Nicholas J. Georgiadis, Dennis A. Yoder, and Manan A. Vyas

Glenn Research Center, Cleveland, Ohio

William A. Engblom

Embry Riddle Aeronautical University, Dayton Beach, Florida

Status of Turbulence Modeling for Hypersonic

Propulsion Flowpaths

NASA/TM—2012-217277

April 2012

AIAA–2011–5917
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Nicholas J. Georgiadis, Dennis A. Yoder, and Manan A. Vyas

Glenn Research Center, Cleveland, Ohio

William A. Engblom

Embry Riddle Aeronautical University, Dayton Beach, Florida

Status of Turbulence Modeling for Hypersonic

Propulsion Flowpaths

NASA/TM—2012-217277

April 2012

AIAA–2011–5917

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

47th Joint Propulsion Conference and Exhibit

cosponsored by the AIAA, ASME, SAE, and ASEE

San Diego, California, July 31–August 3, 2011
Acknowledgments

The authors would like to thank the Test Resource Management Center (TRMC) and the Test and Evaluation/Science and 

Technology (T&E/S&T) Program for their support. This work was sponsored by the TRMC T&E/S&T Program through the High 

Speed Systems Test (HSST) area (formerly APTT) and the NASA Fundamental Aeronautics Program. 

Available from

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Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by technical management. 
This report provides an assessment of current turbulent flow calculation methods for

hypersonic propulsion flowpaths, particularly the scramjet engine. Emphasis is placed on

Reynolds-averaged Navier-Stokes (RANS) methods, but some discussion of newer methods such as Large Eddy Simulation (LES) is also provided. The report is organized by

considering technical issues throughout the scramjet-powered vehicle flowpath including

laminar-to-turbulent boundary layer transition, shock wave / turbulent boundary layer

interactions, scalar transport modeling (specifically the significance of turbulent Prandtl

and Schmidt numbers) and compressible mixing. Unit problems are primarily used to

conduct the assessment. In the combustor, results from calculations of a direct connect

supersonic combustion experiment are also used to address the effects of turbulence model

selection and in particular settings for the turbulent Prandtl and Schmidt numbers. It is

concluded that RANS turbulence modeling shortfalls are still a major limitation to the

accuracy of hypersonic propulsion simulations, whether considering individual components

or an overall system. Newer methods such as LES-based techniques may be promising, but

are not yet at a maturity to be used routinely by the hypersonic propulsion community.

The need for fundamental experiments to provide data for turbulence model development

and validation is discussed.

Nomenclature

Cf skin friction coefficient

Cp specific heat at constant pressure

Dt turbulent species diffusivity

h static enthalpy

H fuel injector ramp height for scramjet test case

k turbulent kinetic energy

kt turbulent thermal conductivity

M Mach number

Mt turbulent Mach number

P static pressure

Pk production of turbulent kinetic energy

q ̇w rate of wall heat flux

P rt turbulent Prandtl number

Ry turbulent Reynolds number based on wall distance = ρy√k/μ

Rt turbulent Reynolds number = ρk/μω

NASA/TM—2012-217277 1

Status of Turbulence Modeling for Hypersonic 

Propulsion Flowpaths 

Nicholas J. Georgiadis, Dennis A. Yoder, and Manan A. Vyas 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

William A. Engblom 

Embry Riddle Aeronautical University 

Dayton Beach, Florida 32114 
Rex plate Reynolds number

Reν vorticity-based Reynolds number = ρy2Ω/μ

s streamline coordinate

Sij rate of strain tensor

Sct turbulent Schmidt number

St Stanton number

t time

T static temperature

Tr temperature ratio

Tt stagnation temperature

Tw wall static temperature

Uj velocity vector

U∞ freestream velocity

Wij vorticity tensor

x, y, z Cartesian coordinates

y

+ wall normal coordinate

 turbulent dissipation rate

s solenoidal turbulent dissipation rate

μ dynamic viscosity

μt dynamic eddy viscosity

ω specific turbulent dissipation rate = /k

Ω vorticity magnitude

φ fuel equivalence ratio

ρ density

ρ∞ freestream density

τ

T

ij turbulent stress tensor

Introduction

Flowpaths within hypersonic propulsion systems, in particular the scramjet engine, are characterized by

several aerodynamic and thermodynamic features which are very difficult for currently available computational fluid dynamics (CFD) methods to calculate accurately. At the same time, experiments cannot provide

all of the details of scramjet flows because of the difficulties in simulating actual operating conditions in

ground experiments and making appropriate measurements in flight tests. As a result, the need for competent CFD methods to be used in conjunction with experiments is high, in order to successfully carry out

hypersonic research and development programs, as outlined in detail by Drummond et al1 and Tishkoff et

al.2 One CFD modeling area that has remained extremely challenging, yet very critical to the success of

hypersonic propulsion system simulations is turbulence modeling, which is the focus of this paper.

Scramjet powered hypersonic vehicles are among the most integrated vehicle / propulsion system configurations of all aircraft. Nevertheless, we may identify flow features specific to the primary propulsion system

components that provide the greatest turbulence modeling challenges, as identified in Fig. 1. For many

hypersonic vehicles, the entire forebody beginning with the leading edge forms part of the engine inlet. Here

boundary layer transition has a large effect on the thickness of the boundary layers that enter the enclosed

portion of the scramjet engine and also on the the thermal properties of the vehicle forebody. Knowledge

of the boundary layer state is important enough that programs such as HyperX / X-43 have implemented

forced transition locations via boundary layer trips.3, 4

Within the inlet and isolator, shockwave/boundary layer interactions severely complicate the flow, and

provide many challenges to turbulence modeling efforts. Within the combustor, the complexities are numerous. These include three-dimensional turbulent mixing in the presence of a chemically reacting flow,

recirculating flow at the locations of fuel injection, heat transfer at the combustor walls, compressibility effects, and continued shockwave / boundary layer interactions. Further downstream in the expansion system,

the flow may still be reacting and the complexities are similar to those at the beginning of the combustor

region.

The practical state-of-the-art in CFD for hypersonic propulsion flowpath analyses during the past two

NASA/TM—2012-217277 2
decades has been Reynolds-averaged Navier-Stokes (RANS) methods, typically using two-equation eddy

viscosity closures such as k-ω or k- models. In terms of more advanced RANS techniques, some limited

attempts have been made to use algebraic Reynolds stress models, but substantial improvement over the

linear two-equation eddy viscosity approaches has not been demonstrated. There have not been many

attempts to use full Reynolds-stress closures for scramjet analyses. The flow features described in the

preceding paragraph all provide major challenges to the most commonly used two-equation eddy viscosity

RANS techniques. Transition models coupled to RANS are highly empirical at best, and have only been

shown to have potential applicability at high freestream turbulence levels, which is not the case in actual

flight simulations. For hypersonic vehicle inlets, RANS methods have been shown to be quite deficient

for calculating the details of shockwave/boundary layer interactions. Isolator flowfields frequently contain

several shock boundary layer interactions. In ramjet mode, this separated flow enters the combustor creating

a complex flowfield that no RANS models can accurately compute.

