Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

Matthew L. Fotia and John Hoke

Innovative Scientific Solutions Inc., Dayton, Ohio

Fred Schauer

U.S. Air Force Research Laboratory, Wright-Patterson AFB, Ohio

Comparison of Numerically Simulated and 

Experimentally Measured Performance 

of a Rotating Detonation Engine

NASA/TM—2015-218835

August 2015

AIAA–2015–1101
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Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

Matthew L. Fotia and John Hoke

Innovative Scientific Solutions Inc., Dayton, Ohio

Fred Schauer

U.S. Air Force Research Laboratory, Wright-Patterson AFB, Ohio

Comparison of Numerically Simulated and 

Experimentally Measured Performance 

of a Rotating Detonation Engine

NASA/TM—2015-218835

August 2015

AIAA–2015–1101

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Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2015-218835 1

Comparison of Numerically Simulated and Experimentally Measured 

Performance of a Rotating Detonation Engine

Daniel E. Paxson

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44130

Matthew L. Fotia and John Hoke

Innovative Scientific Solutions Inc.

Dayton, Ohio 45440

Fred Schauer

U.S. Air Force Research Laboratory

Wright-Patterson AFB, Ohio 45433

Abstract

A quasi-two-dimensional, computational fluid dynamic 

(CFD) simulation of a 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 and other simplifications yield rapidly converging, 

steady solutions. Viscous effects, and heat transfer effects are 

modeled using source terms. The effects of potential inlet flow 

reversals are modeled using boundary conditions. Results from 

the simulation are compared to measured data from an 

experimental RDE rig with a converging-diverging nozzle 

added. The comparison is favorable for the two operating points 

examined. The utility of the code as a performance optimization 

tool and a diagnostic tool are discussed.

Nomenclature

a nondimensional speed of sound

a* reference speed of sound, 1250 ft/s

f air to fuel ratio (by mass)

gc Newton constant, 32.174 ft-lbm/lbf/s2

h nondimensional annulus height

hf fuel heating value, 51,571 Btu/lbm

l circumferential length

p nondimensional pressure

pman nondimensional inlet manifold total pressure

p* reference pressure, 14.7 psia

q0 nondimensional heat addition parameter

t nondimensional time

u nondimensional circumferential velocity

udet nondimensional detonation circumferential velocity

v nondimensional axial velocity

w conserved variables vector

x nondimensional circumferential distance

y nondimensional axial distance

z reactant fraction

F circumferential flux vector

G axial flux vector

K0 nondimensional reaction rate constant

Rg mixture gas constant, 73.92 ft-lbf/lbm/R

S source term vector

T nondimensional temperature

T* reference temperature, 520 R

Tc0 nondimensional reaction temperature

Tman nondimensional manifold temperature

ε ratio of inlet injector area to RDE annulus area at the 

inlet end

φ equivalence ratio

γ mixture ratio of specific heats, 1.264

ρ nondimensional density

ρ* reference density, 0.055 lbm/ft3

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 one-way inlet (i.e., mechanically or 

aerodynamically valved). Fuel and oxidizer enter axially 

through the inlet end. The detonation travels circumferentially, 

pressurizing and heating the majority of the flow that has 

entered. The flow then expands, accelerates, and exits,
NASA/TM—2015-218835 2

predominantly axially, through the open end. The exiting flow 

is at a higher average total pressure than the entering flow and 

can in principle produce more thrust (with a nozzle) or work 

(with a turbine) than a conventional so-called constant pressure 

combustion process. An illustration of a basic RDE cycle can 

be seen in Figure 1 which shows computed contours of 

temperature throughout the ‘unwrapped’ annular region at a 

moment in time.

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, exceptionally high cycle rates (on the 

order of 103 Hz.) governed by the transit time of the detonation 

around the circumference, and extremely compact design 

options.

Although these devices have been demonstrated in the 

laboratory (Refs. 1 to 5), until recently there have been very few 

built that have been instrumented sufficiently to measure 

performance or to compare with modeling efforts. Meanwhile, 

most published modeling efforts have been focused on somewhat 

idealized systems, as there has been very little data available on 

which to tune and test loss models (Refs. 6 to 9).

