Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

Frederick Schauer and David Hopper

Air Force Research Laboratories, Wright-Patterson Air Force Base, Ohio

Performance Impact of Defl agration to Detonation

Transition Enhancing Obstacles

NASA/TM—2012-217629

July 2012

AIAA–2009–502
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Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

Frederick Schauer and David Hopper

Air Force Research Laboratories, Wright-Patterson Air Force Base, Ohio

Performance Impact of Defl agration to Detonation

Transition Enhancing Obstacles

NASA/TM—2012-217629

July 2012

AIAA–2009–502

National Aeronautics and

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Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2012-217629 1 

Performance Impact of Deflagration to Detonation 

Transition Enhancing Obstacles 

Daniel E. Paxson 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Frederick Schauer and David Hopper 

Air Force Research Laboratories 

Wright-Patterson Air Force Base, Ohio 45433 

Abstract 

A sub-model is developed to account for the drag and heat transfer enhancement resulting from 

deflagration-to-detonation (DDT) inducing obstacles commonly used in pulse detonation engines (PDE). 

The sub-model is incorporated as a source term in a time-accurate, quasi-one-dimensional, CFD-based 

PDE simulation. The simulation and sub-model are then validated through comparison with a particular 

experiment in which limited DDT obstacle parameters were varied. The simulation is then used to 

examine the relative contributions from drag and heat transfer to the reduced thrust which is observed. It 

is found that heat transfer is far more significant than aerodynamic drag in this particular experiment. 

Nomenclature 

A non-dimensional cross-sectional area 

A*

 reference cross-sectional area 

a*

 reference speed of sound 

Cf friction coefficient 

D*

 reference diameter 

F flux vector 

gc Newton constant, 32.174 lbm ft/lbf s2 

h obstacle element height 

K0 non-dimensional reaction rate constant 

L tube length 

min  inlet mass flow rate 

p non-dimensional pressure 

p*

 reference pressure 

Pr Prandtl number 

q0 non-dimensional mixture heating value 

ReD* Reynolds number based on the reference diameter 

S source vector 

St Stanton number 

t non-dimensional time 

T non-dimensional temperature 

u non-dimensional velocity 

V volume 
NASA/TM—2012-217629 2 

w conserved flow vector 

z reactant fraction 

 obstacle friction factor correlation coefficient 

p obstacle element pitch 

 ratio of specific heats 

 non-dimensional density 

*

 reference density 

Subscripts: 

e referring to the exhaust plane 

H referring to the head cavity 

wall referring to the tube wall 

Introduction 

Many pulse detonation engine (PDE) configurations rely on regularly spaced obstacles in the flow to 

enhance deflagration to detonation transition (DDT). Without them, impractically long passages (tubes), 

or oxygen enrichment would be required to successfully achieve detonations. However, obstacles can 

exact a severe performance penalty on the engine due, at least in part, to the unavoidable drag that their 

functionality produces. As a result of this simultaneous necessity and penalty, the prevailing approach to 

obstacle design has aimed to simply minimize both the blockage from the obstacles and the fractional 

length over which the obstacles are present. While this is a reasonable approach, it isn’t particularly 

quantitative, and it leads to significant trial and error. For practical PDE design, it is not enough to know 

that there is a performance penalty from obstacles. The magnitude and sensitivity of the penalty as a 

function of some measurable parameters must be known. Most realistic designs are a compromise 

involving competing objectives that are weighed against one another. Without measurable figures of merit 

(or decrement) it is difficult to undertake such a process. 

This paper represents an analysis-based step toward that end. A numerical PDE simulation tool is 

modified to model the impact of obstacles and is then validated on a particular experiment in which the 

extent of the DDT obstacles was varied. The validated simulation allows a detailed interrogation of the 

PDE interior and reveals, among other things that the “loss” from obstacles results at least as much from 

enhanced heat transfer between the working fluid and walls as it does from lost momentum. Heat transfer 

in PDE’s generally has the effect of heating the incoming flow, and cooling the expanding, postdetonative gas. The first of these effects results in overfilling the tube with a lower density, detonable 

mixture. Thrust is therefore reduced, and the spilled fuel is wasted. The second effect implies that 

released chemical energy from combustion is used to heat walls rather than being converted to kinetic 

energy for thrust. 

