Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

A Simplifi ed Model for Detonation Based

Pressure-Gain Combustors

NASA/TM—2010-216821

November 2010

AIAA–2010–6717
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Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

A Simplifi ed Model for Detonation Based

Pressure-Gain Combustors

NASA/TM—2010-216821

November 2010

AIAA–2010–6717

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2010-216821 1 

A Simplified Model for Detonation Based 

Pressure-Gain Combustors 

Daniel E. Paxson 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

A time-dependent model is presented which simulates the essential physics of a detonative or otherwise 

constant volume, pressure-gain combustor for gas turbine applications. The model utilizes simple, global 

thermodynamic relations to determine an assumed instantaneous and uniform post-combustion state in one of many 

envisioned tubes comprising the device. A simple, second order, non-upwinding computational fluid dynamic 

algorithm is then used to compute the (continuous) flowfield properties during the blowdown and refill stages of the 

periodic cycle which each tube undergoes. The exhausted flow is averaged to provide mixed total pressure and 

enthalpy which may be used as a cycle performance metric for benefits analysis. The simplicity of the model allows 

for nearly instantaneous results when implemented on a personal computer. The results compare favorably with 

higher resolution numerical codes which are more difficult to configure, and more time consuming to operate. 

Nomenclature 

a* reference speed of sound 

a/f air to fuel ratio (by mass) 

ff fill fraction, fraction of tube filled with detonable mixture 

hf fuel heating value 

m mass 

p pressure 

PR total pressure ratio of exit to inlet flow 

q0 non-dimensional heat addition parameter 

Rg real gas constant 

T temperature 

TR total temperature ratio of exit to inlet flow

u gas velocity 

U wheel velocity 

W mass averaged work per cycle 

 ratio of specific heats 

ηt non-uniform turbine efficiency 

ρ density 

Subscripts 

3 pertaining to the compressor exit 

4 pertaining to the turbine inlet 

CV constant volume 

e exit plane 

eff effective 

init initial post-combustion state 

initial initial pre-combustion state 

s static 

t total 

Superscripts 

- mass average 

 deviation from mean 
NASA/TM—2010-216821 2 

I. Introduction 

Confined volume combustion cycles, implemented in non-positive displacement devices, are currently under 

investigation for use as pressure-gain combustors in gas turbines (Refs. 1 to 7). They are envisioned as replacements 

for the conventional, constant pressure combustors (which in reality always effect a total pressure loss). Pressuregain combustion converts the operational basis of the gas turbine from a Brayton cycle to something approaching an 

Atkinson cycle which, for the same heat addition and mechanical compression, is thermodynamically superior. 

One concept under consideration utilizes detonative combustion in a series of circumferentially arranged tubes, 

each undergoing a periodic cycle (aka, a multi-tube pulse detonation engine, or PDE). Each tube is filled, or partially 

filled, with a detonable mixture of fuel and air from the upstream compressor. The filling end is then closed, and 

detonation is initiated. The detonation propagates supersonically down the tube, followed immediately by a Taylor 

wave which brings the flow nominally to rest at an elevated pressure and temperature. The entropy produced during 

this portion of the cycle is theoretically only slightly less than what would be produced from a true (deflagrative) 

constant volume combustion event. In principle however, a detonation can propagate down the entire tube 

considerably faster than a deflagration. This may yield higher frequency operation, which in turn can yield greater 

throughflow for a given combustor size. When the detonation reaches the exhaust end of the tube, a blowdown 

period commences. During this time the fluid rapidly exhausts from the tube, and the pressure in the tube drops due 

to expansion. The high velocity exhaust gas, which is at a higher total pressure than the compressor discharge air, is 

directed into the downstream turbine. Eventually, the pressure at the inlet end of the tube drops to the level of the 

compressor outlet, and the tube inlet is opened. The tube is refilled with a fresh fuel/air charge and the cycle is 

completed. In principle, this cycle repeats at high frequency in each of the many tubes comprising the combustor. It 

is a matter of ongoing debate whether the tubes should be fixed or rotary, as well as whether the gas exhausted 

during the blowdown period, which is both temporally and spatially non-uniform, should be mixed and diffused 

prior to entry into a turbine nozzle ring, or sent directly into a turbine rotor. Much debate has also surrounded the 

question of whether the exhaust end of the tubes should be valved as is the inlet end. Theoretically, higher 

performance is achievable by doing so; however, considerable complexity is added. In this paper, only 

configurations that have unvalved (i.e., open) exhaust are considered. 