This paper represents an initial attempt at bridging this gap 

wherein results from a computational fluid dynamic (CFD) 

model with loss sub-models are compared to measurements 

from an instrumented RDE having a converging-diverging 

exhaust nozzle. A simplified, two-dimensional, CFD model has 

been developed which operates in the reference frame of the 

rotating detonation and is therefore steady when converged

(Ref. 6). It is largely idealized with the exception of a loss 

model to account for the effects of typical RDE inlets which 

generally consist of a throat having considerably smaller cross 

sectional area than the RDE annulus. For the present work, axial 

annulus area variation capability has been added (i.e., it is 

quasi-two-dimensional, or Q-2-D), along with loss sub-models 

to account for viscous and heat transfer effects between walls 

and working fluid. A loss sub-model has also been added to 

assess the impact of potential flow reversal at the inlet end 

caused by the high pressure spike behind the detonation. Since 

present RDE’s do not have mechanical inlet valves this is a 

distinct possibility depending upon the manifold pressure with 

which the RDE is fed and the size of the aforementioned inlet 

throat. The added sub-models are directly adapted from 

validated quasi-one-dimensional models used in PDE

simulations (Refs. 10 to 13). As such, only brief descriptions 

will be provided here. Details of the Q-2-D formulation for the 

governing equations will be provided.

The RDE experiment has been described in Reference 14. 

Therefore, only relevant details will be presented in this paper. 

The simulation will be compared to the experiment at two 

operating points. Comparison will be based on thrust, mass flow 

rate, and pressure at several axial locations.

It will be shown that the simulation and experiment compare 

quite favorably, and that the simulation can therefore be used to 

assess the performance impact of various RDE design and flow 

features. 

Figure 1.—Computed contours of nondimensional temperature throughout the 

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

detonation direction

high velocity combustion products out

detonation

x 

low velocity detonable mixture in

y 
NASA/TM—2015-218835 3

2.0 Simulation Description

2.1 Governing Equations

On the assumptions that the annulus radius of curvature is

much greater than the height, and that the working fluid is a

calorically perfect gas, the governing equations of motion may 

be written in nondimensional, detonation frame of reference

form as follows.

S

y

hG

x

hF

t

hw

+ = ∂

∂

+

∂

∂

∂

∂ (1)

where

( )

( )





































ρ

ρ +

+

γ γ −

ρ

ρ

ρ

zh

ph u v h

vh

uh

h

w

1 2

= 2 2 (2)

( )

















































ρ















 ρ +

γ −

ρ

















ρ

γ

ρ

uzh

h p u v

u

uv

u h p

uh

F = 

2

+

( 1)

+

2 2

2

(3)

( )





















































ρ















 ρ +

γ −

















ρ

γ

ρ

ρ

vzh

h p u vv

v h p

uvh

vh

G

2

+

( 1)

= +

2 2

2 (4)

( ) ( )

( )

( ) ( )





























































≤

> − ρ

 − 

 



 + ρ − +













≤

>

ρ

− ρ

− ρ − −

=

γ

h

T T

K T T z

h h u u v T T

T T

K T T

zq

c v v dy

dh

c u u u u

S

c

c

f wall

c

c

f

p

f

0

0 0

0.8

2 2 det

0

0 0

0

0.8

det

0.8

det

0;

;

0;

;

0

(5)

The governing equations are closed with the equation of state, 

which is written as follows.

p = ρT (6)

The nondimensional pressure, p, density, ρ, temperature, T, 

and velocities, u and v have been obtained using a reference state 

p*

, ρ*, T* and the corresponding sound speed a*. The 

nondimensional distances, x and y have been obtained using the 

circumference, l. The nondimensional time, t has been scaled 

using the nominal circumferential wave transit time, l/a*

. The 

nondimensional annulus height, h, represents the cross-sectional 

area normalized by a reference area. The source term vector, 

Equation (5) contains expressions to account for wall friction, 

area variation, heat transfer, heat release and reaction rate. The 

constants cf and hf are correlated friction and heat transfer 

coefficients that depend on fluid properties and geometry

(Ref. 13). The wall temperature Twall is assumed to be room 

temperature since all of the experiments simulated in this paper 

were run for very short periods of time (on the order of 1 sec.).