It is also shown in this paper that the relationship between obstacle length and momentum loss (drag) 

is quite non-linear. This observation could have a significant impact on obstacle design. 

The experiment and simulation will be described, though only briefly, since they have both been 

documented extensively in the literature (Refs. 1 to 9). Only details relevant to the present work will be 

highlighted. Model validation will then be presented through a comparison between the code prediction 

and experimental measurements. The Discussion section which follows will illustrate the utility of the 

model by demonstrating the potential performance impact on a basic PDE, notionally designed for 

Mach 2.0 flight. 
NASA/TM—2012-217629 3 

Experimental Rig Description 

The experimental results are obtained from the PDE at the Air Force Research Laboratory’s Pulsed 

Detonation Research Facility (Ref. 1). This PDE uses the head and valves of an automotive engine to 

control the airflow into the detonation tube. The PDE cycle consists of equal time allotted for: i) filling 

the detonation tube with pre-mixed hydrogen and air at an equivalence ratio of one, ii) ignition, 

detonation, and blow-down, and iii) purging of the detonation tube with air. For these experiments the 

pressure upstream of the automotive poppet valves is controlled such that, during the fill cycle, the mass 

flow of pre-mixed fuel and air is equivalent to the volume of the tube times the density of the fuel-air 

charge at the gas temperature upstream of the poppet valves and atmospheric pressure times the operating 

frequency. The fuel-air mixture is ignited with a 115 mJ spark at the closed end. 

Up to four tubes can be sequentially fired on the rig. The frequency of operation, the fill fraction, and 

the purge fraction can all be varied, along with fuel to air ratio, the fuel type, and the spark delay. 

Numerous measurements are available from the rig, including thrust, and average flow rate. Additionally, 

the rig can accommodate tubes of varying length and diameter. For this work, two different 

configurations are utilized. Parameters for each are given in Table 1. All tests are run until thermal 

equilibrium is reached. 

Simulation Description 

The core of the simulation is a high-resolution, time-accurate, single progress variable, reactive, 

quasi-one-dimensional (Q1D) CFD code which integrates differential equations of motion for a 

calorically perfect gas, thereby capturing the detailed gasdynamic phenomena occurring inside a tube of 

specified cross sectional area distribution. For reference, the equations of motion are written in nondimensional form as follows. 

)x,w(S = 

x

)w(F + 

t

w







 (1) 

where 



















































 







Az 

z+ q A

2

u +

)( 

p

Au 

A 

 = w

0

2

1

(2)

TABLE 1.—OPERATIONAL PARAMETERS FOR TWO CONFIGURATIONS OF THE EXPERIMENT 

Configuration 

number 

Tubes 

used 

Fuel Equivalence 

ratio 

Fill 

fraction1

Purge 

fraction2

Frequency 

(Hz) 

Length 

(in.) 

Diameter 

(in.) 

Spark delay 

(msec) 

1 1 H2 1.0 1.0 0.5 10.0 72.0 2.0 0.0 

2 4 H2 1.0 1.0 1.0 10.0 48.6 2.0 0.0 

1

Volumetric fraction of the tube filled with fueled mixture at the inlet temperature and ambient pressure. 2

Volumetric fraction of the tube filled with air at the inlet temperature and ambient pressure. 
NASA/TM—2012-217629 4 



























































 









 













Azu 

z + q A u + 

)(

p u 

 + u A p

Au 

 = F

0

2

2

21

(3)

   



























































 



 

 









  

 



 

<T T

K T T z A -

TT u

D

LASt

u u A

D

L

C dx

dA

p 

 = x),w(S

*

* f

0

00

wall

;0

;

1

4

2

0

(4) 

The non-dimensional pressure, p, density, ρ, and velocity, u have been obtained using a reference 

state p*

, ρ*, and the corresponding sound speed a*. The distance, x has been scaled by the passage length, 

L. The time, t has been scaled using the nominal wave transit time, L/a*

. The cross-sectional area, A has 

been scaled by a reference value, A*

. The heat of reaction of the reactant gas mixture, q0 is a constant 

which depends on the assumed fuel, oxidizer, and fuel-to-air ratio corresponding to the reactant fraction, 

z = 1.0. 