In order to reasonably assess thermodynamic benefits, and to perform preliminary sizing for pressure-gain 

combustors in a variety of gas turbine engine classes, a performance model is needed. There are numerous 

approaches borrowed from the PDE community that may be considered for this task. These range from complex 

single or multi-dimensional, multi-species, reactive computational fluid dynamic (CFD) codes, to the relatively 

simple, algebraic, state-based models (Refs. 7 to 14). For preliminary benefits analyses, many of the CFD 

approaches are prohibitively costly in terms of configuration (gridding, boundary conditions, event timing, 

chemistry models, etc.), computational resources, and turnaround time. On the other hand, the simplest approaches 

do not always capture the essential physics required to provide realistic results. In particular, the algebraic models 

(Ref. 9) fail to account for the variation in the exhausted flow, which in turn affects the calculated availability of the 

gas supplied to the turbine. Even the so-called zero-dimensional (a.k.a. 0-D, or lumped volume) approaches 

(Refs. 12 and 13), which are time-dependent, can be unreliable for the analysis of many gas turbine pressure-gain 

combustors. In this application, the exhaust static pressure of the combustion device may be quite close to the inlet 

total pressure. In other words, there is no ram pressure to force flow through it, and it must therefore be nearly selfaspirating. Modeling a self-aspirated cycle requires a momentum equation, which the 0-D approaches generally lack. 

In this paper, a hybrid approach is presented for performance modeling of pressure-gain combustors. It 

essentially combines a non-reacting, one-dimensional (1-D) CFD approach for the continuous blowdown and refill 

portions of the cycle with an algebraic approach for the detonative (or otherwise constant volume) event. The result 

is a cycle simulation code that runs in fractions of a second on a personal computer, yet yields realistic estimates of 

performance, size, and operational frequency. 

Difficulties associated with algebraic and lumped volume models will first be presented in order to motivate the 

present approach. This will be accomplished using an example gas turbine engine application for which a pressuregain combustor is considered. The pressure-gain cycle is initially computed using a high-fidelity, reactive CFD code 

(Ref. 10), which for this work serves as the most accurate solution. The output is then compared to algebraic and 

lumped volume models. It is shown that neither model correctly captures details of the exhaust flow, the nonuniformity of which is proved to be essential in estimating performance. The hybrid model just introduced will then 

be described in detail. Results from it will also be presented and compared to those of the higher fidelity, reactive 

code. The assumption of a single, calorically perfect (i.e., fixed properties, fixed ) gas will be used throughout in 

order to simplify equations and comparisons. Though this is not a realistic assumption for a practical detonative 
NASA/TM—2010-216821 3 

combustor environment, using it does not impact the concepts, or methodologies presented. The fixed value of 

 = 1.3 used herein represents an average value (between the maximum and minimum that would actually occur) 

chosen according to the methodology described in Reference 13. 

II. Application Example 

Consider the notional combustor shown in Figure 1 at an operating point defined by the parameters and 

conditions of Table 1. These conditions are representative for sea level static operation of a modern regional 

turbofan engine with overall pressure ratio (OPR) 24. For any periodic cycle undergone by each combustor tube 

(e.g., detonative, Lenoir, etc.), the mass averaged total temperature of the exit flow may be obtained from the 

conservation of mass and energy as follows. 

  0

3

4 11 q T

T

t

t  (1) 

The non-dimensional heat addition parameter, q0, is defined as 





 



  



1 3

0

f TR a

h

q

tg

f (2) 

Equation (1) must hold regardless of the cycle details or analysis method used. It results directly from the 

conservation of mass and energy applied to a control volume. The critical issue for gas turbine applications lies in 

assessing the associated total pressure, pt4. For steady systems this is a straightforward process. It is essentially the 

inlet total pressure less some factor accounting for mixing losses, and those associated with heating at finite 

Mach number. For unsteady pressure-gain systems, assessing the exhaust total pressure is non-trivial, and therefore 

requires separate discussion. 

3 4

Figure 1.—Notional detonative combustor geometry. 