The complexities occurring farther downstream in the combustor and expansion system, namely high

temperatures, compressibility effects, recirculating flow in the regions of fuel injection, and other threedimensional flow features, further overwhelm currently practical CFD capabilities. Baurle5 provides a comprehensive overview of turbulent reacting methods for supersonic combustion while Ebrahami6 addresses

validation challenges for high speed combustion problems from scramjets to rockets. Turbulent chemistry

interactions are obviously crucial in these regions, yet no RANS-based techniques have been shown to accurately represent the actual flow state. Constant values for the turbulent Prandlt and Schmidt number are

typically used to extend the RANS-based eddy viscosity used in the momentum equations to the turbulent

heat transport and species transport in the energy and individual species transport equations, respectively.

This is known to be a significant limitation,5, 7 and while some efforts have been underway in recent years to

develop more sophisticated RANS-based models for such scalar transport, not enough testing and calibration

has been performed to date to determine the potential of these techniques.

Beyond RANS-based methods, large eddy simulation (LES) techniques and hybrid RANS-LES techniques

have received significant attention over the past decade for application to aerodynamic flow analyses. While

progress has been made in developing and using such techniques, especially for flows away from walls,

LES-based techniques cannot be considered a mature science, free of modeling parameters that need to be

set by the analyst. Effects of grid sensitivity are fundamentally different from RANS-based techniques. In

RANS, it is the expectation that as one refines a computational grid, a grid-independent solution is achieved.

In contrast for LES, as the grid is refined, typically more small turbulent structures are resolved instead

of modeled, and the solution can indeed be expected to change. In the subsonic combustor community,

LES techniques have been shown to have some promise,8in particular for flows where wall effects can be

considered small, and where turbulent velocity scales are on the order of the mean flow velocity. In a scramjet

combustor, however, the situation is different, as the core flow velocity is still of appreciable magnitude, and

wall boundary layer effects are important. References 9 and 10 have demonstrated some of the earliest

application of LES-based methods to scramjet combustors. While LES-based methods have promise for

calculating the largest scales of turbulence, chemical reactions occur at the microscale, so the subgrid scale

modeling used in LES for scramjet simulations (or any combustor for that matter) is very important. Some

research into developing more accurate subgrid scale closures to handle turbulence /chemistry interactions

is underway,11 but such methods are far from being considered mature enough to be applicable to practical

hypersonic propulsion system analyses.

In this paper, the chief modeling challenges briefly outlined in this introduction are addressed. The

current practical state-of-the-art in RANS-based techniques is emphasized, utilizing some of our own recent

experiences. Our efforts utilized the Wind-US code12–14 which is a production RANS solver with several

available turbulence models. Discussion of more computationally intensive techniques such as LES, including

hybrid RANS-LES approaches, will be provided. Suggestions for the required research to result in improvements to these methods will be described. A discussion of experimental efforts that would aid modeling

efforts will also be provided.

RANS Turbulence Modeling

For nearly two decades, the practical state-of-the-art in RANS turbulence modeling has employed twoequation eddy viscosity turbulence models. In linear eddy viscosity models, the Boussinesq approximation is

used to relate the turbulent Reynolds stress to the mean rate of strain tensor through a turbulent (or eddy)

NASA/TM—2012-217277 3
viscosity:

τ

T

ij = μt



2Sij −

2

3

∂uk

∂xk

δij−

2

3

ρkδij (1)

The rate of strain is tensor, Sij , is given by:

Sij =

1

2



∂uj

∂xi

+

∂ui

∂xj



(2)

It is beyond the scope of this paper to describe all two-equation models that are in use today. A detailed

list of several RANS turbulence models that have been used in hypersonic simulations is presented in Ref.

15. It should be noted that the vast majority of the literature concerning validation and comparison of

RANS turbulence models has not been in the hypersonic regime, but at much slower speeds, particularly

subsonic and transonic. A majority of the two-equation turbulence models in use are either k −  or k − ω

formulations, which we focus on here. While not discussed here, a model that has received attention for high

speed flows in recent years in the k − ζ (enstrophy) model as described in Refs. 16–18. Here we describe

in some detail one turbulence model that is widely used, and has been utilized for the majority of our work

in recent years. This model is the Shear-Stress Transport (SST) two-equation turbulence formulation of

Menter.19 The SST model has been shown to be robust and relatively accurate for a broad range of flows,

including wall boundary layers and free shear layer regions. The SST model is a two-layer model which

employs the k − ω model of Wilcox20 in the inner region of boundary layers and switches to a k −  model in

the outer region of boundary layers and in mixing regions. The outer k− model is transformed to provide a

second set of k − ω equations with a blending function used to transition between the two sets of equations.

∂ρk

∂t +

∂ρUjk

∂xj

= Pk − β

∗

ρωk +

∂

∂xj



(μ + σkμt)

∂k

∂xj



(3)

∂ρω

∂t +

∂ρUjω

∂xj

=

α

νt

Pk − βρω2 +

∂

∂xj



(μ + σωμt)

∂ω

∂xj



+ (1 − F1)2ρσω2

1

ω

∂k

∂xj

∂ω

∂xj

(4)

Pk = min 2μtSijSij −

2

3

(μtSmm − ρk)Snn ; μtΩ

2



(5)

F

∗

1 = tanh(arg4

1

) (6)

F4 = exp −(Ry/120)8



(7)

F1 = max (F

∗

1

, F4) (8)

arg1 = min max √

k

β

∗ωy

;

500ν

ωy2

!

;

4ρσω2k

CDkωy

2

!

(9)

CDkω = max 2ρσω2

1

ω

∂k

∂xj

∂ω

∂xj

; 1.0 × 10−20(10)

and from these the turbulent viscosity is given as

μt = min α

∗

ρk

ω

;

a1ρk

ΩF2



(11)

F2 = tanh(arg2

2

) (12)

arg2 = min 2

√

k

β

∗ωy

;

500ν

ωy2

!

(13)

The constants (φ1) associated with these equations for the inner, k − ω, model are:

σk1 = 0.85, β∗

1 = 0.09 ·

5/18 + (Rt/8)4

1 + (Rt/8)4

, α∗

1 =

.025 + Rt/6

1 + Rt/6

,

σω1 = 0.5, β1 = 0.075, α1 =

5

9

·

0.1 + Rt/2.7

1 + Rt/2.7

, a1 = 0.31

NASA/TM—2012-217277 4
and the constants (φ2) for the transformed k −  model are:

σk2 = 1.0, β

∗

2 = 0.09, α

∗

2 = 1,

σω2 = 0.856, β2 = 0.0828, α2 = 0.4403, a1 = 0.31

These constants are blended using the same switching function, F1 that is found in the model equations such

that φ = F1φ1 + (1 − F1)φ2 for any of the given parameters.

The SST turbulence model has become popular for its ability to handle separated flows and complex

geometry in the near wall region due to the strengths of the k−ω model, while maintaining the characteristics

of the k −  model to be more accurate in free shear layers.