The finite rate reaction model in Equation (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. The heat of 

reaction of the reactant gas mixture, q0 is a constant which 

depends on the assumed fuel, oxidizer, and air-to-fuel ratio 

corresponding to the reactant fraction, z = 1.0.

( 1) * 0

γ + = R T f

h

q

g

f

(7)

For the remainder of the paper, all quantities displayed or 

discussed are considered nondimensional unless stated 

explicitly otherwise.
NASA/TM—2015-218835 4

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. Second-order spatial 

accuracy (away from discontinuities) is obtained using 

piecewise linear representation of the primitive variable states 

within the cells (MUSCL). Oscillatory behavior is avoided by 

limiting the linear slopes. More details of the method may be 

found in Reference 6, which also demonstrates limited 

numerical validation.

2.3 Boundary Conditions

Referring to Figure 1, the following boundary conditions are 

imposed. At x=0.0 and x=1.0, periodic (aka symmetric)

conditions are used. This means the rightmost boundary image 

cells are assigned the value of the leftmost interior cells at each 

y location. Similarly the leftmost boundary image cells are 

assigned the value of the rightmost interior cells. The symmetry 

boundary insures that the x-dimension of the computational 

space faithfully represents an annulus (which is continuous and 

has no boundary). At y = ymax, constant pressure outflow is used 

along with characteristic equations to obtain ρ, and v for the 

image cells. If the resulting flow is sonic, or supersonic, then 

the imposed pressure is disregarded. If, in addition, the 

upstream flow is supersonic, then p, ρ, and v are extrapolated 

from the interior (Ref. 10). The possibility for a normal shock 

solution whereby supersonic outflow jumps to subsonic is also 

accommodated. The x-velocity component u is set equal to the 

last interior cell at each location.

At y = 0.0 (the inflow face), partially open boundary 

conditions are applied as described and validated in 

Reference 11. This face is presumably fed by a large manifold

at a fixed total pressure, and temperature. The manifold 

terminates at the face and is separated from it via an orifice. The 

ratio of orifice flow area to RDE annulus area, ε is generally 

less than 1. If the interior pressure is less than the manifold 

pressure, pman, then inflow occurs. The boundary condition 

routine determines p, ρ, and v for the inflow face image cells

subject to a momentum (total pressure) loss model which 

depends on the mass flow rate and the value of ε. It is capable 

of accommodating a scenario where the orifice becomes 

choked. 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. If 

the interior pressure along the inlet face is greater than pman, as 

might be found just behind the detonation, then there will be 

backflow into the manifold through the orifice. The boundary 

condition routine can accommodate this as well. Based on the 

aerodynamic shaping of the upstream side of the experimental 

inlet orifice, and the nonaerodynamic shape of the downstream 

side, an effective area (equivalent to a discharge coefficient) of 

60 percent of the prescribed orifice area is used in backflow 

regions (Ref. 15).

In RDE simulations where inlet backflow occurs, the total 

mass and enthalpy that flow backward are averaged over the 

circumferential backflow span (recalling that in the steady 

detonation frame of reference, time is simply span divided by 

detonation velocity). When the interior pressure subsequently 

drops below pman and forward flow resumes, all of the mass that 

flowed backward is sent back into the RDE at the same average 

enthalpy that it exited. Once this mass has re-entered, the 

prescribed manifold premixed air and fuel mixture enthalpy is 

used.

