TM106380. 

Qiu, S. and Simon, T. W. (1994), “Measurements of Heat 

Transfer and Fluid Mechanics Within an Oscillatory Flow in a 

Pipe”, Winter ASME Annual Meeting. 

Recktenwald, G. W. (1989), “A Study of Heat Transfer 

Between the Walls and Gas Inside the Cylinder of a 

Reciprocating Compressor”, University of Minnesota Ph. D. 

Thesis. 

Tew, R.C. Jr. and Ibrahim, M.B. (2001), “Study of TwoDimensional Compressible Non-Acoustic Modeling of Stirling 

Machine Type Components,” Paper IECEC 2001-CT-27, 

Proceedings of 36th Intersociety Energy Conversion 

Engineering Conference, July 29-August 2, 2001, Savannah, 

GA 

Tew, Roy C. , Jr. (1988), “Overview of Heat Transfer and 

Fluid Flow Problem Areas Encountered in Stirling Engine 

Modeling”, NASA TM-100131. 

Tew, R. C., Jr. and Geng, S. M. (1992) , “Overview of NASA 

Supported Stirling Thermodynamic Loss Research”, NASA TM105690. 
NASA/TM—2001-211132 7 

Tew, R. C., Jr., Jefferies, K. and Miao, D. (1978), “A Stirling 

Engine Computer Model for Performance Calculations”, 

DOE/NASA/1011-78/24, NASA TM-78884. 

Tew, Roy C. Jr. (1983), “Computer Program for Stirling 

Engine Performance Calculations”, DOE/NASA/51040-42, 

NASA TM-82960. 

Sesterhenn, Jorn, et. al. (1993), “Flux-Vector Splitting for 

Compressible Low Mach Number Flow”, Computes Fluids, Vol. 

22, No. 4/5, pp. 441-451. 

APPENDIX 

Gas Spring & Two-Space Test Rig Dimensions 

Table A1: Gas Spring Dimensions 

Physical Quantity Symbol Value 

Cylinder Bore (Diam.) D 50.80 mm (2 in.) 

Piston Stroke S 76.2 mm (3 in.) 

Volume Ratio r

ν 2.0 

Table A2: Two-Space Test Rig Dim. (Physical & Simulated) 

Physical Quantity Physical Value Simulation Value 

Cylinder Bore 50.80 mm (2 in) 50.80 mm (2 in) 

Piston Stroke 76.20 mm (3 in) 76.20 mm (3 in) 

Volume Ratio 2.0 2.0 

Annulus O.D. 44.5 mm (1.75 in) 50.80 mm (2 in) 

Annulus I.D. 39.4 mm (1.55 in) 46.4 mm (1.83 in) 

Annulus Gap 2.5 mm (0.10 in) 2.2 mm (0.09 in) 

Annulus Length 445 mm (17.5 in) 445 mm (17.5 in) 

Min Pist/Head Clr. 2.9 mm (0.11 in) 2.9 mm (0.11 in) 
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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13

CFD Modeling of Free-Piston Stirling Engines

Mounir B. Ibrahim, Roy C. Tew, Jr., Zhiguo Zhang,

David Gedeon, and Terrence W. Simon

Stirling cycle; Stirling engines; Simulation; Fluid mechanics; Heat transfer

Unclassified -Unlimited

Subject Category: 34 Distribution: Nonstandard

Prepared for the 36th Intersociety Energy Conversion Engineering Conference cosponsored by the ASME, IEEE, AIChE, ANS, SAE, and AIAA,

Savannah, Georgia, July 29–August 2, 2001. Mounir B. Ibrahim and Zhiguo Zhang, Mechanical Engineering Department, Fenn College of Engineering,

Cleveland State University, 1960 East 24th Street, Cleveland, Ohio 44115–2425; Roy C. Tew, Jr., NASA Glenn Research Center; David Gedeon,

Gedeon Associates: Sage Stirling, Pulse Tube, and Thermoacoustic Software, 16922 South Canaan Road, Athens, Ohio 45701; and Terrence W. Simon,

Thermodynamics and Heat Transfer Division, Department of Mechanical Engineering, University of Minnesota, 111 Church Street, S.E., Minneapolis,

Minnesota 55455. Responsible person, Roy C. Tew, Jr., organization code 5490, 216–433–8471.

NASA Glenn Research Center (GRC) is funding Cleveland State University (CSU) to develop a reliable Computational

Fluid Dynamics (CFD) code that can predict engine performance with the goal of significant improvements in accuracy

when compared to one-dimensional (1-D) design code predictions. The funding also includes conducting code validation experiments at both the University of Minnesota (UMN) and CSU. In this paper a brief description of the work-inprogress is provided in the two areas (CFD and Experiments). Also, previous test results are compared with computational data obtained using (1) a 2-D CFD code obtained from Dr. Georg Scheuerer and further developed at CSU and

(2) a multidimensional commercial code CFD-ACE+. The test data and computational results are for (1) a gas spring

and (2) a single piston/cylinder with attached annular heat exchanger. The comparisons among the codes are discussed.

The paper also discusses plans for conducting code validation experiments at CSU and UMN.