Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

Numerical Analysis of a Rotating Detonation 

Engine in the Relative Reference Frame

NASA/TM—2014-216634

March 2014

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Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

Numerical Analysis of a Rotating Detonation 

Engine in the Relative Reference Frame

NASA/TM—2014-216634

March 2014

AIAA–2014–0284


	



	


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NASA/TM—2014-216634 1 

Numerical Analysis of a Rotating Detonation Engine 

in the Relative Reference Frame 

Daniel E. Paxson 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

A two-dimensional, computational fluid dynamic (CFD) simulation of a semi-idealized rotating 

detonation engine (RDE) is described. The simulation operates in the detonation frame of reference and 

utilizes a relatively coarse grid such that only the essential primary flow field structure is captured. This 

construction yields rapidly converging, steady solutions. Results from the simulation are compared to 

those from a more complex and refined code, and found to be in reasonable agreement. The performance 

impacts of several RDE design parameters are then examined. Finally, for a particular RDE configuration, 

it is found that direct performance comparison can be made with a straight-tube pulse detonation engine 

(PDE). Results show that they are essentially equivalent. 

Nomenclature 

a non-dimensional speed of sound

a*

 dimensional reference speed of sound 

AFR air to fuel ratio (by mass) 

ratio of inlet (or exit) slit height to annulus height 

F non-dimensional conserved circumferential flux vector 

gc Newton constant 

G non-dimensional conserved axial flux vector 

 ratio of specific heats 

hf fuel heating value 

I impulse 

K0 non-dimensional reaction rate constant 

l circumferential length 

L tube length 

My axial Mach number 

p non-dimensional pressure 

p*

 dimensional reference pressure 

q0 non-dimensional heat addition parameter 

Rg real gas constant 

{ non-dimensional density 

{* dimensional reference density 

S non-dimensional source term vector 

T non-dimensional temperature 

T*

 dimensional reference temperature 

Tc0 non-dimensional reaction temperature 

Tsp specific thrust 

t non-dimensional time 

u non-dimensional circumferential velocity 

v non-dimensional axial velocity 

w non-dimensional conserved state vector 
NASA/TM—2014-216634 2 

x non-dimensional circumferential distance 

y non-dimensional axial distance 

z reactant fraction 

Subscripts 

e exit plane 

b back (as in back or ambient pressure) 

det referring to the detonation 

tr total (pressure or temperature) in reservoir 

1.0 Introduction 

The rotating detonation engine (RDE) represents an intriguing approach to achieving pressure gain 

combustion (PGC) for propulsion and power systems. The RDE essentially consists of an annulus with 

one end open (or having a nozzle) and the other end having a valve system (possibly using nonmechanical, fluidic actuation). Fuel and oxidizer enter axially through the valved end. The detonation 

travels circumferentially. Combustion products exit, predominantly axially, through the open end. The 

process is illustrated in Figure 1, which shows computed contours of normalized temperature throughout 

the ‘unwrapped’ annulus at a moment in time during the cycle. Also shown in the figure are crude, but 

roughly scaled inlet and exit axial velocity vectors. The majority of the fluid entering the device is passed 

over or ‘processed’ by the rotating detonation wave which substantially raises the pressure and 

temperature. As the fluid continues to move predominantly axially, it is expanded and accelerated. When 

it leaves the device, the fluid pressure and momentum are elevated relative to the inlet. As such, it 

produces thrust and/or average total pressure gain. 

Figure 1.—Computed contours of non-dimensional temperature throughout the annulus of 

an ‘unwrapped’ RDE, at a moment in time during the cycle. 

x/l

y/l

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

2

4

6

8

10

detonation direction

low velocity detonable mixture in

high velocity combustion products out

detonation

y 

x 
NASA/TM—2014-216634 3 

The potential advantages of this approach compared to conventional pulsed detonation engines 

(PDE’s) include the elimination of initiation and deflagration to detonation transition (DDT) devices, and 

an exceptionally high cycle rate governed by the transit time of the detonation around the circumference. 

Although these devices have been demonstrated in the laboratory (Refs. 1 and 2), they are difficult to 

instrument. Additionally, since nearly all laboratory devices do not have mechanical valves, they cannot 

be safely operated with premixed fuel and air. As such there is significant uncertainty about local 

stoichiometry and therefore detonation properties. The net result of these difficulties is that, while RDE’s 

are operational, their fundamental physical principles are not well understood. Not surprisingly, neither 

are methodologies to optimize performance. 