Although the model assumes that premixed air and fuel enter 

through the inlet, the reality of most RDE experiments is that 

fuel and air are injected separately. The dynamics and feed 

pressures of the two injection systems can be quite different, 

particularly during lean operation. This raises the possibility 

that as the rotating detonation passes a given point, it may stop 

or even reverse the flow in both injectors; however, the 

resumption of inflow may occur at different times in each. This, 

in turn, may raise the possibility of a purging period where air 

enters the RDE for part of the cycle without having fuel added 

to it. In order to crudely explore the effects of this possibility, 

the prescribed inlet reactant fraction of the model may be set to 

zero over various portions of the circumference. This will be 

utilized in the results section of the paper.

2.4 Solution Procedure

The prescribed detonation speed is not known a priori. It must 

be found iteratively. An initial guess is made for udet and the 

simulation is run for the amount of time corresponding to three 

annular revolutions of the detonation. The domain is then 

examined to determine if the detonation has migrated from its 

initial position. If it has moved to the right of where it started, 

then 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 

these results, a new guess is made for udet, and the simulation is 

run for another 3 cycles. The process continues until the 

detonation front remains stationary and the entire domain stops 

changing. As described in Reference 6, the simulation 

deliberately uses a very course grid. For the simulations in this 

paper, a maximum of approximately 25,000 grid points define 
NASA/TM—2015-218835 5

the domain. This fact, combined with other simplifications 

(e.g., a one species reaction equation) mean that each iteration 

takes on the order of one minute on a laptop computer.

The process of initializing the simulation such that the 

flowfield contains a detonation on which speed iterations can 

be made is outlined in Reference 6.

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 can occur. This is because fresh charge (fuel and air) 

is introduced to the RDE 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 deflagration zone. In order to control 

this, the reaction rate constant, K0 is reduced in specified 

regions of the computational space as detailed in Reference 6.

Briefly, and with reference to Figure 1, the regions of low K0

are two rectangles defined as 0.00 ≤ y ≤ 0.10, 0 ≤ x≤ 0.48 and 

0.00 ≤ y ≤ 0.10, 0.60 ≤ x ≤ 1.00. The fraction of the incoming 

flow which deflagrates rather than detonates has an impact on 

RDE performance. To date however, it has not been possible to 

experimentally determine what this fraction is. For most of the 

simulations shown in this paper, rate parameters were set so that 

approximately 8 percent of the flow deflagrates. The impact of 

other deflagration fractions will be briefly discussed in the 

discussion section. 

3.0 Experiment Description

A detailed description of the experimental setup may be 

found in Reference 14. In brief, the annular, axisymmetric rig 

is shown schematically and to scale in Figure 2. The profile 

shown in blue represents the actual RDE shape. The profile in 

black is the Q-2-D equivalent (i.e., same annulus area, and 

effective hydraulic diameter, but fixed mean radius). The rig is 

21.86 in. long, with a mean diameter at the RDE annulus of 

5.76 in. Air and fuel (H2) are injected separately through 

manifolds that feed orifices of known size. Each manifold is fed 

by a calibrated choked flow orifice supplied at a very high 

pressure. During operation therefore the mass flow rate of fuel 

and air into the manifolds are constant and known. The mass 

flow rates of fuel and air into the RDE are highly unsteady 

(though periodic). However, once the manifold pressures, 

which are measured, become steady on a cycle-averaged basis, 

this setup insures that the time average mass flow rates into the 

RDE are identical to the constant mass flow rates into the 

manifolds. It is noted here that the simulation described above 

assumes a premixed flow entering the RDE through a single slit 

orifice. As such, the inlet orifice area used in the simulation is 

comparable to the combined areas of the experimental fuel and 

air orifices. The fluid properties used in the simulation are based 

on a stoichiometric mixture of air and hydrogen.

The rig has a series of static pressure taps placed at the 

locations shown in Figure 2. The taps are connected to 3 ft long 

tubes of 0.023 in. inner diameter, terminated with a pressure 

transducer. The long narrow tube damps the cyclic pressure 

oscillations and thermally protects the transducer. The 

arrangement has been given the name capillary tube averaged 

pressure (CTAP) and is thought to provide a close 

approximation to a time average (Ref. 16). The entire rig is 

mounted on a thrust stand such that gross thrust can be 

measured at any operating point. It is operated in an open air 

facility, so the exit pressure is atmospheric.