The source vector, S(w, x) includes contributions from the chemical reaction rate, viscous wall shear 

forces and heat transfer at the walls. Contributions from turbulent diffusion can also be assessed in the 

source term (Ref. 3); however, they are not listed here as they are not utilized in the present work. 

Similarly, a mass source term to assess leakage is normally included in S(w, x) (Ref. 7), but is not utilized 

here. 

The form of the wall shear coefficient Cf, in the momentum source term is based on a semi-empirical 

correlation for smooth walls similar to that described in Reference 5. It is written as follows. 

20

3520

24130 Re .

D

. *

f *

L

D .C 















  (5) 

Here, D*

 is the reference tube diameter, and ReD* is the local Reynolds number, based on the 

reference tube diameter. 

The Stanton number, St, appearing in the Equation (4) energy source term is assessed using the 

Reynolds-Colburn analogy, 2

PrSt 32 Cf  , where Pr is the fixed Prandtl number (Ref. 6). This source 

term also requires a wall temperature distribution, Twall. For simplicity, heat transfer is assumed to occur 

between the working fluid and the passage walls only (i.e., no axial conduction along the tube, and 

negligible heat transfer on the outside of the tube). Each wall section corresponding to a numerical cell is 

assumed to maintain a fixed temperature over the course of one detonative cycle. The net heat transferred 

to or from the wall is monitored over the course of the cycle, and at the end, the wall temperature is 

updated via simple Euler integration. The detonative cycle is then repeated with the updated wall 

temperature. This process continues until the net heat transfer from each numerical wall section is zero. 

Alternatively, a wall temperature distribution can simply be prescribed, on the supposition that some type 
NASA/TM—2012-217629 5 

of thermal management system can maintain it. If this approach is taken however, the code no longer 

conserves energy, since a path for net heat transfer to or from the working fluid is established. 

The one-step reaction mechanism of the code is relatively simple. As long as there is reactant present, 

a prescribed temperature threshold has been reached, and the non-dimensional reaction rate constant, K0, 

is high enough, detonation will occur. Initiation and deflagration to detonation details cannot be captured 

with this mechanism; however, this is not considered a drawback for the present work. The focus is to 

study the performance impact of DDT obstacles, not their effectiveness at achieving detonations. 

Boundary conditions are supplied to the code as total conditions at the inlet, and static conditions at 

the exit. The conditions are used along with characteristic equations to determine appropriate states for 

“ghost” numerical cells which lie just outside the computing domain at each end of the tube (Ref. 9). The 

exit boundary condition is an open one for this study. The ambient static pressure is imposed. However, a 

logic structure exists within the code such that when the exit flow is supersonic (and cannot sustain a 

normal shock), the imposed pressure is neglected and exit conditions are appropriately extrapolated from 

the interior. Furthermore, a logic structure also exists such that if the imposed static pressure cannot 

sustain outflow, the pressure is treated as a total value and, together with a supplied total temperature is 

used to determine an appropriate inflow state. The inlet (head end) boundary condition is normally either 

open, with total conditions specified, or closed, with reflective wall conditions specified. This allows the 

simulation of ideal valve-type operation. For simulation of the AFRL experiment, the boundary was left 

permanently open, but was coupled to a sub-model of the automotive cylinder head as described below. 

Numerical integration of Equation (1) is done according to high-resolution methodology described in 

References 5, 7, and 9. The present simulation results utilize 200 numerical cells. The non-dimensional 

time step is t = 0.0005. 

Automobile Cyclinder Head Sub-Model 

For simulations of the AFRL PDE experimental rig, a permanently open boundary condition is 

maintained at the head end of the tube; however, the pressure and temperature for this boundary are 

supplied by a coupled, lumped-volume, well- stirred reactor sub-model of the cylinder head which 

comprises the actual head end of the engine. The governing equations may be written non-dimensionally 

as follows. 