Computational fluid dynamic (CFD) studies have been performed (Refs. 3 and 4); however, they are 

relatively few in comparison to those for PDE’s. As such there is a need for further and varied 

computational studies in order to compare against each other, and with the growing body of experiments. 

Furthermore, many of the existing computational models are constructed with the goal of capturing 

tremendous detail in the flow field. This implies fine grid spacing, and complex reaction models that, 

while yielding valuable data, require vast computational resources and time in order to reach 

convergence. For the sort of parametric evaluations needed to understand fundamental operational 

physics, a simpler, less refined modeling approach is beneficial. 

This paper presents the preliminary results of just such an approach. A CFD-based RDE model is 

described whereby the reference frame of the rotating detonation is adopted. As such, converged solutions 

are obtained which are steady in time. This is insured by utilizing a relatively coarse grid and a simple 

finite rate reaction model which respectively eliminate the otherwise unstable shear flow, and the cellular 

detonation structure associated with more detailed analyses. Comparison with the more detailed CFD 

models will show that these simplifications preserve performance prediction capability, and by extension, 

the capture of essential RDE physics. The use of a coarse grid also yields converged solutions rapidly 

while utilizing modest computational resources. This feature makes the model useful for parametric 

studies related to performance. 

The flexibility of the model, and its methodological similarity to a quasi-one-dimensional CFD code 

for investigating conventional pulse detonation engines (PDE) (Ref. 5), allow for a direct comparison of 

semi-idealized RDE and PDE performance. The conditions under which this comparison is made, and the 

results will be described. The results will show that under these conditions, the performance of RDE’s and 

PDE’s is nearly the same. 

A simplified configuration of the model will also be described which is used to investigate 

explanations for the reduced detonation speeds (compared to one-dimensional theory) observed in 

experiments. It will be shown that the lack of axial confinement associated with RDE’s is one possibility. 

2.0 Model Description 

The model is developed with the assumption that the working fluid is a calorically perfect gas (CPG) 

of fixed composition. This is a significant simplification which is far from reality, particularly in the 

vicinity of rotating detonation itself. However, it has been shown that with judiciously chosen average gas 

properties (Ref. 6), remarkably good agreement with experiments can be obtained. More importantly 

however, the simplification does not fundamentally impact performance trends or fundamental physics, 

the capture of which are the goals of the present effort. In addition, the CPG assumption greatly simplifies 

the mathematics associated with code development. 

2.1 Governing Equations 

On the further assumptions that working fluid is inviscid and adiabatic, and that the RDE annulus 

radius of curvature is much greater than its height, the governing equations of motion may be written in 

non-dimensional form as follows. 
NASA/TM—2014-216634 4 

 = S

y

G

x

F

 + 

t

w













 (1) 

where 

 

 















	

























 



  







z

p u v

v

u

1 2

w = 2 2 (2)

 





















	













































  

 









uz

u v

uv

u

u

2

+

( 1)

p u 

+ p

F =

2 2

2

(3) 

 

























	

















































  

 









vz

u v

v

v

uv

v

G

2

+

( 1)

p

+ p =

2 2

2 (4) 



















	









































  

















 

0

0 0

0

0 0

0

0;

;

0;

;

0

0

0

c

c

c

c

T T

K T T

z

T T

K T T

zq

S (5) 

The non-dimensional equation of state needed to complete the governing equation set is written as 

follows. 

p  T (6) 
NASA/TM—2014-216634 5 

The non-dimensional pressure, p, density, {, temperature, T, and velocities, u and v have been 

obtained using a reference state p*

, {*, T*, and the corresponding sound speed a*. The distances, x and y

have been non-dimensionalized by the circumference, l. The time, t has been non-dimensionalized using 

the reference wave transit time, l/a*

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

which depends on the assumed fuel heating value, the oxidizer (air in this work), and fuel-to-air ratio 

corresponding to the reactant fraction, z = 1.0. 

  1 * 0

   R T AFR

h

q

g

f (7) 

For the remainder of the paper, all quantities displayed or discussed are considered non-dimensional 

unless stated explicitly otherwise. 