Figure 3 shows the nondimensional cross-sectional area of 

the rig (referenced to the area at the inlet end) as a function of 

the axial distance. The actual cross-section is shown, as is 

the smooth approximation (i.e., having a continuous first

Figure 2.—Experimental RDE rig schematic.
NASA/TM—2015-218835 6

Figure 3.—Cross sectional area profile as a 

function of normalized axial distance. 

Dashed red profiles represent potential 

‘effective’ profile caused by flow separation.

TABLE 1.—TEST POINT PARAMETERS

Test 

point

φ Total mass

flow rate,

lbm/s

Tman pman Thrust,

lbf

1 1.0 4.22 1.0 35.99 620.3

2 0.42 3.95 1.0 34.29 388.0

derivative) used in the simulation. With this view, it is clear that 

at y = 0.42 there is a rapid increase in area. As will be discussed 

in the next section, the data indicates flow separation in this 

area. In order to test that possibility, the alternate profile shown 

in Figure 3 was implemented in the simulation. This profile 

represents a notional streamline of the main flow created by the 

separated and recirculating flow.

4.0 Results

Two operating points were compared representing 

stoichiometric, and lean operation. Relevant quantities are shown 

in Table 1. The simulated inlet manifold pressure and 

temperature were set to match the experimental air manifold 

conditions. These high feed pressures result in very little 

backflow through the air injector. The mixture heating value q0

corresponds to stoichiometric hydrogen and air. The ratio of 

specific heats used is γ = 1.264. Dimensional reference conditions 

are T* = 520 R, p* = 14.7 psia, ρ* = 0.055 lbm/ft3

, a* = 1250 ft/s. 

The inlet orifice-to-annulus area ratio, ε was adjusted until the

Figure 4.—Computed contours of temperature throughout 

the annulus of the experimental RDE at the Test Point 1 

flow conditions.

simulated mass flow rate matched the measured value to within 

1 percent. The simulation was then compared to the experiment 

on the basis of: required vs. actual ε (0.079), thrust, detonation 

speed (where available) and CTAP distribution.

4.1 Test Point 1: Stoichiometric (φ = 1.0)

The converged flow field solution for Test Point 1, using the 

original smooth fit to the RDE profile of Figure 3, is shown in 

Figure 4. The same temperature contour format used for 

Figure 1 is used here. Also shown in the figure are several 

streamlines (in white) indicating the path of select particles 

from where they enter the domain until they reach the right hand 

side (in the detonation reference frame). All of the characteristic 

features of Figure 1 are seen in Figure 4. Additionally, the 

presence of a throat near the exit causes the majority of the 

upstream flow to be subsonic such that the oblique shock 

generated by the detonation is reflected from the throat and 

travels back upstream. The reflection is just visible and labelled 

in Figure 4. The required value of ε to match the mass flow rate 

was just 5 percent above the actual value. The value of udet

required to achieve a steady solution was 14 percent below the 

theoretical Chapman-Jouguet (CJ) velocity for a classical 1-D 

detonation. This compares favorably with measured values 

which are often 20 percent below the CJ value as reported in 

Reference 14. The computed gross thrust was only 1 percent 

below the measured value.
NASA/TM—2015-218835 7

Figure 5 shows the computed axial distribution of timeaveraged dimensional pressure (recall that a circumferential space 

average is identical to a time-average in this frame of reference) 

and compares it to experimental measurements made using 

CTAP’s. The match is remarkably good with the exceptions of 

points A, and to lesser degrees, points B & C. The point A 

disparity may be due to a mixing delay which causes a 

downstream axial shift of the detonation. This will be discussed 

later. Also shown in the figure is the time-averaged pressure 

distribution computed using the alternate, separation-induced area 

profile of Figure 3. Temperature contours for this solution are not 

shown as they are indistinguishable from Figure 4. Other 

computed results were also nearly the same: ε was 4 percent above 

actual; thrust was 2 percent below measured. However, it is seen 

that the match to the CTAP data in the vicinity of the proposed 

separation region (near B &C) is much better. While this result 

does not prove that separation is occurring, it does lend credence 

to the possibility. As such, for the remainder of the paper, the 

alternate profile will be utilized in all simulation results.