  

















   Au 000 Aa

m

V

LA

dt

d

***

in

H

* H  (6) 

  *** H HH

inin

H

* H zKq)(TAu

Aa

Tm

V

LA

dt

Pd





















  000 00 1  (7) 

  *** H HH

inin

H

* HH zKzAu

Aa

zm

V

LA

dt

zd



















  

000 0

 (8) 

Here, VH is the head volume and min  is the mass flow rate into the head volume. The perfect gas 

relationship, PH = HTH closes the model. The inflow rates of premixed detonable mixture, or purge air 

are prescribed (temporally and quantitatively) to match those of the rig. Outflow from the sub-model 

(the product of terms with the subscript 0) at any given instant of time is equal to the inflow predicted by 

the head-end numerical cell of the CFD code. Thus, the state in the head-end volume is updated via first 

order Euler integration of Equations (6) to (8), using the latest code inflow values. Then, the CFD code 

inflow is updated using the new head-volume stagnation state. 
NASA/TM—2012-217629 6 

In order to crudely mimic the initiation process which follows spark ignition in the actual rig, the 

head-end reaction rate is set several orders of magnitude lower than the value used by the CFD code in 

the tube. 

Generally, the reaction is not allowed to proceed unless the temperature is above a prescribed 

threshold value. However, in order to initiate the reaction at a desired “spark” time, the threshold criterion 

is temporarily removed. 

Sub-Model for DDT Obstacles 

For regions where obstacles exist, the friction coefficient takes the following form. 





















p

f

h C

2 (9) 

Referring to Figure 1, h is the obstacle height, and p is the spacing between obstacles. Equation (9) is 

independent of local Reynolds number. This is characteristic of bluff bodies in cross-flow, and fully 

roughened internal flows in the moderately high Reynolds number regime (Ref. 10). Conceptually, the 

discreet drag force associated with each obstacle is transformed into a continuous, enhanced wall shear 

stress. The force is estimated to be one half that of a cylindrical ring in cross flow, a configuration which 

has a well know drag coefficient. Only one half of this force is considered since the outer portion of the 

ring is near a wall, where the flow is at a much lower velocity than the freestream. The variable  is free 

parameter that is expected to be on the order of 1.0 and is used to correlate the model with experimental 

results. The value used in this work is  = 0.82. Equation (9) assumes axi-symmetric obstacles, with 

regular spacing. For the experimental system under investigation in this paper, the value of Cf from 

Equation (9) is on the order of 5 times the smooth walled value from Equation (5). As with Equation (5), 

the friction coefficient gives rise to an associated heat transfer coefficient which is similarly enhanced 

over the smooth-walled case via the Reynolds-Colburn analogy. 

Rig Simulation Parameters 

Unless otherwise noted, all simulation results use the following parameters. The hydrogen fuel 

heating value is 51,571 Btu/lbm. The single ratio of specific heats,  = 1.264. This value is established 

using the methodology described in Reference 11. The methodology selects a single  which minimizes 

disparities with the actual PDE environment in which  varies substantially over the course of a cycle. 

The simulation also requires a single real gas constant, Rg. A value of 73.92 ft-lbf/lbm/R is used which 

corresponds to a stoichiometric hydrogen/air mixture. The reference pressure and temperature are 

14.7 psia, and 520 R, respectively. The assumed viscosity is 1.0×10–5 lbm/ft/s. The Prandtl number is 0.7. 

Figure 1.—Schematic DDT obstacle layout. 

=

h

p
NASA/TM—2012-217629 7 

Validation Results 

Data from a limit cycle simulation of the Configuration 1 setup (Table 1) is shown in Figure 2. 

Obstacles are present over the first 22 percent of the tube length (Ref. 12). The figure shows contours of 

non-dimensional pressure, temperature, Mach number, and reactant fraction in the tube, over one period. 

Also shown are simulated pressure and temperature distributions in the head cavity. The horizontal axis of 

each contour represents distance along the tube. The vertical axis is non-dimensional time. For reference, the 

fill, thrust, and purge portions of the cycle are labeled. 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 measured gross thrust in the experiment is 5.2 lbf. The simulated thrust, 

obtained by integrating the exhaust momentum flux and pressure over one cycle, is 5.7 lbf. 

The disparity of less than 10 percent is considered small in light of the simplicity of the simulation. For 

example, it is evident that substantial inflow to the exhaust end occurs during the so-called thrust portion of 

the cycle (i.e., when both purge and fill inlet valves are closed). This is most easily seen in the temperature 

contour plot. The simulation results shown assumes that all fluid outside the PDE is at the inlet temperature. 