TABLE 1.—COMBUSTOR PARAMETERS 

Tt3, R 1426 

pt3, psia 352.8 

a/f 42.2 

hf, ft-lbf/lbm 14.3×106

 1.3 

Rg, ft-lbf/lbm-R 53.57 

ps4, psia 352.8 

a*, ft/s 1787 
NASA/TM—2010-216821 4 

A. The Effect of Non-Uniformity 

For ideally expanded, but fundamentally unsteady propulsion systems, the mass averaged kinetic energy emitted 

per period is always greater than the square of the gross specific impulse divided by two (Ref. 15). In steady 

systems, these two quantities are equal. Defining the instantaneous exit velocity as the sum of a mass averaged and a 

fluctuating component, uuu eee   , this statement may be formally written as 

   

e me

ee

e

m

o

e ee

e

dmu

m

u dmu

m

0

2

2

2

2

1

2 2

1 (3) 

or 

222

222

uuu eee

  (4) 

The left hand side of Equation (4) is essentially the kinetic energy. The first term on the right hand side is the 

square of the specific impulse divided by two. The second term on the right hand side, the mass averaged kinetic 

energy of the fluctuating component, is only zero when the flow is steady. For all unsteady (and periodic) exhaust 

flows it is positive. 

Besides applying to thrust production, Equation (4) is relevant for pressure-gain combustion systems in the 

context of work extraction by turbines. Consider an idealized, Pelton-style impulse turbine fed by a non-steady 

combustor tube, as shown schematically in Figure 2. The mass averaged specific work extracted (done by the gas to 

the wheel) per period is 

 UuUW  2 e  (5) 

This has a maximum if 

2

e u U  , at which point 2

2

max e u W  . From Equation (4) therefore, it is evident that the 

maximum possible work extracted from a non-uniform flow is always less than the available kinetic energy. This 

result may be imposed on the system as either a mandatory efficiency loss applied to an unsteady turbine (i.e., 

2

2

e

e t

u

u

 ) or as an availability loss applied to the combustor exhaust flow. In this paper, the latter is adopted. In 

particular, the exhaust total pressure is assigned to be that which, when ideally expanded to the exhaust static 

pressure at the exhaust total temperature, yields the mass averaged specific thrust (not the kinetic energy (Ref. 9)), 

viz. 

U

ue

tube

wheel

Figure 2.—Idealized Pelton type 

turbine schematic. 
NASA/TM—2010-216821 5 

 

 









































 



  



1

34

2

4

4

2

1 1

tt

e

s t

TT

au

p

p

*

 (6) 

Here, the velocity has been normalized by a reference speed of sound, a*

, calculated from the total temperature at 

the inlet (station 3). 

Whether a turbine loss or Equation (6) is used, it is clear that an assessment of the mass averaged velocity is 

needed. This, in turn, implies the need for representative distributions of fluid states (i.e., pressure, density, velocity) 

in the exit plane of the combustor tubes. The mass averaged velocity is, after all, an integral evaluated over one 

period. 

Figure 3 shows the computed, normalized exit velocity distribution from one tube of the Figure 1 notional 

combustor, subjected to the Table 1 parameters, and undergoing a detonative cycle. The results were obtained using 

a high resolution, reacting, computational fluid dynamic (CFD) code (Ref. 10). It is plotted as a function of the 

fraction of the initial mass ejected rather than as a function of time. 

For reference, a complete wave diagram of the cycle is shown in Figure 4. Here, color contours of normalized 

pressure, temperature, velocity, and reactant fraction in the tube, over one period are plotted. The horizontal axis of 

each contour represents distance along the tube. The vertical axis is non-dimensional time. Next to each contour are 

numbers which represent the minimum and maximum value of the contoured variable plotted in the x-t space. These 

may be used to scale the spectrum shown. The black rectangle in the lower left of each contour represents temporal 

location of a wall boundary condition simulating a closed valve. 