Of the two test points examined, this was the only one to 

computationally exhibit the inlet backflow scenario described

earlier. This is shown in Figure 6 where the inlet mass flux is 

plotted as a function of circumferential distance. With reference 

to Figure 4, it can be seen that the backflow occurs just behind 

the detonation front. The total mass that flows backward is only 

0.6 percent of the total inflow, so the impact is minimal. For 

other RDE experiments with lower manifold pressures, this 

may not be the case.

4.2 Test Point 2: Lean (φ = 0.42) 

As discussed earlier, the measurement of fuel and air flows 

(and thereby equivalence ratio, φ) in the experiment is done on 

a time-averaged basis. There are no diagnostics by which to 

ascertain the dynamic behavior through the injectors. As such, 

there is no way to know if the low equivalence ratio of this test 

point is achieved via the flow of a mixture at a uniform φ=0.42, 

or whether it may be achieved through pure air flowing over 

part of the cycle (i.e., purge flow) and the rest being a mixture 

closer to φ = 1.0.

What is known is that the manifold pressure feeding the fuel 

is approximately 1/4 of that feeding the air. This suggests that 

as the high-pressure detonation passes a given point, the fuel 

flow is more likely to be interrupted (and for a longer fraction 

of the cycle) than the air. As such, the purge scenario just 

described is suspected to be active. This is bolstered by 

considering the known difficulty of sustaining detonations of 

lean hydrogen/air mixtures in devices at the scale of this 

experiment. Nevertheless, both the uniform φ and purge 

scenarios are considered below. For the purge scenario, it is

Figure 5.—Computed time-averaged, and 

measured CTAP pressures along the axis 

of the experimental RDE using the original 

and alternate area profiles.

Figure 6.—Inlet mass flux profile as a function of 

circumferential distance for Test Point 1.

assumed that whatever reactive mixture enters the device does 

so at φ = 1.0. The remaining fluid entering is pure air (φ = 0.0). 

The split between the two is adjusted by trial and error until the 

cycle averaged φ is equal to 0.42.

Figure 7 and Figure 8 show temperature contours for the two 

scenarios in a manner similar to Figure 1. Note that there are 

two detonation waves within the annular channel. The number 

of waves present cannot be predicted by this simulation since it 

is determined by physical details too fine to be captured under 

the present approach. As such, the number of waves is 

prescribed based on experimental observations. In Figure 8, the 

prescribed distribution of φ is shown at the bottom. The 

comparison with experiment of required inlet area and thrust for 

0 50 100 150

0

0.2

0.4

0.6

0.8

1

1.2

Pressure, psia

y

computed-orig.

computed-alt.

CTAP

A

B

C

D E

F

G

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

1

1.5

2

x

mass flux

backflow

Mass flux rate
NASA/TM—2015-218835 8

each case is summarized in Table 2. As with Point 1, either 

scenario matches the experiment remarkably well.

The Test Point 2 pressure distributions for the two flow 

scenarios are shown in Figure 9. Again, the match with 

experiment is quite good. Discerning which of the two proposed 

flow scenarios is actually occurring in the experiment is not 

obvious given the data at hand. Additional instrumentation in 

subsequent experiments (e.g., high frequency ion gages, 

emission measurements) may provide answers.

4.3 Mixing Delay

The computational model assumes premixed fuel and air 

enters the RDE through the inlet, and allows a reaction to 

commence as soon as the fluid temperature exceeds a threshold 

value, Tc0 (see Eq. (5)). As described earlier, the experimental 

fuel and air are injected separately and are likely to require 

some finite time (and associated convection distance) to mix 

before they will react. In order to cursorily examine the effect 

of this reaction delay, the simulation was modified such that a 

user specified number of computational cell rows near the inlet 

would not react, even if the threshold temperature was reached.