Thus, when there is inflow at the exhaust end, there is a sharp interface between hot gases in the tube and 

cold gas which is being drawn in. The interface follows the motion of the fluid and it is seen that the exhaust 

inflow penetrates more than halfway up the tube. That such inflow occurs during low frequency PDE 

operation has been experimentally confirmed (Ref. 13). The actual temperature of the exhaust inflow gas is 

not known; however, it is reasonable to expect that it is higher than the inlet temperature. This is so because 

some fraction of the initially expelled hot gas can be re-ingested, and because it is evident that the PDE is 

overfilling with a combustible mixture (see the portion identified as “spill” in Fig. 2), which is ultimately 

ignited by the detonation, creating a potential region of ingestible hot gas near the PDE exit. As a test of this 

notion, it is found that assuming an exhaust inflow temperature of twice the inlet value yields a thrust 

prediction of 5.4 lbf, which is within 4 percent of the measured value. 

Figure 2.—Computed limit cycle contours of pressure, temperature, Mach number, and reactant fraction in the 

tube, over one period of the AFRL PDE. Also shown are simulated pressure and temperature distributions in 

the head cavity. 

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

log(p) Log(T) Mach #

Reactant

Fraction

0

5

10

15

20

0 5

PH, TH

t

P

T

time

High

Low

1.19

-0.49

1.01

-0.02

1.28

-0.77

1.00

0.00

Fill

Thrust

Purge

detonation

spill
NASA/TM—2012-217629 8 

 

A second cause for thrust disparity with the experiment lies in the interaction between the time at 

which detonation actually commences, and the various, spurious waves which are evident in the Figure 2 

pressure contour. The waves are generated by reflections from boundaries and by the dynamics of the 

head cavity. If detonation is approximately coincident with a reflected compression wave, it can cause an 

increase in thrust. If it is an expansion wave, the thrust may be less. A slight delay in the time at which 

detonation occurs can determine which of these interactions occur. In the simulation, this implies that a 

change in the combustion rate constant of Equation (8) (which is heuristically chosen, as explained 

earlier) can impact the measured thrust. In fact, a 10 percent reduction in the rate constant is found to 

reduce the measured thrust of this configuration by approximately 2 percent. 

Figure 3 compares the measured and simulated limit-cycle wall temperature distributions for the same 

configuration used for Figure 2. Simulation results are presented for both the ambient and the elevated 

exit temperature boundary conditions described earlier. The trends of the simulation and experiment are 

similar, particularly in the vicinity of the obstacles. The largest disparity occurs in the latter portion of the 

tube, where the noted exhaust inflow occurs. The disparity is less in the Te = 2.0 case, which gives some 

credence to the supposition that the exhaust inflow gas is actually at a higher temperature than the 

ambient, or inflow air. Given the uncertainty of the exhaust inflow gas temperature just mentioned, 

measured and simulated wall temperature disparities in this region are not surprising. Furthermore, it is 

well documented that the interface observed in the Figure 2 temperature contour is not actually sharp. It is 

highly deformed due to multi-dimensional flow effects. In fact, the flow can become highly stratified 

during such inflow scenarios, thereby altering the wall temperature distribution. In practical PDE 

configurations, such back flow is unlikely due to the probable presence of a nozzle, and significantly 

higher operating frequencies. 

Figure 3 also shows the simulated, time-averaged gas temperature in the tube, with the Te = 1.0 

boundary condition. The distribution is different from that of the wall temperature, particularly in the 

obstacle region. This demonstrates the fact that the heat transferred to the wall depends as much on the 

heat transfer coefficient as on the temperature difference between gas and wall. 