In eddy viscosity models such as the SST formulation, the ultimate quantity that is used in the momentum

equations to model the turbulent stresses is the eddy viscosity, μt. In most RANS codes, a constant turbulent

Prandtl number is used to extend the eddy viscosity to model the turbulent thermal transport via the relation:

P rt = μtCp/kt (14)

The turbulent species diffusivity is extended from the eddy viscosity in a similar manner via the turbulent

Schmidt number:

Sct = μtCp/Dt (15)

In most applications at low Mach numbers and without reactions or significant heat transfer effects, the

air is treated as a calorically perfect gas; a constant turbulent Prandtl number on the order of unity is

typically used and no species transport equations are solved. However, the turbulent Prandtl and Schmidt

numbers have significant effects for supersonic combustion problems as will be discussed in a later section.

Beyond two-equation models, full Reynolds-stress closures, or second moment closures as they are sometimes referred to, offer a more complete representation of the three-dimensional turbulent stress field. However, the improvements in accuracy over linear two-equation models have not been shown to be sufficiently

high enough to justify the additional computational cost.21 As a result, the international CFD community

has not broadly accepted the usage of full Reynolds stress closures, even with the known limitations of linear

eddy-viscosity models. As a compromise beyond two-equation viscosity models, there has been some work

to develop and investigate non-linear explicit algebraic stress models (EASMs). Unlike linear two-equation

models, EASM formulations are sensitive to turbulent stress anisotropies and have a direct relation to the

full Reynolds stress model.22 As a result, EASM models have the capability to include more relevant flow

physics than the linear models. However, they are also solved using a two-equation approach and as a result

are not significantly more computationally expensive than linear two-equation models.

The turbulent stress tensor of the EASM (for both k −  and k − ω) is:

τ

T

ij = 2μt{Sij −

1

3

Skkδij + [a2a4 (SikWkj − WikSkj )

−2a3a4



SikSkj −

1

3

SklSlkδij 

]} − 2

3

ρkδij

(16)

where Sij was defined previously in Eq. 2 and Wij is:

Wij =

1

2



∂ui

∂xj

−

∂uj

∂xi



(17)

The eddy viscosity, μt is:

μt = C

∗

μρkτ = −ρkα1 (18)

where the turbulent time scale is τ = 1/ω = k/. The quantity α1/τ is equivalent to −C

∗

μ and is obtained

from the solution to a cubic equation at every point in the flow field. The closure constants, a2, a3, a4,

and entire solution procedure are described in Refs. 23 and 24. The form of the k − ω model used as the

underlying two-equation model for the EASM described here is provided in Ref. 25.

In the following sections, we consider the performance of RANS turbulence modeling for component

or subset problems that exhibit the turbulence modeling-dominated features found within the hypersonic

propulsion system flowpath as described in the introduction and shown schematically in Fig. 1. We begin

with a discussion of laminar-to-turbulent boundary transition. Consideration of shock wave / turbulent

NASA/TM—2012-217277 5
boundary layer interactions, which dominate the flow within the scramjet isolator but are pervasive to the

entire scramjet propulsion flowpath, are considered next. Moving toward the rear of the propulsion flowpath,

consideration of turbulence modeling factors in the combustor and exhaust system are presented in the last

section.

Forebody/Inlet Transition Modeling

Hypersonic flight vehicles frequently have boundary layers with significant laminar regions on vehicle

forebodies and inlet surfaces because of the low freestream disturbances and low densities characteristic of

the altitudes that such vehicles fly within. While flight tests can reproduce these atmospheric conditions,

scaling issues due to Reynolds number affect the ability to reproduce the effects of boundary layer state,

and in particular the point of transition. Ground test facilities frequently have freestream turbulence levels

that are high enough to affect the transition point. Whereas the mechanism of transition in free flight is

usually modal growth, the high freestream turbulence experienced by ground test articles frequently leads

to bypass transition, in which the transition to turbulent flow is dictated by large freestream disturbances.

In this section, we consider a RANS-based bypass transition model. It is generally agreed that RANS-based

techniques are not applicable to modal growth situations.

Details of the formulation presented here are provided in Ref. 26. The model is based on the SST turbulence model and was built starting from a previous SST-based transition model.27–29 Several modifications

were made to enable: (1) consistent solutions regardless of flow field initialization procedure and (2) fully

turbulent flow beyond the transition region. In the following, we highlight only the key aspects of the implementation of the bypass transition model. Building upon the SST turbulence model, the key modification

is made to the production term in the turbulent kinetic energy equation:

∂ρk

∂t +

∂ρUjk

∂xj

= P TM · Pk − β

∗

ρωk +

∂

∂xj



(μ + σkμt)

∂k

∂xj



(19)

where P TM is termed the “production term multiplier.” The ω equation is not modified.

The final form of the transition model that is recommended based on the work of Ref. 26 is:

P TM = 1 − 0.94(P TM1 + P TM2) F3 tanh (y

+/17)2



(20)

F3 = e

−(

Rt

3 )

2

(1 − P(Rt)) + 1

2

P(Rt) (21)

P(Rt) = 2.5

√

2π

e

−(Rt−3)2

2 (22)

P TM1 = 1 − CP TM1









(3.28 × 10−4)Rev − (3.94 × 10−7)Re2

v + (1.43 × 10−10)Re3v



; Rev < 1000



0.12 + (1.00 × 10−5Rev



; Rev > 1000

(23)

P TM2 =







− |K|

0.4 Rev

80 ; K < 0

0 ; K > 0

(24)

1.0 ≤ CP TM1 ≤ 2.0 (Recommended range) (25)

where Rev is the vorticity-based Reynolds number, Rt is the turbulent Reynolds number, and the pressure

gradient parameter, K, is given by

K = −

μ

ρ

2U3



1 − M2

 dp

ds (26)

In Refs. 28 and 29, P TM2 was formulated for flows with significant internal flow pressure gradients,

and specifically for flows within low pressure turbine stages. In all cases examined in this work and that

of Ref. 26, where flows with significant pressure gradients were not examined, P TM2 was not found to be

significant.

This model is intended for flows where bypass transition is the key transition mechanism as opposed to

transition dictated by modal growth. Validation of the new transition model was performed for flows ranging

from incompressible to hypersonic conditions.

NASA/TM—2012-217277 6
Figure 2 shows a comparison of simulations for a widely used data set for incompressible transition flow

over a flat plate(referred to as the T3A test case30). The fully turbulent SST solution shows the rapid

transition (much closer to an entirely turbulent flow) that provided the motivation for the present work. It

does a poor job of accurately capturing the early laminar behavior and the transition location which would

lead to incorrect results for the drag on this flat plate. For the transition model solutions, the effect of the

key parameter, CP TM1, is to control the transition onset location. As with nearly all RANS-based transition

prediction schemes, the width of the calculated transition region is shorter than indicated by experimental

data. The solution with CP TM1 = 2.0 captures the transition onset location best while decreasing CP TM1

to 1.0 (the default value from original formulation) captures the end of the transition zone (i.e. where fully

turbulent flow is realized) best.