TABLE 2.—TEST POINT 2 COMPARISON

Scenario εrequired/εactual – 1,

%

Fcomputed/Factual – 1,

%

Uniform φ –4 0

Purge –3 –5

Figure 7.—Computed contours of temperature 

throughout the annulus of the experimental RDE, at 

the Point 2 flow conditions with uniform φ.

Figure 8.—Computed contours of temperature 

throughout the annulus of the experimental RDE, at 

the Point 2 flow conditions with purge.

Figure 9.—Computed time-averaged, 

and measured CTAP pressures 

along the axis of the experimental 

RDE using the uniform φ and purge 

scenarios.

0 50 100 150

0

0.2

0.4

0.6

0.8

1

1.2

Pressure, psia

y

computed-uniform

computed-purge

CTAP

A

B

C

D

E

F

G
NASA/TM—2015-218835 9

The results of applying this modification to the first 6 cell 

rows (i.e., 0 < y < 0.06) for the Test Point 1 case are shown in 

Figure 10 and Figure 11 (similar to Figure 4 and Figure 5). As 

with other simulations, the value of ε required to match the mass 

flow rate was just 1 percent below the actual value. The 

computed gross thrust was only 1 percent below the measured 

value.

Comparing Figure 10 to Figure 4 it is seen that the detonation 

is shifted in the downstream direction. This results in a much 

better match between computed pressure and CTAP 

measurement near point A of Figure 11. As with other 

simplified sub-models presented in this paper, this result does 

not prove a mixing delay, but it does give it some measure of 

likelihood. It is also interesting to note (though it is not shown) 

that the shifting of the detonation reduces the backpressure on 

the inlet behind the detonation. This, in turn results in no 

predicted inlet backflow. No sensors were available in the 

experiment to validate this prediction.

5.0 Discussion

Favorable comparison between the present simulation and 

two test points from just one experiment does not constitute 

sufficient validation. More comparisons with varied 

experiments are necessary, and modification (as well as 

rejection) of the sub-models is inevitable. However, the results 

presented do suggest value in utilizing the code to examine the 

effect of various fluidic or geometric aspects on engine 

performance. This is something that is exceedingly difficult to 

do experimentally. For example, interrogating the Test Point 1 

simulation reveals that 27 percent of the chemical energy is 

absorbed by the cold walls. The code can be run adiabatically 

(and inviscidly) to show that the heat transfer reduces gross 

thrust by 16 percent.

As a second example, the impact of deflagrated vs. detonated 

flow can be examined. It was mentioned earlier in the paper that 

the deflagration reaction rate parameter is variable, and that all 

of the simulations presented deflagrated approximately 

8 percent of the incoming mixture. With the Test Point 1 

simulation the reaction rate was adjusted so that approximately 

17 percent of the mixture deflagrated. Doing so resulted in a 

4 percent reduction in thrust.

Many other parametric variations could be considered with 

this simplified simulation approach. Doing so provides insights 

into both the physics of RDE operation, and possibilities for 

optimizing performance.

Figure 10.—Computed contours of temperature throughout the 

annulus of the experimental RDE at the Test Point 1 flow 

conditions with reaction delay.

Figure 11.—Computed time-averaged, and 

measured CTAP pressures along the axis 

of the experimental RDE using reaction 

delay.

0 50 100 150

0

0.2

0.4

0.6

0.8

1

1.2

Pressure, psia

y

computed-delay

CTAP

A

B

C

D E

F

G
NASA/TM—2015-218835 10

6.0 Conclusion

A quasi-two-dimensional, computational fluid dynamic 

(CFD) simulation of a rotating detonation engine (RDE) has 

been described. The computational approach is one of 

simplification, utilizing a relatively coarse grid, a single 

progress variable reaction mechanism, and operation in the 

detonation frame of reference. Real world effects such as those 

due to viscosity and heat transfer are added through the use of 

validated sub-models. The result is a rapidly converging 

simulation which provides steady-state solutions that capture

the essential physics of an RDE. An experimental RDE was 

simulated and the output was compared with measurements at 

two operating points in terms of mass flow rate, thrust 

produced, and time-averaged pressure distributions. The 

comparison was shown to be favorable at both operating points.