Figure 4 compares the simulated and measured specific thrusts based on the fill flow (not including 

purge flow) as functions of the fraction of the tube over which there are obstacles. For this 

Configuration 2, of Table 1 is used. Again, the agreement is quite good, suggesting that the simulation has 

successfully captured the essential physics associated with this PDE. Evidently, the impact of the obstacles 

can be quite large. Even with a reasonable obstacle extent of xobstacles/L = 0.36, the specific thrust is reduced 

from the no obstacle case by 19 percent. Furthermore, a direct-connect test such as this one does not reflect 

500

700

900

1100

1300

1500

0 0.2 0.4 0.6 0.8 1

x/L

Temperature, R

Exp. -wall

Sim.-wall, Te=1.0

Sim.-wall, Te=2.0

Sim.-gas average

obstacle

region

40

50

60

70

80

90

100

110

120

0 0.2 0.4 0.6 0.8 1

xobstacles/L

Fill Specific Thrust, lb f-s/lbm

0.01

0.02

0.03

0.04

0.05

0.06

0.07

PH-pamb

Tsp-Exp.

Tsp-Sim.

PH-p PH-pamb

Figure 3.—Experimental and simulated wall 

temperatures, and simulated time-averaged 

gas temperatures during PDE operation. 

Figure 4.—Experimental and simulated fill-specific 

thrust as functions of obstacle length. The nondimensional head cavity pressure is also shown. 
NASA/TM—2012-217629 9 

the full extent of the loss. The obstacles cause drag during the filling stage as well as during the detonation 

and blowdown stages. However, the filling drag is not manifested in the gross thrust measurement because 

the system automatically compensates for the drag by raising the inlet pressure. This is demonstrated in 

Figure 4, which also shows the simulated, non-dimensional, time-averaged head cavity pressure (less the 

ambient pressure) during the fill stage as a function of obstacle length. As the obstacle length increases, 

higher total pressure is required in order to force the same amount of flow through the tube within the same 

period of time. The impact of this excess total pressure can be assessed by using it to calculate an effective 

flight speed, which in turn can be used to calculate a specific ram drag force. This is then subtracted from 

the gross specific thrust. The outcome of such an assessment shows a 6.5 to 11 percent reduction in the 

specific thrust values shown in Figure 4, over the xobstacles/L range plotted. 

Discussion 

With the simulation and obstacle model somewhat validated, it is instructive to differentiate between 

thrust reduction arising from lost momentum, and that from heat transfer. This is not possible with an 

experiment, since a frictionless, or adiabatic rig cannot be built; however, it is relatively straightforward 

with a simulation. 

Figure 5 repeats the simulated results of Figure 4, and compares them to results obtained by changing 

various model parameters. The curve labeled “adiabatic” in the legend is generated by assuming that there 

is drag from the obstacles (and wall shear), but no heat transfer. The striking feature of this curve is that 

aerodynamic drag from obstacles seems to play a relatively small part in the overall gross thrust 

reduction, though it is noted that the filling drag discussed earlier is neglected. For reference, the result 

from a simulation of this configuration in which all friction related terms are set to zero is shown with the 

legend heading of “frictionless”. The addition of wall shear drag evidently leads to an approximate 

9 percent thrust reduction from this ideal scenario. The addition of obstacles further reduces the thrust by 

another 12 percent, but only when they occupy the full length of the tube. 

The adiabatic results are quite non-linear with regard to obstacle extent. This is not surprising if the 

Mach number contour of Figure 2 is considered. Much of the drag is generated during the early part of the 

thrust stage (7.5  t  9.0). During this period it can be seen that the velocity, responding to the leftrunning expansion wave, is very high in the latter 50 percent of the tube. Since the aerodynamic loss from 

obstacles is proportional to the square of the velocity, it stands to reason that when the obstacles extend 

into this region, their impact will be greater. 

Figure 5.—Simulated fill-specific thrust as functions of 

obstacle length for several parametric model variations. 

40

50

60

70

80

90

100

110

120

130

0 0.2 0.4 0.6 0.8 1

xobstacles/L

Fill Specific Thrust, lb f-s/lb

m

all effects

adiabatic

cooled

frictionless
NASA/TM—2012-217629 10 

The curve in Figure 5 with the legend labeled “cooled” represents a case where the friction and heat 

transfer sub-models are active, but the wall of the tube is artificially maintained at the inlet temperature. 