Returning to Figure 3, two velocity distributions are shown: the actual exit velocity ue, and the effective exit 

ue_eff. The latter is defined as 

 

ee

se eeffe u

pp

uu



  4 _ (7) 

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.2 0.4 0.6 0.8 1.0

ue/a*

me/minitial

uef 

avg 

rms 

ue

* e au

*2e au

ue_eff /a*

ue /a*

Figure 3.—CFD computed, normalized exit velocity distribution 

as a function of normalized ejected mass for a detonative 

cycle. 
NASA/TM—2010-216821 6 

p/p* T/T* Mach #

Reactant

Fraction

time

detonation

High

Low

4.59

0.69

2.63

0.95

1.00

-0.12

1.00

0.00

x/L x/L x/L x/L

Figure 4.—Contours of normalized pressure, temperature, Mach number, and reactant fraction in a tube 

over the course of one computed detonative cycle. 

The effective velocity accounts for the contribution to momentum from pressure forces when the flow is over or 

under expanded in the exhaust plane. In the present cycle, this situation arises immediately after the detonation 

reaches the exhaust end of the tube (see Fig. 4). Also shown in Figure 3 are ue (essentially the area under the ue_eff

line), and 2

ue . Using Equations (1), (2), (6), and the computed value of .6650 * _effe au  , it is found that 

002

3

4  .

t

t

T

T

TR , and 161

3

4  .

t

t

p

p PR 1

CFD based calculations of ue are arguably the most accurate for obtaining pressure-gain combustor performance 

estimates such as those just presented. As mentioned however, they are cumbersome and time consuming to use, and 

may be considered impractical for performance assessments requiring many operating points or engine 

configurations. On the other hand, the much simpler algebraic methods cannot convey any information about ue

since they are neither time nor mass dependent. Interestingly, with some simplifying assumptions about the nature of 

entropy production (Refs. 9 and 15), they can be used to find 2

ue . However, Equation (4) and a glance at Figure 3 

will show that if this is used in place of 2

ue in Equation (6), it will result in a value of PR = 1.35 for the example 

operating point and cycle under consideration here. This unreasonably high pressure ratio, combined with a lack of 

available information regarding flow rates (i.e., sizing information) or cycle periods makes the algebraic methods 

impractical for performance assessment. 

B. The Lumped Volume Approach 

Lumped volume models represent a time-dependent approach; however, there is no spatial dependence (Ref. 10). 

The PDE tube is considered as a single vessel (lump) which, at any moment in the cycle to be described, can be 

represented by a single state. The exhaust nozzle is modeled in quasi-steady fashion meaning that, at each moment 

in time, the exit conditions can be found from the specified nozzle geometry, the exit static pressure, and the 

upstream PDE vessel state. They are relatively simple to construct, produce rapid results (compared to CFD), and 

can be quite accurate for certain configurations. 

 1

For the detonative cycle, TR = 1.98. This is slightly below the value of 2.00 found with Equations (1) and (2) and Table 1. 

The reduction arises from the necessity of a small amount of unfueled buffer gas which must be inserted between the hot exhaust

gas in the tube and the new detonable change that is entering. The impact on PR is negligible. 
NASA/TM—2010-216821 7 

In the model process, the vessel is first filled with a reactive mixture of known stoichiometry at the inlet state. A 

constant volume combustion process then proceeds instantaneously, leaving the vessel fluid in a uniform high 

pressure, high temperature state to begin the blowdown process. Blowdown is a time-dependent, isentropic 

expansion process. The exit plane quantities are determined by the specified nozzle geometry, the ambient static 

pressure and the instantaneous tube pressure and temperature. Blowdown continues until the tube pressure matches 

the inlet total pressure, after which the fill portion of the model cycle commences. During the fill portion of cycle, a 

new detonable mixture enters through the inlet while the remaining post-detonative gas exits through the nozzle. The 

fresh charge and hot gas are assumed to stay separated by an interface, across which the pressure is constant, but the 

density varies. Fill proceeds until the total mass which has flowed from the exit plane matches the mass originally in 

the tube when the cycle began. 

Lumped volume models may be appropriate for pressure-gain combustor configurations where ps4 is 

substantially below pt3 and where the notional combustor contains a throat (constriction) at the exhaust. For other 

configurations, such as the one considered in this paper, the lumped volume approach is problematic. Without 

spatial resolution, or a momentum equation, critical gasdynamic phenomena, and features related to fluidic inertia 

are lost. The results are often poor predictions of cycle periods, inaccurate distributions of ue, and therefore improper 

measures of pt4. 