Hypersonic transitional flows present unique challenges in both modeling and experimentation. To baseline the present model’s ability to predict transitional behavior in hypersonic conditions, it was validated

against transition data taken on sharp nose cones in the AEDC tunnel B at Mach 7.93.31 The simulation is

performed on a 7◦ half-angle cone. As in the experiment, the wall temperature was set at 0.42Tt, where Tt

is the freestream stagnation temperature. Several unit Reynolds numbers (Re/m) were evaluated in Ref. 31

to provide a complete scan of the transition region. These data show good agreement for transition location

across a range of Re/m, allowing the simulation to use only one, ≈ 6.8E6 Re/m, corresponding closely with

the center of the experimental range. Inflow conditions are calculated to match those of Ref. 31. Insight

from Ref. 32 indicated that an inlet freestream turbulence intensity of 1.25% was appropriate for AEDC

tunnel B.

Figure 3 shows the Stanton number (St), as defined by Ref. 31 and as shown in Eq. 27, versus Reynolds

number for the SST model alone, the SST transition model using CP TM1 = 1.0 and CP TM1 = 2.0, and the

experimental data.

St =  ̇ qw/(ρ∞U∞(h(Tt) − h(Tw))) (27)

As in the incompressible case, the predicted behavior is overly abrupt. However, the model accurately

captures the minimum heat transfer value, corresponding to transition onset for CP TM1 = 2.0 and the

location where the flow becomes fully turbulent for CP TM1 = 1.0.

The transition model solutions are a marked improvement over the SST model alone, which vastly overpredicts the heat transfer by indicating a fully turbulent state from the leading edge of the test article. For a

model in a high-speed wind tunnel that has a significant laminar region, the error due to transition onset can

be quite large. The total heat transfer integrated over the length of the cone differs by 38.7% between the

standard (fully turbulent) SST model and the SST-based transition model with CP TM1 = 1.0. Accurately

capturing this behavior is especially important when trying to evaluate thermal properties and heat transfer

behavior near the tip of a hypersonic vehicle. The large discrepancy in heat transfer rates indicated by the

two models would substantially alter the vehicle’s predicted temperature profile.

While the transition model presented here demonstrates some promise for bypass transition, it is emphasized that a general RANS-based predictive technique for the modal growth situation that is more important

in free flight has not been demonstrated nor is likely feasible. LES is likely not a candidate either, because

the transition process is dominated by amplification of initially very small disturbances, which is in contrast

to the LES philosophy of calculating the largest turbulent scales while modeling the smallest scales. In

the long run, direct numerical simulation (DNS) probably offers the best prospects for such a predictive

capability. In the interim, techniques such as e

N methods, in which stability theory is used to predict the

location of transition,33–35 will remain the realistic option.

Shock Wave Turbulent Boundary Layer Interactions

The shock wave / turbulent boundary layer interaction (SWTBLI) is a very common phenomena that

has significant effects throughout the hypersonic propulsion system flowpath. The problem has been studied

extensively over the past few decades in laboratory experiments and in computational fluid dynamics (CFD)

efforts. While gains in understanding of the underlying fluid dynamics of SWTBLI have been made, research

continues into understanding the complex interaction of a shock wave with a turbulent boundary layer. In

a typical SWTBLI, the adverse pressure gradient induced by the shock system causes a flow separation that

frequently is unsteady and three-dimensional. Without question, control of SWTBLI has yet to be mastered.

While there have been numerous CFD studies, there is not a single approach that has been identified as

optimal for calculation of SWTBLI.

NASA/TM—2012-217277 7
For practical aerodynamic analyses that contain one or more SWTBLIs as is the case for a scramjet

isolator, RANS is still the only feasible approach. More recently, large eddy simulation (LES) and hybrid

RANS-LES techniques have been investigated for application to the SWTBLI problem, but typically have

been restricted to small unit problems, as applying an LES-based technique to a more complex system (i.e.

aircraft inlet) involving SWTBLIs is still largely prohibitive due to the range of scales involved. Knight and

Degrez36 and Knight et al37 provide comprehensive overviews of a broad range in CFD methods as applied

to SWTBLIs, in particular those investigated under AGARD and RTO working groups. Consideration of

both RANS and LES methods is made. Reference 38 focuses on an assessment of RANS-based methods

while a survey of LES-based approaches as applied to SWTBLIs is presented in Edwards.39 The overall conclusions of these survey papers is that RANS methods are inherently unable to calculate some of the crucial

features of SWTBLI, in particular the unsteadiness of the shock system and separated flow. In addition, the

three dimensional features are also troublesome for RANS-based techniques. LES-based techniques may be

promising, but neither sufficient maturity of these techniques nor experience using them has been realized.

In an inlet or isolator the shocks closest to the leading edge of the vehicle have the highest approach Mach

number. Further downstream, the shocks may occur at lower approaching Mach number, but the boundary

layers are typically thicker and may have already been subject to effects of upstream SWTBLIs. Such is the

case in scramjet isolators.

We applied several two-equation models including eddy-viscosity models and EASMs to a Mach 5 SWBLI

problem in the current work. Mach number contours for an SST Wind-US solution are provided in Fig. 4.

Comparisons of several solutions using linear two-equation turbulence models and EASMs with experimental

data from Ref. 40 are provided in Fig. 5. It may be observed that the linear k −  model solution provides

poor agreement with experimental data. While the k −  based EASMs, with and without compressibility

corrections, marginally improve upon the linear k −  solution, the largest improvement is obtained by using

the k − ω formulations, including the Menter SST model. The k − ω EASMs provide improvement just past

the shock impingement point. The solution obtained with the SST model also provides reasonable agreement

in the post-shock region, with somewhat lower calculated skin friction than the experimental data and the

other k − ω models (linear and EASM).

A recent workshop considering CFD calculations for a set of SWTBLI cases was held in conjunction

with the 48th American Institute of Aeronautics and Astronautics (AIAA) Aerospace Sciences Meeting. An

overview of the workshop objectives is presented in Benek41 with a summary of overall findings presented in

Benek.42 Several investigators contributed solutions to this workshop. Contributions included RANS and

LES-based computations. DeBonis et al43 provides a comprehensive assessment of the CFD calculations,

including uncertainty analysis of the submitted CFD results and experimental data obtained for the same

configurations. Hirsch44 examined some of the CFD trends, in particular effects of turbulence modeling,

RANS versus LES, and numerical schemes submitted by several investigators and includes the results from

our efforts described here.

One of two SWTBLI configurations considered in the workshop was a test case obtained at the Institut

Universitaire des Systemes Thermiques Industriels (IUSTI) in Marseille, France.45 The experimental data

was utilized in the European Union SWTBLI research project referred to as UFAST.46 The UFAST experiments utilized an 8 degree shock generator which spanned the entire width of the tunnel with an approaching

Mach 2.25 flow. The supply stagnation pressure was 50.5 kPa and the stagnation temperature was 293 K. A

schematic of the experimental configuration representing that used in the UFAST experiments is shown in

Fig. 6 as taken from DeBonis et al.43 Particle Image Velocimetry (PIV) was used in all cases to characterize

the interaction region, with both mean flow velocities and turbulent statistics obtained.