Fixing the wall temperature yields an estimate of the performance impact of thermal management systems 

without knowing their details. If the wall temperature is set to the inlet temperature, this scenario can be 

used to estimate the performance of short duration rig experiments (including single shot), or the early 

portions of longer experiments. In both cases, the gasdynamic processes reach limit cycle behavior much 

sooner than the walls reach thermal equilibrium. In other words, the wall temperature is essentially fixed 

over many cycles. The performance shown in this curve is better than that seen when the walls are 

allowed to reach thermal equilibrium; however, it is still well below that of the adiabatic scenario. If the 

heat transfer sub-model is correct, then the performance impact from cooling the hot, post-detonative flow 

is nearly as significant as that from heating the cool, inlet flow. As evidence that the model is correct, it is 

noted that for xobstacles/L = 0.22, the cooled wall specific thrust is approximately 26 percent higher than the 

thermal equilibrium case (i.e., all effects). This difference compares favorably with thrust measurements 

of the Reference 12 experiment which shows 6.5 lbf thrust in the early moments of a long duration test, 

and 5.2 lbf at thermal equilibrium. 

Flight Considerations 

Presuming that the model described above is reasonably valid, consideration is now turned to the 

implications for a notional flight configuration. The particular PDE examined is optimized for Mach 2.0 

flight, at a dynamic pressure of 1500 lbf/ft2

. An ideal inlet feeds the tube (or tubes), and an ideal valve 

(i.e., loss free, opens instantly the moment that the tube pressure is below the inlet stagnation pressure) 

controls the flow. The PDE is fully fueled with a stoichiometric hydrocarbon/air mixture. Purge flow is 

kept to a minimum of approximately 2 percent of the total throughflow. The simulated tube has a constant 

cross section over 80 percent of its length. This is followed by a contraction section for the next 7 percent 

and an expansion section for the remaining 13 percent. The geometry is shown at the top of Figure 6. The 

hypothetical PDE tube has an L/D*

 ratio of 10, where the reference diameter is at the head-end of the 

tube. The value of ReD* used is 3.3×107

. Obstacles are present over 50 percent of the length with, 

h/p = 0.094. 

Figure 6 illustrates the limit cycle for the ideal version of this PDE in the same fashion as Figure 2. 

Here, the word “ideal” indicates that all wall friction, heat transfer, and obstacle terms are inactive. The 

time period when the valve is closed is indicated by the black rectangles aside each contour. The net fuel 

specific impulse for this PDE configuration is calculated to be 1796 sec. 

The loss terms are then activated, and a uniform wall temperature of 4.0 times the inlet is imposed. 

This value is chosen because it yields no net heat transfer from the walls. In other words, it crudely 

represents a thermal management system in which relatively cool regions of the PDE flow path are used 

to cool relatively hot regions. The resulting limit cycle is shown in Figure 7. Note that the time scale on 

this figure is the same as Figure 6. Thus, it is seen that the cycle time required to achieve the same true fill 

fraction (filling the entire tube with reactant) is less than the ideal. The time at which the inlet valve opens 

is also earlier than the ideal case because the tube pressure stays elevated after the detonation for a shorter 

period. The calculated specific impulse is 1605 sec., approximately 11 percent below the ideal value. 

Without obstacles (smooth walls), the specific impulse is 1722 sec. Examining the temperature contour, it 

is clear that the post detonative gas is being cooled and that the inlet flow is heated. 
NASA/TM—2012-217629 11 

Figure 6.—Computed contours of pressure, temperature, Mach number, and reactant fraction 

for one limit cycle of an ideal stoichiometric, hydrocarbon/air, Mach = 2.0 flight PDE. The 

value of  is 1.28. 

Figure 7.—Computed contours of pressure, temperature, Mach number, and reactant fraction 

for one limit cycle of a stoichiometric, hydrocarbon/air, Mach = 2.0 flight PDE with losses. The 

value of  is 1.28. 

time

log(p/pt_in) T/Tt_in Mach #

Reactant

Fraction

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

reflection

detonation

High

Low

inflow

exhaust

1.46

-1.48

9.48

0.59

3.01

-0.67

1.00

0.00

time

log(p/pt_in) T/Tt_in Mach #

Reactant

Fraction

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

reflection

detonation

High

Low

inflow

exhaust

1.13

-1.13

11.44

0.94

2.81

-0.41

1.00

0.00

Twall=4.0
NASA/TM—2012-217629 12 

Conclusions 

A simple model has been developed to assess the impact of deflagration-to-detonation (DDT) 

enhancing obstacles on pulse detonation engine (PDE) performance. The model was implemented in a 