Figure 5 shows the distribution of ue for the same notional combustor tube, as predicted by a lumped volume 

simulation. Since the exit static pressure, ps4 is assumed to be the same as the inlet total pressure, pt3, the lumped 

volume method will never fully exhaust all of the flow (there is no pressure to drive the flow out and it has no 

inertia). In order to avoid this, ps4/pt3 has been set to 0.9995. The resulting cycle period is approximately 15 times 

longer than that predicted by the CFD based simulation. The distribution of ue in Figure 5 looks quite different than 

that of Figure 3. The mass averaged value, at .5440 * e au  , is also different (and lower). This results in a lower 

available total pressure ratio of 101

3

4  .

t

t

p

p PR . 

The disparity between the lumped volume and CFD performance predictions can be attributed to two factors. 

First and foremost, the differences in ue_eff distribution alter 2

ue

 from Equation (4). Second, the available kinetic 

energy from the constant volume combustion assumption of the lumped volume model is less (entropy production is 

greater) than from detonation based combustion used in the reactive CFD model. 

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.2 0.4 0.6 0.8 1.0

ue/a*

me/minitial

ue 

avg 

rms 

*

e au

*2e au

ue_eff /a*

Figure 5.—Lumped volume computed exit velocity distribution as 

a function of normalized ejected mass for a constant volume 

cycle. 
NASA/TM—2010-216821 8 

III. The Hybrid Approach 

In the present model, a sort of hybrid approach is taken to simulate the complete cycle. The tube is assumed to be 

initially filled with a stationary, uniform mixture of air and fuel at the inlet pressure and temperature. Like the 

lumped volume method, it is assumed to undergo an instantaneous, uniform, zero-velocity, constant volume reaction 

that results in a peak pressure and temperature as follows. 

  0

3

11 q T

T

t

CV  (7) 

3 t3

CV

t

CV

T

T

p

p  (8) 

This state is then used as an initial condition for a quasi-one-dimensional (or in the case at hand, strictly one 

dimensional) Euler flow solver. Because the reaction has already taken place, the solver need not contain species 

equations or a reaction mechanism. Furthermore, with reactions removed from the computing domain, the potential 

for discontinuities (i.e., shocks) has largely been removed as well. With a continuous flowfield, the need for a 

computationally intense, high resolution, monotonicity preserving numerical scheme vanishes. As such, any number 

of relatively simple (though still second order in time and space) schemes can be used, and the grid spacing may be 

made quite large. Both the Euler equations and the numerical schemes for their solution are well documented in the 

literature (e.g., Ref. 16). As such, no details concerning them will be presented in this paper. For this work, 

MacCormack’s integration method (Ref. 17) was used, with a computational grid comprising 50 cells. For reference, 

the grid for the detonative CFD simulation of Figure 3 contained 200 cells. Because of the Courant-Freidrich-Lewy 

(CFL) convergence criteria, fewer grid points also allows for larger time steps. The net result is a very rapid scheme 

for computing the gas state in the tube exit plane for the blowdown and fill portions of the cycle. 

Starting from the post-combustion CV state, the left end of the tube has a solid wall boundary condition 

imposed, while the right end has a constant pressure boundary. Descriptions of these and other boundary conditions 

can readily be found in numerous CFD references, as well as in Reference 18. These conditions are imposed until 

the pressure at the left end of the tube matches pt3. This point in time constitutes the beginning of the fill period and 

the left wall boundary condition is replaced with an open inflow condition. The temperature supplied for this 

boundary is Tt3. The simulation continues integrating until the total mass which has flowed from the exit plane is 

equal to the mass originally in the tube when the integration began. 

Figure 6 shows contours of normalized pressure, temperature and Mach number for the example cycle using the 

present hybrid method. This should be qualitatively compared to the detonative cycle of Figure 4. It can be seen that 

several of the same gasdynamic features exist (with the obvious exception of the detonation itself). These include a 

large, left-running expansion fan at the commencement of the blowdown period, and a largely steady, rightward 

flow during the fill period, despite the existence of a weak adverse pressure gradient. Not surprisingly, the predicted 

period of the hybrid cycle is within 25 percent of the computed period of the detonative cycle. If the transit time of 

the detonation is accounted for, the predictions for total cycle period are even closer. Such accuracy is acceptable 

when preliminary performance and sizing estimates are being investigated. It represents substantial improvement 

over that obtained with either the algebraic or lumped volume modeling methods. 