For this test case, we utilized the Menter SST k-ω and k-ω based EASM two-equation models and the

one-equation model due to Spalart and Allmaras.47, 48 Axial velocity contours obtained for the UFAST test

case are shown in Fig. 7. The oblique wave originating from the sharp leading edge of the shock generator

may be observed, along with the SWTBLI region centered about x = 320mm. The complex interaction

on the top of the tunnel actually results in a flow separation that is much larger than that in the focused

SWTBLI region on the bottom.

The experimental data was collected within the region focused around the bottom wall interaction. Axial

velocity contours for the UFAST test case using the three turbulence modeling approaches are compared to

the experimental PIV measurements in Fig. 8. The CFD solution slices represent the same physical domain,

with the same contour levels as the experimental data. The CFD solutions were interpolated to the same

physical locations where the PIV data were taken, as was required by the organizers of the AIAA workshop,

NASA/TM—2012-217277 8
for purposes of computing differences between solutions and experimental data.43 Examining these contours,

the extent of the adverse pressure gradient effects indicated by the Menter SST solution are larger than the

other solutions and experimental data. Further comparisons of the CFD solutions and experimental data

are made for the axial velocity profiles at four axial locations in Fig. 9. One may observe that the CFD

solutions plotted do not go all of the way to the wall, and again this is due to interpolation of the CFD

results to the PIV measurement locations.

The comparisons in Fig. 9 show the same trends as indicated by the contour plots of Fig. 8. In

particular, the size of the flow separation indicated by the SST solution is greater than that indicated by the

experimental data or the other turbulence models. It appears that the SA and ASM models do not predict

as large a SWTBLI region as the experimental data. These results are very similar to those reported for SA

and SST by Bhagwandin and DeSpirito.49 They also considered a k- model which produced solutions that

highly underestimated the effect of the SWBLI. This is generally the expected performance of k- models in

separated or adverse pressure gradient flows.21

Combustor / Exhaust System Modeling

For several years, the University of Virginia (UVA) has conducted research on a benchmark scramjet

combustor, shown schematically in Fig. 10, to investigate dual-mode ramjet/scramjet engine behavior.50–52

This configuration consists of a two-dimensional convergent-divergent nozzle providing a Mach 2 flow to a

rectangular isolator. An electric heater is used to provide clean air at approximately 1200 K to the test article.

To examine test media effects, water vapor and carbon dioxide delivery systems may be used to replicate

the vitiated air levels typical of combustion heated facilities simulating Mach 5 flight. For reference, the

key stations through the UVA configuration are listed in table 1. Downstream of the isolator, an unswept

ramp contains a single round fuel injector which provides unheated hydrogen to a combustor section. The

burned flow then enters a divergent nozzle and exhausts to the freestream. The fuel injector geometry is a

convergent-divergent nozzle. We have applied RANS based calculations to this configuration,12, 53, 54 with

much of the focus on determining the capability of CFD to replicate differences between clean air and vitiated

air on engine performance as a function of fuel equivalence ratio. Gupte et al55 used finite element analysis to

examine the thermal-structural response of the UVA test article and found substantial effects of the thermal

deformation on the predicted flowfield when comparing CFD solutions from the baseline to the thermally

deformed configuration.

Station x/H

Isolator Entrance -45

Fuel Injection Plane 0

Nozzle Exit 57

Table 1. Key positions for UVA scramjet configuration

Most recently, the results of Vyas et al54 demonstrated the ability to qualitatively reproduce the equivalence ratios where the combustor operation switched from ramjet to scramjet mode in the experiments of

Rockwell et al.52 In all of these experiments, the equivalence ratios were relatively low, on the order of 0.5

or less, in comparison to the typical equivalence ratios on the order of unity that a scramjet intended for

actual flight will experience. Variants of the baseline UVA scramjet combustor flow have been developed, as

discussed in Goyne et al,56 to enable higher equivalence ratios in anticipation of planned flight tests.

A major difficulty in drawing solid conclusions of CFD-to-experiment comparisons of scramjet predictions

is that frequently the only quantity available for comparison is surface pressure measurements as a function

of axial position in the scramjet flowpath. In CFD, several factors can contribute to significant changes in

the pressure distribution, i.e. turbulence model selection, settings for the turbulent Prandtl and Schmidt

numbers, chemical kinetics choice, wall temperature and/or heat transfer modeling. Typically an analyst will

make a selection for all of these model settings, and then vary one of the parameters, such as the turbulent

Schmidt number, to optimize the CFD generated pressure distribution until it passes most closely through

the experimental pressure distribution. While this is the current “state-of-practice,” optimizing such settings

for one scramjet configuration at one operating point does not guarantee the same settings will be optimal

NASA/TM—2012-217277 9
at other operating points, say with different freestream inflow conditions and fuel equivalence ratios. Over

a period of several years, the efforts discussed in Refs. 12, 53, 54 considered effects of turbulence model

selection, settings for the turbulent Prandtl and Schmidt numbers, and chemical kinetics modeling. We also

worked closely with the experimentalists, who made significant efforts to carefully measure heat transfer

through the scramjet walls, to model the wall thermal conditions as closely as possible.

The discussion here focuses on findings specific to turbulence modeling. Figure 11 shows Mach number

contours in the UVA scramjet flowpath for the fuel off (φ = 0.0) case using the Chien k −  model. The

top plot shows contours from the facility nozzle entrance through isolator, combustor and exhaust nozzle.

The middle plot focuses on the region near the end of the isolator and combustor, while the bottom plot

extends from the combustor to the exhaust nozzle exit plane. Supersonic flow exits from the facility nozzle,

through the combustor, and into the expansion nozzle with no pressure rise in the combustor, which would

occur for fueled cases. In contrast to a flight vehicle nozzle, where by design the flow would continue to

expand through the exhaust nozzle, the flow in the UVA experiments is overexpanded due to limitations in

the overall system nozzle pressure ratio, and a shock system with flow separation may be observed near the

exhaust nozzle exit.

Figure 12 shows a comparison of RANS solutions obtained using the Menter SST model and the Chien

k − model. Through most of the flowpath, the pressure distributions are very similar, but in the last half of

the exhaust nozzle, the SST pressure distribution deviates from the experimental data and the k− solution.

This is the result of the SST solution separating much farther upstream in the nozzle. As discussed in the

SWTBLI section, typically k − models are not considered a good choice for flows with separations, and the

SST model has developed a broad reputation for performing better in such situations. However, as shown in

the SWTBLI results, it appears that SST overreacts in moderate to highly separated flows situations. This

may be in part due to the actual shear stress limiter term active within the Menter SST model, but the

performance is generally similar to other k − ω models.