The finite rate reaction model (Eq. (5)) is a very simple one which simply states that the reaction rate 

is equal to the product of a rate constant, K0, and the amount of reactant present. However, the reaction is 

not allowed to proceed unless the reactants are above a reaction temperature, Tc0. 

2.2 Numerical Treatment 

The governing equations are integrated numerically in time using an explicit, second-order, two-step, 

Runge-Kutta technique. Spatial flux derivatives are approximated as flux differences, with the fluxes at 

the discrete cell faces evaluated using Roe’s approximate Riemann solver (Ref. 7). Second-order spatial 

accuracy (away from discontinuities) is obtained using piecewise linear representation of the primitive 

variable states within the cells. Oscillatory behavior is avoided by limiting the linear slopes (Ref. 8). 

2.3 Boundary Conditions 

Boundary conditions are imposed by assigning time-dependent values to so-called image (aka 

“ghost”) cells, which are just outside of the interior computing domain. Referring to Figure 1, the 

following techniques are used to determine values. At x = 0.0 and x = 1.0, symmetry is invoked. This 

means the rightmost boundary image cells are assigned the value of the leftmost interior cells at each ylocation. Similarly the leftmost boundary image cells are assigned the value of the rightmost interior cells. 

The symmetry assignment insures that the x-dimension of the computational space faithfully represents 

an annulus (which is continuous and has no boundary). 

At the exhaust end, y=0.4, constant pressure outflow is used along with characteristic equations to 

obtain , and v for the image cells. If the resulting flow is sonic, or supersonic, then the imposed pressure 

is disregarded. If it is exactly sonic then the sonic values of p, , and v are assigned. If it is supersonic, 

then p, , and v are extrapolated from the interior (Ref. 9). The x-velocity component, u is set equal to the 

last interior cell at each location. This is a crude form of extrapolation which simply insures that  0





y

u , 

thereby allowing slip between exiting flow and that outside the computing domain (about which nothing 

is known). 

At y=0.0 (the inflow face), two different boundary conditions are used depending on the interior flow 

state. This face is presumably fed by a large reservoir at a fixed total pressure, and temperature. If the 

interior pressure along this face is greater than the reservoir pressure, ptr, as might be found just behind 

the detonation, then a solid wall type boundary conditions are applied with slip in the x direction. This 

prevents backflow of hot combustion products into the inlet reservoir. If the interior pressure is less than 

the reservoir pressure, then partially open inflow boundary conditions are applied as described and 

validated in Reference 10. These determine p, , and v for the inflow face image cells. This type of 

boundary condition assumes that flow enters through a slit which is some fraction, of the annulus 
NASA/TM—2014-216634 6 

height. The slit may be imagined as an infinitesimally short converging nozzle. The flow between the 

reservoir and the nozzle throat is isentropic. For values of < 1 however, there is a total pressure mixing 

loss that takes place across a notional mixing plane separating the nozzle throat from the interior 

computing domain. In the mixing plane, the magnitude of the total pressure loss varies with the flow rate 

through the slit and the value of . 

The x-velocity component, u, is prescribed during inflow, and it is here that a reference frame change 

is implemented. Rather than specify u=0 (i.e., no swirl) which is the laboratory or fixed frame condition, 

the negative of the detonation speed, udet is prescribed instead. As a result of this change to the detonation 

reference frame, the computational space becomes one where a steady-state solution is possible. This is 

the computational equivalent to placing an aircraft model in a wind tunnel and moving the air rather than 

the model. 

2.4 Solution Procedure 

The detonation speed is not known a priori. The classical calculation associated with the so-called 

Chapman-Jouguet (CJ) point is no longer reliable, as it is based on the assumption of strictly onedimensional flow (with heat addition (Ref. 11). As such, an initial guess is made for udet and steady cold 

flow is established in the domain. The value of q0 is temporarily set to zero, allowing the potential for 

reaction without heat release. A steady heat addition rate prescribed at x=0.5, and extending from y=0 to 

y=0.1 is then applied. The simulation is restarted (from the established cold flow state), and run until 

steady state is again reached. This step allows the internal flow to become heated, thereby allowing the 

reaction to proceed, even though the heat release rate (and location) is fixed. At this point, the fixed heat 