The computed distribution of exhaust for the notional combustor using the hybrid method is shown in Figure 7. 

For comparison, the detonative velocity distribution of Figure 3 is also shown. Although the hybrid velocity profile 

misses the detonation induced spike from 0.06<me/minitial<0.28, it remains at a higher level from 

0.28<me/minitial<0.64. The two profiles are subsequently quite comparable during the fill phase which extends from 

0.64<me/minitial<1.00. Most importantly however, the hybrid method value of .7100 * e au  is quite close to the 

value of 0.665 obtained with the detonative CFD simulation. Consequently, the pressure ratio of PR = 1.18 

calculated from Equation (6) and ue from the hybrid method also nearly matches that obtained from the detonative 

CFD. Evidently then, the much simpler hybrid simulation method outlined here successfully captures the essential 

physics required for preliminary performance and combustor sizing at the example conditions of Table 1. 
NASA/TM—2010-216821 9 

p/p* T/T* Mach #

time

High

Low

2.301

0.627

2.301

0.853

0.710

0.000

x/L x/L x/L

Figure 6.—Contour plots of normalized pressure, temperature, and Mach number, for the 

computed cycle using the hybrid method. 

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.2 0.4 0.6 0.8 1.0

ue/a*

me/minitial

ue 

avg 

rms 

uepdc 

*

e au

*2e au

ue_eff /a*

(ue_eff /a*)PDC

Figure 7.—Computed exit velocity distribution as a function of 

normalized ejected mass for the example cycle using the 

hybrid method. 
NASA/TM—2010-216821 10 

IV. Additional Results 

The preceding discussion has demonstrated that the hybrid simulation method reasonably matches a higher 

fidelity detonative CFD simulation at a single operating point. In this section, attention is turned to additional 

operating points. In particular, the same notional combustor configuration and inlet conditions are maintained, but 

the air/fuel ratio is varied. By Equation (2), a change in a/f alters TR, which alters ue , and thus PR. The computed 

results of a/f variations are shown in Figure 8. Here PR according to Equation (6) is plotted as a function of TR from 

Equation (2) using the various modeling methods described. The two lines corresponding to the algebraic methods 

use 

2

ue instead of 2

ue in Equation (6), since the latter cannot be computed. The difference between the two 

algebraic lines is intended to illustrate the impact on available kinetic energy that arises from assuming constant 

volume versus true detonative combustion. It simply shows graphically that detonation generates less entropy. In 

either case, the predicted values of PR are well above those from the detonative CFD, which is considered the 

benchmark result for the purposes of this paper. As stated earlier, these results indicate that algebraic methods are 

not appropriate for performance analyses of pressure gain combustors. 

Also shown in Figure 8 are results using the present hybrid, and the lumped volume model approaches. The 

former tends to match the detonative CFD at low TR, while over-predicting performance as TR increases. The 

lumped volume method under-predicts throughout the TR range examined (which spans the a/f range 

15.7<a/f<78.5). This trend, combined with generally large over-predictions of cycle period, suggest that the lumped 

volume technique is also inappropriate for combustor configurations of the type described here. 

A. Fill Fraction Variations 

There is a second conceptual technique by which the overall a/f and related TR may be varied. The combustible 

mixture entering the tube is held at a low a/f (near stoichiometric), but the volume fraction of the tube actually filled 

with a combustible mixture is less than one. The remainder of the gas filling the tube is unfueled air. Such a scenario 

is often considered when fully filling the tube with a lean mixture would result in poor detonability or a generally 

slow combustion process. The post-combustion flow exiting the tube is thermally stratified, but the mass averaged 

total temperature still obeys Equation (2), provided all of the entering fuel and air are used in determining a/f. 