Q-1-D code and validated on experimental results. It was found that observed performance reductions are 

often due more to the enhanced heat transfer associated with the obstacles than from drag. Analysis 

indicates that for practical PDE configurations, DDT obstacles can reduce specific impulse by more than 

10 percent. These results obviously suggest that DDT obstacles should be kept to a minimum in terms of 

size, and axial extent. More importantly however, they suggest that efforts to reduce heat transfer to and 

from tube walls may prove more fruitful for optimizing PDE performance, particularly since obstacles 

may be unavoidable. 

References 

1. Schauer, F.R., Bradley, R.P. and Hoke, J.L., “Detonation Initiation of Hydrocarbon-Air in a Pulsed 

Detonation Engine,” AIAA–2005–1343, Jan., 2005. 

2. Paxson, D.E., “A Performance Map for Ideal Air Breathing Pulse Detonation Engines,” AIAA–

2001–3465, July, 2001. 

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

ASME Journal for gas Turbines and Power, Vol. 119, No. 3, July, 1997, pp. 668–675. 

4. Paxson, D.E., Optimal Area Profiles for Ideal Single Nozzle Air-Breathing Pulse Detonation 

Engines, AIAA–2003–4512, July, 2003. 

5. Paxson, D.E., Wilson, J., “Recent Improvements to and Validation of the One-dimensional NASA 

Wave Rotor Model,” NASA TM-106913, 1995. 

6. Holman, J.P., Heat Transfer, 5th ed., McGraw-Hill, New York, 1981, pp 195. 

7. Paxson, D.E., “Comparison Between Numerically Modeled and Experimentally Measured WaveRotor Loss Mechanisms,” AIAA Journal of Propulsion and Power, Vol. 11, No. 5, 1995, pp. 908–

914. 

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

Turbine Engines,” AIAA–2004–3396, July, 2004. 

9. Paxson, D.E., “A General Numerical Model for Wave Rotor Analysis,” NASA TM 105740, 1992. 

10. Schlichting, H., Boundary Layer Theory, 7th ed., McGraw-Hill, New York, 1979, pp. 17 and 617. 

11. Paxson, D.E., Brophy, C.M. and Bruening, G.B., “Performance Evaluation of a Pulse Detonation 

Combustion Based Propulsion System Using Multiple Methods,” in Proc. JANNAF 42nd 

Combustion Subcommittee, 30th Airbreathing Propulsion Subcommittee, 30th Exhaust Plume 

Technology Subcommittee, 24th Propulsion Systems Hazards Subcommittee, 12th SPIRITS User 

Group Joint Meeting, Newton, MA, JSC CD-53 (CD ROM) CPIAC Abstract No. 2008-0006BW, 

May 2008. 

12. Hoke. J.L., Bradley, R.P. Schauer, F.R., “Impact of DDT Mechanism, Combustion Wave Speed, 

Temperature, and Charge Quality on Pulsed-Detonation-Engine Performance,” AIAA–2005–1342, 

Jan., 2005. 

13. Opalski, A.B., Paxson, D.E., Wernet, M.P., “Detonation Driven Ejector Exhaust Flow 

Characterization Using Planar DPIV,” AIAA–2005–4379, July, 2005. 
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Performance Impact of Deflagration to Detonation Transition Enhancing Obstacles 

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14. ABSTRACT

A sub-model is developed to account for the drag and heat transfer enhancement resulting from deflagration-to-detonation (DDT) inducing 

obstacles commonly used in pulse detonation engines (PDE). The sub-model is incorporated as a source term in a time-accurate, quasi-onedimensional, CFD-based PDE simulation. The simulation and sub-model are then validated through comparison with a particular 

experiment in which limited DDT obstacle parameters were varied. The simulation is then used to examine the relative contributions from 

drag and heat transfer to the reduced thrust which is observed. It is found that heat transfer is far more significant than aerodynamic drag in 

this particular experiment. 

15. SUBJECT TERMS

Combustion; Pulse detonation engine 

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