addition rate is turned off, and q0 is restored to the value corresponding to the fuel/oxidizer mixture of 

interest. The simulation is then restarted and run for the amount of time corresponding to three annular 

revolutions of the detonation at the assumed detonation speed. The domain is then examined and if it is 

seen that the detonation front has moved to the right of x=0.5, then it is apparent that the initial guess at 

udet is too high. If the detonation front has moved to the left, then the initial guess is too low. Based on this 

determination, a new guess is made for udet, and the simulation is re-run for another three detonation 

revolutions. The process continues until the detonation front remains stationary and the entire domain 

stops changing. Because the simulation is typically only 200×80 numerical cells (several orders of 

magnitude coarser than previous work (Refs. 3 and 4), this can be accomplished in only minutes on a 

modern laptop computer. When the simulation appears converged, it is then run for the equivalent of 10 

additional cycles as a check on whether the detonation is stationary. After this the total mass and energy 

fluxes at boundaries are checked to insure conservation (i.e., that residuals are acceptably small). 

2.5 Deflagration and Detonation Zones 

As illustrated in Figure 1, the primary heat release mechanism in an RDE is a detonation. However, 

there is also a region where deflagration occurs. In the absence of a so-called buffer layer (pure air with 

no fuel) premixed charge is introduced adjacent to hot combustion products. If the flow field, and 

numerical approximation thereof, were truly inviscid, no reaction would occur in this region. In real flows 

however, there is always diffusion; and the best CFD schemes cannot completely avoid numerical 

diffusion even when physical diffusion terms are neglected. Diffusion results in combustion products 

heating the premixed charge, thereby allowing the chemical reaction to proceed. 

The particularly simple reaction mechanism used in the present model, together with the coarse grid 

which generates relatively high numerical diffusion, can lead to unrealistically high flame speeds in the 

region labeled ‘deflagration zone’ in Figure 1. In fact, so much reaction can occur in the deflagration zone 

that there is no reactant left to detonate. In order to control this, the reaction rate constant, K0 is reduced in 

specified regions of the computational space. The regions can be altered depending on the configuration 

simulated; however, they are generally defined as shown in Figure 2. 
NASA/TM—2014-216634 7 

Figure 2.—Deflagration and detonation zones in the computing domain 

Figure 3.—Configuration and boundary conditions for detonation speed investigation. Relevant parameters: =0.00, 

=1.264, T*

=520 R, p*=14.7 psia, *=0.055 lbm/ft3, a*=1250 ft/s, ucj=5.47, q0=23.4. The detonable mixture is 

stoichiometric h2 and air. The computational grid is 200x10 cells. 

3.0 Results 

3.1 Preliminary Detonation Speed Investigation 

Before proceeding to a full RDE simulation, the code was used to investigate a simpler configuration. 

It has been noted that the propagation speed of the detonation in an RDE is slower than in a conventional 

PDE. The reasons for this are not clear; however, one possibility may be the lack of confinement at the 

‘top’ of the RDE detonation (i.e., y  0.1 in Figure 1). Immediately behind the detonation front, this 

provides a sort of ‘leakage path’ for energy and momentum that would otherwise propel the detonation 

forward. In order to test this explanation, the code was run using the boundary conditions and geometry 

illustrated in Figure 3. 

If the upper surface slit width is set to =0.0, then it becomes another wall. No leakage is possible. If 

the mixture is ignited, and the numerical scheme is working correctly, a purely one dimensional stationary 

(i.e., steady) detonation should form. In other words, the leftward detonation wave velocity exactly 

matches the rightward inflow velocity. 

The expected outcome of such a test is shown in Figure 4. The upper portion of the figure shows 

contours of temperature throughout the domain after two non-dimensional time units of simulation. The 

simulation was initiated (at t=0) by setting all numerical cells between 0.0<x<0.5 to the predetonative 

inlet state, and all those between 0.5<x<1.0 to the Chapman-Jouguet (CJ) state. The lower portion of 

Figure 4 shows the distribution of pressure, temperature, and Mach number at y=0.025. Also shown as 

dashed lines, are the theoretical, post-reactive CJ values for pressure and temperature. It is clear that a 

stationary detonation has formed, that the flow is one-dimensional, and that the post reactive CJ state is 

properly computed. 