The utility of the code as a performance optimization tool and 

a diagnostic tool was illustrated by two example parametric 

variations. To the authors’ knowledge, this represents the first 

publicly available direct comparison of simulated and measured 

RDE performance. Additional validation of the simulation is 

needed and planned.

References

1. Dyer, R., Naples, A., Kaemming, T., Hoke, J., Schauer, F., 

“Parametric Testing of a Unique Rotating Detonation 

Engine Design,” AIAA 2012-121, Jan. 2012.

2. Naples, Hoke, J., Karnesky, J., Schauer, F., “Flowfield 

Characterization of a Rotating Detonation Engine,” AIAA 

2013-0278, Jan. 2013.

3. Sonwane, C., Claflin, S., Lynch, E., Stout, J., “Recent 

Advances in Power Cycles Using Rotating Detonation 

Engines with Subcritical and Supercritical CO2,” SWRI 4th 

International Symposium - Supercritical CO2 Power Cycles, 

Pittsburgh, PA, Sep. 9-10, 2014.

4. Kato, Y., et al., “Thrust Measurement of Rotating Detonation 

Engine by Sled Test,” AIAA 2014-4034, Jul. 2014.

5. Welsh, D.J., King, P.I., DeBarmore, N.D., Schauer, F.R., 

Hoke, J.L., “RDE Integration with T63 Turboshaft Engine 

Components, AIAA 2014-1316, Jan. 2014.

6. Paxson, D.E., “Numerical Analysis of a Rotating Detonation 

Engine in the Relative Reference Frame,” AIAA-2014-

0284, Jan. 2014, also NASA/TM—2014-216634, 2014.

7. Towery, C.A., et al., “Examination of Turbulent Flow Effects 

in Rotating Detonation Engines,” AIAA 2014-3031, June

2014.

8. Schwer, D., Kailasanath, K., “Fluid Dynamics of Rotating 

Detonation Engines with Hydrogen and Hydrocarbon 

Fuels,” Proceedings of the Combustion Institute, Vol. 34, 

No. 2, 2013, pp. 1991–1998.

9. Kim, T., Choi, J. “Numerical Study of Detonation Wave 

Propagation in 2-D Channels of Arbitrary Radius of 

Curvature,” AIAA 2014-3903, Jul. 2014.

10. Paxson, D.E., “A General Numerical Model for Wave Rotor 

Analysis,” NASA TM 105740, 1992.

11. Paxson, D.E., “An Improved Numerical Model for Wave 

Rotor Design and Analysis,” AIAA-93-0482, Jan. 1993.

12. Perkins, H.D., et al., “An Assessment of Pulse Detonation 

Engine Performance Estimation Methods Based On 

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

13. Paxson, D.E., Schauer, F.R., Hopper, D., “Performance 

Impact of Deflagration to Detonation Transition Enhancing 

Obstacles,” AIAA-2009-0502, Jul. 2009, also, NASA/TM—

2012-217629.

14. Rankin, B.A., Hoke, J.L., Schauer, F. R., “Periodic Exhaust 

Flow through a Converging-Diverging Nozzle Downstream 

of a Rotating Detonation Engine,” AIAA 2014-1015, Jan.

2014.

15. White, F.M., Fluid Mechanics, McGraw-Hill, 1979, p. 381.

16. Paxson, D.E., Hoke, J.L., “Time Averaged Pressure Measurement in Fundamentally Unsteady Pressure Gain Combustion 

Systems,” in Proc. JANNAF 45th Combustion Subcommittee, 33rd Airbreathing Propulsion Subcommittee, 27th 

Propulsion Systems Hazards Subcommittee, Monterey, CA, 

December, 2012, also NASA/TM—2013-217826, Jan. 2013.