Our initial set of calculations simulating an entire sweep in fuel equivalence ratios, as reported in Bhagwandin et al,53 used the SST model and 7-species, 8-step H2-air kinetics due to Evans and Shexnayder57

modified to include third body efficiencies other than unity, as taken from the model of Jachimowski.58

After obtaining the result shown in Fig. 12, which was not expected, we completed an additional sweep in

equivalence ratios for both clean and vitiated air as reported in Vyas et al,54 using 9-species, 13-step H2-air

kinetics due to Peters and Rogg59 and the Chien k- model. The Peters and Rogg model was selected for

the subsequent studies because the complete kinetics set was developed to investigate vitiated air. These

latter calculations were able to reproduce the experimentally observed “mode transition” or point at which

the core flow entering the combustor region changed from subsonic (ramjet) to supersonic (scramjet). Figure

13 provides a comparison of Mach number contours for two cases, both at the same fuel equivalence ratio,

φ = 0.26, but in one case the supply flow was clean air and the other has vitiated air with 10 percent water

vapor per mole. It may be observed that the clean air case is in ramjet mode while the vitiated air case is in

scramjet mode. These results are largely due to chemical kinetics effects, although to obtain this qualitative

agreement with experimental results on the equivalence ratio where mode transition occurred required close

calibration of the choices in turbulence model settings, including turbulent Prandtl and Schmidt numbers.

Figure 14 shows the strong sensitivity of the solutions to turbulent Schmidt number.

We emphasize that these choices may not be optimal for all scramjet configuration simulations. In

particular, the performance of the k − model may be considered somewhat fortuitous, and in general is not

considered an optimal choice for flows with separations. The conclusion here is that the “state-of-practice”

in RANS as applied to complete scramjet systems requires calibration of the modeling parameters for one

case, i.e. selecting fixed values for the turbulent Prandtl and Schmidt numbers and then trusting that the

trends of the solutions will remain qualitatively accurate at other operating points. As mentioned previously,

it is difficult to draw definitive conclusions on the validity of a particular CFD modeling choice for a complex

system such a scramjet when only pressure distributions as a function of axial position are available. For

this reason, usually unit problems are considered to investigate the behavior of individual model settings.

Scalar Transport Modeling

A broadly used experiment for validation of supersonic reacting flows is that of Burrows and Kurkov.60

A schematic of their experimental setup is shown in Fig. 15. Despite the fact that these experiments

were performed a number of years ago, the data obtained for this hydrogen-vitiated air test case remains

widely used today. Using the 8-step kinetics of Evans and Shexnayder discussed previously, we present

NASA/TM—2012-217277 10
results here that exhibit the sensitivity of the solutions to settings for the turbulent Prandtl and Schmidt

numbers. Figures 16 and 17 provide temperature contours in the mixing section which illustrate the effects

of variation in these scalar transport settings on the ignition point. The ignition point in the experiment

was approximately x = 22cm. Exit plane contours are shown in Fig. 18. While both quantities have

considerable effects, the turbulent Schmidt number effects are more pronounced. The Schmidt number

controls the turbulent species transport, and considering the Arrhenius rate form of the chemical kinetics

used in most RANS combustion simulations as well as the actual physical process requiring sufficient reactants

to be present for combustion to occur, it is not surprising that the turbulent Schmidt number has such an

important effect on the combustion behavior.

Unfortunately the optimal turbulent Schmidt number and turbulent Prandtl number for one case are not

likely optimal for another. Further, the current common practice of setting constant values to P rt and Sct for

modeling scalar transport is very likely inadequate for the complex flows occurring in scramjet systems. This

finding has been been reported elsewhere, such as in Ref. 7. Work has been underway to investigate variable

Prandtl/Schmidt number formulations, or “scalar variance methods” to be used for improving the modeling

of turbulent heat and mass transport in scramjets and high speed reacting jets such as those described in

Refs. 61–65. The treatment of turbulent-chemistry interactions, such as those involving probability density

function (PDF) formulations66, 67 is beyond the scope of this paper. However, one study that demonstrated

a variable Schmidt number model incorporated with a PDF approach is described in Ref. 68. While these

efforts are promising, the methods are not mature enough yet to be in broad use by the scramjet community.

Sufficient experimental data to thoroughly develop and validate these methods is also not available, yet very

much needed.

Compressible Mixing

In the scramjet combustor and into the exhaust nozzle, turbulent mixing is highly three dimensional and

subject to compressibility effects. In this section we briefly discuss the status of modeling compressible

mixing. A number of experiments may be found in the literature concerning planar mixing layers and effects of

compressibility on mixing layer growth rates. In recent years, most computational and experimental research

efforts has focused on jet flows, driven by aircraft noise considerations. For the high speed compressible jet

problem, one of the most widely used data sets is from the benchmark experiments performed at the NASA

Langley Research Center (LaRC) Jet Noise Laboratory by Seiner et al.69 These experiments investigated an

axisymmetric water-cooled Mach 2 nozzle. The nozzle operated at fully expanded conditions for this series of

cases. In our studies, the turbulence models examined were the Menter SST and Chien k- formulations. In

addition, these two models were used with and without the Sarkar compressibility correction,70, 71 as shown

in Eq. 28.

 = s(1 + αM2

t

) (28)

and the turbulent Mach number is defined as:

Mt =

√

2k/a (29)

where s is the solenoidal dissipation rate solved in the dissipation equation for the k- model where

ω = /k. The coefficient, α, is set to the default value of 1.0 here for use with both turbulence models.

The Sarkar compressibility correction increases the rate of turbulent kinetic energy dissipation and then

lowers the resultant turbulent viscosity. Compressibility corrections such as the Sarkar formulation, are

most commonly employed for free shear layers or jets. For wall bounded flows, compressibility effects are

not considered to be important for Mach numbers less than about 5, as discussed by Wilcox.21 Rumsey

examined compressibility corrections applied to k − ω models for hypersonic boundary layers in Ref. 72.

In Dembowski and Georgiadis,73 a comparison of computations to experiments were made for all of the

operating points, with variations in jet stagnation temperature that were measured in the experiments of Ref.

69. Results are presented for the highest temperature case with nozzle plenum stagnation temperature set

to 1370 K in Fig. 19. For this and other operating points, the Chien k- solution and SST solutions with no

compressibility corrections provide similar solutions with faster jet mixing than indicated by the experimental

data. The Chien and SST model results obtained with the Sarkar compressibility correction indicate much

longer potential cores and slowest initial mixing of any of the solutions for all of the temperatures investigated.

In all of the cases shown in Ref. 73 where several jet operating temperatures were investigated, solutions

NASA/TM—2012-217277 11
obtained with the compressibility correction overpredict the potential core length, as shown for the highest

temperature case considered here. In results for supersonic jet calculations shown by Gross et al74, 75 the

performance of SST with and without the Sarkar correction is analogous to the results obtained here. As

shown in Ref. 76, the potential core length behavior observed here for the uncorrected models is not the

same for lower Mach number jets, where two-equation turbulence model results generally indicated longer

potential core lengths, and implied slower initial mixing rates, than found experimentally. As a result, while

compressibility corrections could be tailored, for example via the Sarkar coefficient, α, for a particular case,

there are not enough physics in such models to accurately capture compressibility effects. The conclusion

here is that compressible mixing is yet another area in which RANS turbulence modeling improvements are

still necessary.