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

x

K0_def  0.15*K0_det

K0_det = 260.0

K0_def  0.15*K0_det

Air and fuel in at CJ velocity

and ambient static conditions

Wall with slip

Partially open outflow (i.e. a slit) with slip 

at ambient exit pressure

Constant pressure or 

extrapolated supersonic outflow

Standing det.

x

y

0 1 0

0.05

u=5.47

v=0.00

p=1.00

=1.00
NASA/TM—2014-216634 8 

(a) 

(b) 

Figure 4.—Computed, steady contours of temperature throughout the Figure 3 

computational test space with =0.0 (a); distributions of pressure, temperature 

and Mach number at y=0.025 for the same simulation (b). 

(a) 

(b) 

Figure 5.—Computed, steady contours of temperature throughout the Figure 3 

computational test space with =0.5 (a); distributions of pressure, temperature and 

Mach number at y=0.025 for the same simulation (b). 

When the slit width , is set to a non-zero value it is observed (though not shown) that the detonation 

migrates downstream. Given sufficient simulation time, it will ultimately exit the computing domain. This 

indicates that the detonation speed does indeed decrease when confinement is reduced. In order to recover 

a stationary wave, the inflow velocity must be reduced. The results of such an exercise are shown in 

Figure 5. Here, the same information as Figure 4 is conveyed; however, the slit width is set to =0.5. The 

inflow velocity required for a stationary detonation is 4.7. This is approximately 14 percent below the CJ 

velocity of 5.47 required for the =0.0 case. Figure 5(a) also shows two particle streamlines (in white) so 

that the y-velocity component behind the detonation can be seen. It is interesting to note in the 

distribution of Figure 5(b) that the CJ pressure, temperature and Mach number are never achieved. 

However, the entropy and total enthalpy rise are exactly the same as Figure 4. In other words, the reduced 

pressure rise, and detonation speed do not represent lost availability for the fluid to do work. 

x

y

0 1

0

0.05

2

4

6

8

0

5

10

15

20

25

30

0.0 0.2 0.4 0.6 0.8 1.0

p,T, M

x

p T

M p_CJ

T_CJ

x

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.05

2

4

6

8

0

5

10

15

20

25

30

0.0 0.2 0.4 0.6 0.8 1.0

p,T, M

x

p T

M p_CJ

T_CJ
NASA/TM—2014-216634 9 

Figure 6.—Computed contours of temperature throughout the annulus of an 

‘unwrapped’ RDE, at steady state. Relevant parameters: ptr=10, Ttr=1.038, 

pb=1, =0.20, =1.264, T*

=520 R, p*=14.7 psia, *=0.055 lbm/ft3, a*=1250 ft/s, 

udet=4.7, q0=23.4. 

3.2 Full RDE Simulation 

The results of a converged simulation of the full RDE domain, under the boundary conditions 

described earlier, are shown in Figure 6 in the form of temperature contours. This is similar to Figure 1. 

The inlet reservoir pressure and temperature (in the fixed frame) are ptr=10 and Ttr=1.038 respectively. 

The exit static pressure is pb=1. The inlet slit-to-annulus throat height ratio is =0.20. The mixture 

heating value q0=23.4 corresponds to stoichiometric hydrogen and air. The ratio of specific heats used is 

=1.264. Dimensional reference conditions are T*

=520 R, p*=14.7 psia, *=0.055 lbm/ft3, a*=1250 ft/s. 

The value of udet required to achieve a steady solution is udet=4.7. This is 14 percent lower than the purely 

one-dimensional CJ detonation speed for these reference conditions of 5.47. The computational grid is 

200×80 cells. The features of Figure 6, including the ‘tilt’ of the detonation, the height of the detonation, 

the oblique shock, the deflagration region, and the shear or vortical region appear similar to high fidelity 

CFD simulations of the same geometry and conditions (Refs. 3 and 4). This does not prove the validity of 

the simulation; however, it is at least encouraging, particularly given the mere 18 sec that it takes to 

compute the flow field for one revolution of the detonation on an ordinary laptop computer. 

Also shown in Figure 6 are select streamlines (in white) which show the paths of particles through the 

computational domain. It can be seen that most particles enter, pass through the detonation (in this frame 

of reference), and leave directly through the exit plane. However, some will pass through the oblique 

shock generated by the detonation wave and will exit at a substantially different state than the other 

particles. This indicates a non-uniform exhaust flow which will be illustrated later in the paper. 