Examining the effects of this form of overall a/f variation is straightforward (though time consuming) using the 

detonative CFD code described above. It is also possible to account for it using in the present hybrid model. The key 

to doing so lies in choosing an appropriate post-reactive state with which to initiate the numerical blowdown 

process. There are several options available which will conserve mass and energy over the entire cycle. In this work, 

1.0

1.2

1.4

1.6

1.8

2.0

1.0 2.0 3.0 4.0

Total Pressure Ratio

Total Temperature Ratio

Algebraic w/ detonation

Algebraic w/ CV

Detonative CFD

Hybrid

Lumped Volume

Figure 8.—Total pressure ratio as a function of total temperature ratio computed using 

various methods. TR is varied by directly varying a/f. 
NASA/TM—2010-216821 11 

the state which provides the best match to the detonative CFD simulation is as follows. The density is held at the 

same value as the inlet (i.e., whatever processes occur during the instantaneous “combustion” event do so at constant 

overall volume). The pressure is uniform and chosen such that the total entropy generated is the same as what would 

be generated by constant volume combustion of only the reactant filled tube fraction. 



















































 













  11

1

3 3

init

t

CV

t p

p ff

p

p (9) 

Here, the term ff is the fraction of the tube which is filled with the air and fuel mixture. The total internal energy of 

the system is also preserved, which, unlike Equations (7) and (8), requires the imposition of an initial velocity. This 

is written as follows. 

  1

2 1 + 1 

3

init

3

init

 













  













  

t t

CV

T

T

T

T ff

a

u

* (10) 

Clearly, the imposition of a uniform post-combustion state on the hybrid method will not result in the 

appropriate thermal stratification of the exit flow that is observed in true partial fueling scenarios. Nevertheless, the 

computed mass averaged velocity appears to match the detonative CFD quite well. This is shown in Figure 9 where 

PR is plotted as a function TR for the two methods. The variation in TR for this figure represents a range in fill 

fraction from 0.2<ff<1.0. 

1.0

1.2

1.4

1.6

1.8

2.0

1.0 2.0 3.0 4.0

Total Pressure Ratio

Total Temperature Ratio

Detonative CFD

Hybrid

Figure 9.—Total pressure ratio as a function of total temperature ratio computed using 

detonative CFD and hybrid simulation methods. TR is varied by varying fill fraction. 
NASA/TM—2010-216821 12 

V. Discussion 

The hybrid pressure-gain combustor model described in this paper reasonably matches the performance 

predictions and cycle times of a higher fidelity, detonative, CFD model over a range of operating conditions. The 

results are rapidly generated (fractions of a second, versus minutes), and require less input data and progress 

monitoring than detonative CFD. These features make the hybrid approach well suited for insertion as a pressuregain module in a gas-turbine cycle deck. Furthermore, the fact that the hybrid approach does use a form of CFD 

suggests that real world loss models are straightforward to apply. The model presented is ideal in the sense that the 

only entropy generated comes from combustion and non-uniformity. However, many unavoidable losses, (heat 

transfer, viscous, leakage, etc.) are relatively easy to include as source terms in the governing equations (Refs. 19 

and 20). This can enhance the real-world accuracy of the simulation without appreciably altering the cycle 

computing time. 

Though the present model is strictly one-dimensional, modification to compute quasi-one dimensional flows is a 

trivial exercise. This would extend the model capability to include combustors with exhaust nozzles. 

VI. Conclusion 

An efficient model for computing the performance of detonative pressure gain combustor configurations has 

been described. The approach focuses on correctly capturing the exhaust velocity distribution, while conserving 

overall mass and energy flows. Using a hybrid combination of algebraic combustion models and simple CFD 

algorithms appropriate for smooth, non-reacting flows, the new model can compute closed cycle combustor 

performance in fractions of a second on a typical personal computer. The results, including flow rates, cycle times, 

and pressure ratios, compare favorably with higher fidelity, reacting CFD simulations. The code is appropriate, and 

intended, for use in cycle decks so that pressure-gain benefits assessments may be performed on a variety of gas 

turbine platforms. 

References 

1. Caldwell, N., and Gutmark, E., “Performance Analysis of a Hybrid Pulse Detonation Combustor/Gas Turbine 

System,” AIAA–2008–4880, July, 2008. 

2. Perkins, H.D., et al., “An Assessment of Pulse Detonation Engine Performance Estimation Methods Based On 

Experimental Results,” AIAA–2005–3831, July, 2005. 

3. Paxson, D.E., Perkins, H.D., “Thermal Load Considerations for Detonative Combustion-Based Gas Turbine 

Engines,” AIAA–2004–3396, July 2004. 

4. Rasheed, A., Tangirala, V.E., Vandervort, C.L., Dean, A.J., Haubert, C., “Interactions of a Pulsed Detonation 

Engine with a 2D Blade Cascade,” AIAA–2004–1207, January, 2004. 