LES-based Methods for Combustor and Exhaust System

With all of these limitations of RANS-based techniques for the scramjet combustor and exhaust nozzle regions, there is hope that LES and hybrid RANS-LES may offer better prospects for improving the ability to

calculate regions dominated by turbulent mixing. There have been significant efforts underway in developing and applying LES for mixing problems, especially turbulent jets issuing from aircraft exhaust nozzles.

However there are a number of unresolved issues in LES that significantly affect the accuracy of turbulent

jet flows and other mixing dominated problems. There have been some limited attempts to use LES for

scramjet combustors, as in Refs. 9 and 10. In the combustor, the LES problem of accurately capturing

turbulent-chemistry interactions is analogous to the RANS problem. Some research into techniques such as

filtered density functions, the LES analogy to the RANS usage of PDFs, are being examined.11, 77

Considering the scramjet system as a whole, the need to accurately treat both the wall bounded regions

and the mixing layers away from the walls provides difficult challenges. At first impression, hybrid RANSLES methods seem to be a promising solution to this problem, and indeed they should be and are being

explored.78–80 However the interface region where RANS switches to LES, is one of the most significant challenges. In aerodynamic flows, hybrid RANS-LES has found success where the RANS regions are demarcated

from the LES regions and some physical feature is responsible for the instabilities that generate and sustain

turbulence in the LES regions.81 In the scramjet, frequently a flameholder with a rearward facing step can

serve as the physical mechanism behind which large scale eddies may be generated, to drive the necessary

unsteadiness of the LES region. Countering this, however, are the large values of eddy viscosity convected

from the thick isolator boundary layers calculated with RANS, which serve to damp the unsteadiness in the

computation.

Experimental Validation Data

The need for fundamental experiments to provide data for method development and validation is very

high. As discussed previously, while full scramjet simulations are informative, it is very difficult to perform

validation of specific CFD modeling features for complete scramjet systems. One reason is that historically

only static pressure instrumentation is employed in scramjet experiments. Further, it is nearly impossible

to parametrically isolate the benefits of one particular modeling feature, whether it be turbulence modeling,

chemical kinetics modeling, or heat transfer, when the effects of the modeling limitations in the others is

unknown. For these reasons, unit problems are more useful in model development and validation. While there

are a number of such unit problem experiments reported in the literature, most of the data available consists of

mean flow measurements. Improvements in turbulence modeling for hypersonic propulsion requires turbulent

statistical data from simultaneous measurements of velocities, temperatures, and species concentrations.

One current effort addressing this need is the US National Center for Hypersonic Combined Cycle Propulsion.82 This center is developing and using advanced experiments and diagnostics such as Coherent AntiStokes Raman Scattering (CARS), PIV, Rayleigh techniques, and Planar Induced Flourescence (PLIF) to

obtain more detailed data including turbulent statistics and the simultaneous measurements of velocities,

temperatures, and species concentrations. This center is also developing advanced turbulent calculation

methods targeted at the supersonic combustion problem, including RANS, LES, and hybrids. NASA’s

Fundamental Aeronautics Program, the United States Air Force Office of Scientific Research and the Test

Resource Management Center’s Test and Evaluation/Science and Technology (T&E/S&T) Program are also

sponsoring experimental research efforts from geometrically simple unit flow problems to fully integrated

systems and also sponsoring computational method development and validation efforts. With hypersonic

NASA/TM—2012-217277 12
funding historically very cyclical in the United States, it is hoped that a sustained effort will be possible, as

the research needs are great.

Some recommendations for specific experimental data needs that are not currently being broadly addressed are as follows: In the area of SWTBLIs, advanced measurement techniques such as PIV have been

recently applied to fundamental experiments, such as those described earlier in this paper. However, nearly

all of these experiments have been conducted in facilities with small tunnel cross sectional areas, where

sidewall effects frequently contaminate the focus SWTBLI region. It would be highly desirable to utilize an

experimental rig where such sidewall effects could be eliminated. PIV has become a powerful and broadly

used experimental technique. Obtaining high accuracy in regions such as the beginning of a jet shear layer,

where the turbulent flow state is far from equilibrium, is quite a challenge. However, it is this region where

both RANS turbulence models and direct calculation methods (i.e. DNS and LES) need data for method

development and validation.

Thermal boundary layers obviously have significant effects on hypersonic propulsion flowpath aerothermodynamics. While heat transfer and surface temperature measurements are being made in some select

configurations, this is the exception rather than the rule. Such measurements are very difficult to make, but

crucial in attempts to accurately represent the thermal boundary layer states as well as identifying potential

thermal deformation which in turn can alter the flowpath cross sectional area and aerodynamics. For the

combustor, detailed mappings of the turbulent flow state for velocities, temperatures, and species concentrations are necessary. While some fundamental experiments are attempting to obtain these quantities, perhaps

revisiting an experiment such as the Burrows-Kurkov test case (which only provided mean flow quantities)

with these more modern experimental techniques (i.e. PIV, CARS, and PLIF) would greatly assist modelers in determining if a particular modeling enhancement was actually improving the fidelity of the correct

turbulent flow physics.

Conclusions

This paper presents one group’s perspective on the current state of turbulent flow calculation methods for

hypersonic propulsion flowpaths. It is not intended to serve as a comprehensive review of all computational

techniques in use today, but does discuss key issues throughout the vehicle propulsion flowpath. These

include laminar-to-turbulent boundary layer transition, shock wave / turbulent boundary layer interactions,

and modeling of the combustor and exhaust system. Emphasis was placed on Reynolds-averaged NavierStokes (RANS) methods, although prospects of newer methods such as Large Eddy Simulation (LES) based

techniques were also provided. It is concluded that RANS turbulence modeling shortfalls are still a major

limitation to the accuracy of hypersonic propulsion simulations, whether considering individual components

or an overall system. Newer methods such as LES-based techniques may be promising, but are not yet at a

maturity to be used routinely by the hypersonic propulsion community.

RANS turbulence model development in the past decade has been minimal for all aerodynamic regimes.

Most of the programmatic support for turbulent calculation techniques has been toward LES-based techniques. While the broader CFD community will likely be using LES more in the future as computing power

continues to improve, there is still a need in the foreseeable future for better RANS-based techniques. The

subsonic aircraft community, while also examining hybrid RANS-LES techniques, is currently beginning

to revisit pure RANS models because of the importance to aerodynamic predictions of quantities such as

vehicle drag. Support for improving some of the modeling shortfalls identified in this paper for hypersonic

propulsion flowpaths is also warranted. Finally, more detailed experimental data to help guide modeling

improvements is also needed.

NASA/TM—2012-217277 13
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NASA/TM—2012-217277 14
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NASA/TM—2012-217277 15
68Keistler, P. G., Gaffney, R. L., Xiao, X., and Hassan, H. A., “Turbulence Modeling for Scramjet Applications,” AIAA

Paper 2005-5382, June 2005.

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Figure 1. Schematic of scramjet-powered hypersonic vehicle with key turbulent flow physics.