Figure 7 shows the computed gross specific impulse as a function of the ratio of inlet total to exit 

static pressure. The geometry and working fluid properties are all identical to those used for Figure 6. 

Also shown are the computed results from Reference 3. The Reference 3 model used a much finer grid 

resolution (approximately 2 orders of magnitude more cells), a non-calorically perfect gas, and a more 

sophisticated reaction mechanism than the present model. Furthermore, the partially open inlet was 

modeled in a different fashion than the present work (a mass source term was used) as were the exit 

boundary conditions (non-reflecting conditions were imposed). It was also run in the laboratory reference 

frame. Nevertheless, the performance computed by the two models is quite similar. This provides a 

measure of confidence in the present approach as a tool for interrogating the flowfield, gaining 

understanding, and performing rapid parametric analyses. 

x

y

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

2

4

6

8

10
NASA/TM—2014-216634 10 

4.0 Discussion 

4.1 Semi-Idealized Behavior 

The partially open inlet model represents a crude attempt to account for the losses that may be 

encountered in an actual RDE; however, it has not yet been validated against any experiment. 

Furthermore, it introduces a loss that is not fundamental. That is to say it can, in theory, be eliminated. As 

such, it is worth exploring simulations where =1.0. 

Under this scenario, the performance increases substantially from that shown in Figure 7. This is 

evident in Figure 8 where gross specific impulse at =1.0 and 0.20 are compared. This is an expected 

result as the mixing losses associated with the partially open inlet model can be substantial. However, the 

performance shown is still considered semi-ideal because there is no exit nozzle modeled and much of the 

exit flow (in fact all in many cases) is under-expanded. 

Figure 7.—Gross specific impulse as a function of RDE inlet total to 

exit static pressure ratio for the same geometry and properties used 

in Figure 6. 

Figure 8.—Gross specific impulse as a function of RDE inlet total to 

exit static pressure ratio for two values of , the ratio of inlet throat 

to annulus height. The geometry and properties are the same as 

those used in Figure 6. 

2500

3000

3500

4000

4500

5000

5500

6000

6500

0 5 10 15 20 25

Gross specific impulse, sec.

ptr/pb

Ref. 3

Present model

2500

3000

3500

4000

4500

5000

5500

6000

6500

0 5 10 15 20 25

Gross specific impulse, sec.

ptr/pb

e=1.0

e=0.2



NASA/TM—2014-216634 11 

A detailed examination of the interior indicates that the flow through a portion of the =1.0 RDE seems 

to attain a sort of similarity regardless of the inlet pressure. The axial-extent of the detonation becomes 

fixed. The distribution of axial Mach number and temperature in this region also become fixed. The reason 

for this appears to be an internal fluidic choking of the flow that occurs downstream (axially) from the 

detonation. This has the same effect as a choked flow through a physical throat in that Mach number and 

other distributions become fixed and the exit pressure no longer affects the flow upstream of the throat. An 

example of the interior axial Mach number distribution is shown in the contour of Figure 9 for =1.0 and 

ptr=2.75. All other parameters are the same as those of Figure 6. The maximum contour shown (dark red) 

corresponds to My=1.0. In other words, My values greater than 1.0 still appear as dark red. It can be seen that 

there is a somewhat convoluted, but nevertheless nearly continuous (over all x locations) contour line 

representing the ‘fluidic throat’ whereby the fluid has accelerated from the subsonic values along the inlet 

plane to sonic values along the contour. This contour is shown as a dashed white line in the figure. The 

formation of this throat (i.e., choking) helps to explain why the length of an RDE of the configuration shown 

in this paper has almost no impact on performance. For example, Figure 10 shows contours of temperature 

for a semi-ideal RDE under exactly the same conditions as Figure 9, except the computational domain is 

only half as long. The computed fuel specific impulse was the same as that of Figure 9 to within 0.2 percent. 

This result may have implications for the design of compact RDE’s and for appropriate back-pressurization 

strategies; however, considerably more investigation is needed. 