5. Petters, D. and Felder, J., “Engine System Performance of Pulse Detonation Concepts Using the NPSS 

Program,” AIAA–2002–3910, July, 2002. 

6. Snyder, P. Alparsian, B., and Nalim, R., “Gas Dynamic Analysis of the CVC, A Novel Detonation Cycle,” 

AIAA–2002–4069, July, 2002. 

7. Nalim, R.M., and Paxson, D.E., “A Numerical Investigation of Premixed Combustion in Wave Rotors,” ASME 

Journal of Engineering for Gas Turbines and Power, Vol. 119, No. 3, 1997, 

pp. 668–675. 

8. Kailasanatha, K., Patnaika, G., and Lia, C., “The Flowfield and Performance of Pulse Detonation Engines,” 

Proceedings of the Combustion Institute, Vol. 29, Issue 2, pp. 2855–2862, 2002. 

9. Heiser, W.H., Pratt, D.T., “Thermodynamic Cycle Analysis of Pulse Detonation Engines,” Journal 

of Propulsion and Power, Vol. 18, No. 1, pp. 68–76, 2002. 

10. Paxson, D.E., “Performance Evaluation Method for Ideal Airbreathing Pulse Detonation Engines,” Journal of 

Propulsion and Power, Vol. 20, No. 5, pp. 945–950, 2004. 

11. Tangirala, V.E., Murrow, K., Fakunle, O., Dean, A.J., “Thermodynamic and Unsteady Flow Considerations in 

Performance Estimation for Pulse Detonation Applications,” AIAA–2005–226, January, 2005. 

12. Mitrofanov V.V., Zhdan S.A. In: Roy G., Frolov S., Santoro R., Tsyganov S., editors., Confined Detonations 

and Pulse Detonation Engines, Moscow: Torus Press, 2003, pp. 177–84. 

13. Paxson, D.E., Brophy, C., Bruening, G., “Performance Evaluation of a Pulse Detonation Combustion Based 

Propulsion System Using Multiple Methods,” JANNAF Journal of Propulsion and Energetics, Vol. 3, No. 1, 

May, 2010, pp. 44–55. 

14. Kaemming, T., Dyer, R., “The Thermodynamic and Fluid Dynamic Functions of a Pulsed Detonation Engine 

Nozzle,” AIAA–2004–3916, July, 2004. 
NASA/TM—2010-216821 13 

15. Foa, J.V., Elements of Flight Propulsion, Wiley, New York, 1959, pp. 274–283. 

16. Anderson, J.D., Computational Fluid Dynamics, McGraw-Hill, 1995. 

17. MacCormack, R.W., “The Effect of Viscosity in Hypervelocity Impact Cratering,” AIAA Paper, 

69–354, 1969. 

18. Paxson, D.E., “A General Numerical Code for Wave Rotor Analysis,” NASA TM 105740, 

July, 1992. 

19. Paxson, D.E., Schauer, F., and Hopper, D., “Performance Impact of Deflagration to Detonation Transition 

Enhancing Obstacles,” AIAA–2009–502, January, 2009. 

20. Paxson, D.E., Wilson, J., Welch, G.E., “Comparison Between Simulated and Experimentally Measured 

Performance of a Four Port Wave Rotor,” AIAA–2007–5049, July, 2007. 
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A Simplified Model for Detonation Based Pressure-Gain Combustors 

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14. ABSTRACT

A time-dependent model is presented which simulates the essential physics of a detonative or otherwise constant volume, pressure-gain 

combustor for gas turbine applications. The model utilizes simple, global thermodynamic relations to determine an assumed instantaneous 

and uniform post-combustion state in one of many envisioned tubes comprising the device. A simple, second order, non-upwinding 

computational fluid dynamic algorithm is then used to compute the (continuous) flowfield properties during the blowdown and refill stages 

of the periodic cycle which each tube undergoes. The exhausted flow is averaged to provide mixed total pressure and enthalpy which may be 

used as a cycle performance metric for benefits analysis. The simplicity of the model allows for nearly instantaneous results when 

implemented on a personal computer. The results compare favorably with higher resolution numerical codes which are more difficult to 

configure, and more time consuming to operate. 

15. SUBJECT TERMS

Pulse detonation engines; Combustion chambers; Pulsed jet engines 

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