NASA/TM—2012-217277 16
Re x

Cf

0 200000 400000 600000 0

0.002

0.004

0.006

T3A Data

SST - fully turbulent

SST - transition, CPTM1=1.0

SST - transition, CPTM1=1.5

SST - transition, CPTM1=2.0

Laminar Correlation

Turbulent Correlation

Figure 2. Incompressible transition comparison.

Rex

St

1E+06 1E+07

0.001

0.01

Re/m = 3.3E6

Re/m = 3.9E6

Re/m = 6.6E6

Re/m = 8.2E6

SST - fully turbulent

SST - transition, CPTM1 = 1.0

SST - transition, CPTM1 = 2.0

Figure 3. Hypersonic transition comparison.

NASA/TM—2012-217277 17
x (mm)

0 100 200 300 400 500

M: 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Figure 4. Mach number contours for shock-wave boundary-layer interaction problem.

x (mm)

Skin Friction, Cf

300 350 400 450 500

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Schulein, et al. (1996)

Schulein (2004) [optical]

SST

Wilcox k-ω (1988 model)

Chien k-ε

ASM k-ε

ASM k-ω

Figure 5. Skin friction along bottom wall for shock-wave boundary-layer interaction problem.

NASA/TM—2012-217277 18
θ 

M∞ 

PIV Plane 

y 

x 

Figure 6. Schematic of typical SWTBLI experimental configuration (courtesy of DeBonis et al43).

Figure 7. Velocity contours for UFAST test case using SST.

NASA/TM—2012-217277 19
(a) Experiment

(b) Menter SST

(c) Spalart-Allmaras

(d) k − ωASM

Figure 8. Axial velocity contours for UFAST test case in vicinity of SWTBLI.

NASA/TM—2012-217277 20
u (m/s)

y (mm)

0 200 400 600 0

5

10

15

Expt

SST

SA

k-t ASM

(a) x = 280 mm

u (m/s)

y (mm)

0 200 400 600 0

5

10

15

Expt

SST

SA

k-t ASM

(b) x = 300 mm

u (m/s)

y (mm)

0 200 400 600 0

5

10

15

Expt

SST

SA

k-t ASM

(c) x = 320 mm

u (m/s)

y (mm)

0 200 400 600 0

5

10

15

Expt

SST

SA

k-t ASM

(d) x = 340 mm

Figure 9. Axial velocity profiles for UFAST test case in vicinity of SWTBLI.

NASA/TM—2012-217277 21
Figure 10. Schematic of UVA direct connect dual-mode combustor (courtesy of Rockwell et al52).

Figure 11. Mach number contours for scramjet flow case, fuel off.

NASA/TM—2012-217277 22
x / H

P / Pref

-40 -20 0 20 40 60 0

0.5

1

1.5

2

2.5

3

Expt., Clean Air

Wind-US, SST

Wind-US, Chien k- ¡

Figure 12. Pressure distributions for scramjet flow case, fuel off.

NASA/TM—2012-217277 23
(a) Clean air

(b) Vitiated air

Figure 13. Mach number contours for scramjet case, φ = 0.26.

NASA/TM—2012-217277 24
x / H

P / Pref

-40 -20 0 20 40 60 0.5

1

1.5

2

2.5

3

3.5

4

Expt., Clean Air

Wind-US, Prt=.9, Sct=.5

Wind-US, Prt=.9, Sct=.7

Wind-US, Prt=.9, Sct=.9

Figure 14. Turbulent Schmidt number effect on clean air scramjet case, φ = 0.26.

Figure 15. Schematic of Burrows-Kurkov experiment (taken from Burrows and Kurkov60).

NASA/TM—2012-217277 25
(a) Prt = 0.5, Sct = 0.7

(b) Prt = 0.7, Sct = 0.7

(c) Prt = 0.9, Sct = 0.7

Figure 16. Temperature contours for Burrows-Kurkov test case illustrating effect of P rt on ignition point.

NASA/TM—2012-217277 26
(a) Prt = 0.7, Sct = 0.5

(b) Prt = 0.7, Sct = 0.7

(c) Prt = 0.7, Sct = 0.9

Figure 17. Temperature contours for Burrows-Kurkov test case illustrating effect of Sct on ignition point.

H2O Mole Fraction

y (cm)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

3.5

4

Experiment

Sct = 0.7, Prt = 0.5

Sct = 0.7, Prt = 0.7

Sct = 0.7, Prt = 0.9

(a) Turbulent Prandtl number variations

H2O Mole Fraction

y (cm)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

3.5

4

Experiment

Sct = 0.5, Prt = 0.7

Sct = 0.7, Prt = 0.7

Sct = 0.9, Prt = 0.7

(b) Turbulent Schmidt number variations

Figure 18. Exit profiles for Burrows-Kurkov test case.

NASA/TM—2012-217277 27
x / D

u / Ujet

5 10 15 20 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Seiner Data

SST

SST with Sarkar

Chien k-¡

Chien k-¡ with Sarkar

(a) centerline axial velocity

x / D

Mach No.

5 10 15 20 0

0.5

1

1.5

2

2.5

Seiner Data

SST

SST with Sarkar

Chien k-¡

Chien k-¡ with Sarkar

(b) centerline Mach number

x / D

Tt / TT-jet

5 10 15 20 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Seiner Data

SST

SST with Sarkar

Chien k-¡

Chien k-¡ with Sarkar

(c) Centerline stagnation temperature

Figure 19. Comparison of solutions for heated Mach 2 jet.

NASA/TM—2012-217277 28
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Status of Turbulence Modeling for Hypersonic Propulsion Flowpaths 

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Georgiadis, Nicholas, J.; Yoder, Dennis, A.; Vyas, Manan, A.; Engblom, William, A. 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

This report provides an assessment of current turbulent flow calculation methods for hypersonic propulsion flowpaths, particularly the 

scramjet engine. Emphasis is placed on Reynolds-averaged Navier-Stokes (RANS) methods, but some discussion of newer meth- ods such 

as Large Eddy Simulation (LES) is also provided. The report is organized by considering technical issues throughout the scramjet-powered 

vehicle flowpath including laminar-to-turbulent boundary layer transition, shock wave / turbulent boundary layer interactions, scalar 

transport modeling (specifically the significance of turbulent Prandtl and Schmidt numbers) and compressible mixing. Unit problems are 

primarily used to conduct the assessment. In the combustor, results from calculations of a direct connect supersonic combustion experiment 

are also used to address the effects of turbulence model selection and in particular settings for the turbulent Prandtl and Schmidt numbers. It 

is concluded that RANS turbulence modeling shortfalls are still a major limitation to the accuracy of hypersonic propulsion simulations, 

whether considering individual components or an overall system. Newer methods such as LES-based techniques may be promising, but are 

not yet at a maturity to be used routinely by the hypersonic propulsion community. The need for fundamental experiments to provide data 

for turbulence model development and validation is discussed.

15. SUBJECT TERMS

Hypersonics; Turbulence; Transition; Heat transfer; Scramjet; Large Eddy Simulation (LES) 

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