This fluidic fixing of the flowfield is similar in many ways to the behavior of PDE’s when they are 

equipped with a physical throat at the exhaust end. In order to illustrate this, and to perhaps provide a 

basis of comparison between RDE’s and PDE’s, a validated quasi-one-dimensional PDE simulation code 

(Ref. 5) was run using the same idealizations, parameters, and even computational approach as the present 

RDE code. The PDE simulation contained provisions for a converging exhaust nozzle, and the throat size 

was adjusted until the time averaged inlet fill Mach number of the PDE matched the space averaged axial 

inlet fill Mach number of the RDE (approximately 0.54). The converged PDE cycle is shown in Figure 11 

in the form of contours of pressure and temperature. In this figure the vertical axis represents time. The 

horizontal axis is distance along the PDE tube. 

Figure 9.—Computed contours of axial Mach number throughout the annulus of 

an ‘unwrapped’ RDE, at steady state. The same parameters as Figure 6 are 

used except: ptr=2.75, =1.0. 

Figure 10.—Computed contours of temperature throughout the annulus of an ‘unwrapped’ 

RDE, at steady state. The same parameters as Figure 9 are used. 

x

y

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0

0.2

0.4

0.6

0.8

x

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

2

4

6

8

10
NASA/TM—2014-216634 12 

Figure 11.—Contours of pressure and temperature over one 

cycle of a PDE under the same conditions and with the same 

average inlet Mach number as Figure 9. 

Figure 12.—Gross specific impulse as a function of total to static 

pressure ratio for semi-idealized PDE and RDE at the same 

average inlet Mach number.

The performance of the RDE and PDE are compared in Figure 12 which, like Figure 7 and Figure 8, 

shows gross specific impulse as a function of inlet total-to-exit static pressure. It can be seen that they are 

quite similar. The slightly lower RDE performance (approximately 12 percent maximum at low pressure 

ratios) has not been fully explored as of this publication; however, it is likely due to three effects. First, 

approximately 10 percent of the incoming RDE mixture is consumed through less productive deflagration 

rather than through detonation in the RDE simulation. No deflagration occurs in the PDE simulation. 

Second, the PDE simulation must include a buffer zone of unfueled inlet air (purge flow) in order to avoid 

premature ignition of the reactant mixture by the hot gases remaining in the tube during filling. Purge 

flow often raises fuel specific impulse (while reducing specific thrust) (Refs. 5 and 12). Approximately 3 

to 4 percent of the PDE inflow is purge, while there is none in the RDE. Third, the RDE has a unique 

2500

3000

3500

4000

4500

5000

5500

6000

6500

0 5 10 15

Gross specific impulse, sec.

ptr/pb

RDE PDE
NASA/TM—2014-216634 13 

circumferential velocity element that does not contribute to thrust, but does consume chemical energy. 

This will be discussed in the next section. 

Before doing so however, it is worthwhile to examine the inflow and outflow of the two semiidealized cycles just described in order see their similarities. This is done in Figure 13. Figure 13(a) 

shows the inflow (axial) Mach number as a function of the fraction of the total cycle mass flow that has 

entered each device. Figure 13(b) shows the fraction of the cycle impulse as a function of the fraction of 

the total cycle mass flow that has exited each device. The origin of both plots coincides with the moment 

in time when the inlet closes. For reference, the impulse is defined as follows. 

u dt PDE p p

v dx RDE p p

u

I

t

o

e e e b

x

o

e e e b

 















 





 















 







 

 

2

2

det

1

(8)

Comparing the profiles shown, it is seen that for this semi-idealized cycle, the evolution of thrust and 

the ingestion of mass are quite similar despite the quite different approaches (RDE vs. PDE) to 

detonation. 

(a) 

(b) 

Figure 13.—Inflow axial Mach number as a function of cycle inflow mass flow fraction (a); 

cycle impulse fraction as a function of cycle exit mass flow fraction (b). 

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Inflow Mach Number

Fraction of Cycle Mass Flow

PDE

RDE

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Fraction of Cycle Impulse

Fraction of Cycle Mass Flow

PDE

RDE
NASA/TM—2014-216634 14 

4.2 Swirl 

4.2.1 Exit Swirl 

Another contributing factor to the mildly reduced RDE performance compared to the PDE may be 

due to the swirl (circumferential velocity components) associated with the RDE exit flow. Although the 

primary flow direction is axial, the transverse direction of the detonation produces swirl. This is shown in 

Figure 14 where computed laboratory frame velocity components are plotted as functions of 

circumferential position in the RDE exit plane for the semi-ideal cycle of Figure 9. The circumferential 

component does not contribute to thrust. In fact, it can be shown mathematically that unless swirl is 

introduced at the inlet to the RDE, no net swirl can exit. In other words, there is no torque present. In 

terms of the formulation in this paper, this can be stated as follows. 

  ! " 0 1

det

det

  

x

o

e e e v u u dx

u

(9) 

When the profile of Figure 14 is used in Equation (9), the equality is found to hold to within the 

numerical accuracy of the code (i.e., the left hand expression of Eq. (9) divided by the impulse from 

Eq. (8) is less than 0.009). 

Although there is no net momentum associated with the locally swirling flow, there is kinetic energy 

(ultimately converted from the added chemical energy). The computed mass averaged circumferential 

kinetic energy for the Figure 14 profile is approximately 2 percent of the chemical energy added. This 

may be written as: 

  ! "

 

0.022

2 0

2

det





 

 

 

x

o

e e

x

o

e e e

q v dx

v u u dx

(10) 

Figure 14.—Computed laboratory frame velocity components as 

functions of circumferential position in the RDE exit plane. 

Parameters are the same as Figure 9. 

-1.5

-0.5

0.5

1.5

2.5

3.5

4.5

0.0 0.2 0.4 0.6 0.8 1.0

Velocity (laboratory frame)

x

v

u+u_det
NASA/TM—2014-216634 15 

Figure 15.—Computed contours of vorticity throughout the annulus of an 

‘unwrapped’ RDE, at steady state. The same parameters as Figure 9 are 

used. 

4.2.2 Internal Swirl 

The coarse grid of the present simulation causes it to be dissipative. One of the results of this is that 

lines of slip (which are allowed in inviscid formulations) are manifested as shear layers. The dissipation is 

numerical; however it may be argued that the effect is physical since real flows are not inviscid. For the 

RDE simulation, the dissipation of interest occurs at the interface of the flow that has passed through the 

detonation and is subsequently accelerated (axially and circumferentially), and the flow that is entering 

through the inlet. This is illustrated in Figure 15 which shows contours of vorticity for the Figure 9 

conditions. Vorticity is a measure of fluid rotation, but it can also measure shear as it is essentially a 

combination of spatial velocity gradients. Figure 15 illustrates that the incoming flow retards the postdetonative flow in a dissipative fashion. In other words, it is a form of drag (albeit relatively small). 

In closing this section, it is important to emphasize that the results and discussion just presented are 

preliminary and specific to the configurations used, as well as to the assumptions made. They are intended 

to demonstrate the utility of the modeling approach described for understanding the physics of RDE’s and 

examining the sensitivity of performance to parametric changes. They are not meant as statements on the 

practicality of the RDE. 

5.0 Conclusion 

A relatively simple, two-dimensional, computational fluid dynamic (CFD) based approach to the 

simulation of rotating detonation engines (RDE’s) has been presented. The simplification is achieved by 

computing in the detonation frame of reference and by utilizing a course computational grid. These result 

in rapid convergence to steady solutions. Results from the code were shown to compare favorably with 

those from a more complex and refined code. The capability to assess the impact of RDE configuration 

changes was demonstrated. So too was the retention of sufficient flowfield detail such that the 

understanding of fundamental physical processes governing RDE operation can be ascertained. As part of 

the latter demonstration, a methodology was described which may allow direct performance comparison 

of a particular RDE and a pulse detonation engine (PDE). Under this methodology, it was shown that 

their performance is essentially equivalent. The simplicity of the code means that it can easily be 

modified to include various well-known loss mechanisms (e.g., heat transfer, wall friction), and 

configuration modifications such as an exit throat. Such modifications are planned for future versions. 

References 

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Detonation Engine Design,” AIAA 2012-121, January 2012. 

x

y

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

-4

-3

-2

-1

0

Accelerated flow retarded 1

by slower moving inflow
NASA/TM—2014-216634 16 

2. Naples, Hoke, J., Karnesky, J., Schauer, F., “Flowfield Characterization of a Rotating Detonation 

Engine,” AIAA 2013-0278, January 2013. 

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