NASA/TM–2014–218353

On Parametric Sensitivity of

Reynolds-Averaged Navier-Stokes SST Turbulence Model:

2D Hypersonic Shock-Wave Boundary-Layer Interactions

James L. Brown, Ph.D.

NASA Ames Research Center, Moffett Field, CA

May 2014
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NASA/TM–2014–218353

On Parametric Sensitivity of

Reynolds-Averaged Navier-Stokes SST Turbulence Model:

2D Hypersonic Shock-Wave Boundary-Layer Interactions

James L. Brown, Ph.D.

NASA Ames Research Center, Moffett Field, CA

National Aeronautics and

Space Administration

Ames Research Center

Moffett Field, CA 94035-1000

May 2014
Acknowledgments

This work was sponsored by NASA’s Hypersonic Entry Descent and Landing

(HEDL/STMD) projects.

The author gratefully acknowledges discussions with Joseph G. Marvin on free

interaction theory.

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does not constitute an offical endorsement, either expressed or implied, of such products or

manufacturers by the National Aeronautics and Space Administration.

Available from:

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7115 Standard Drive

Hanover, MD 21076-1320

443-757-5802
Abstract

Examined is sensitivity of separation extent, wall pressure and heating to variation of primary input flow parameters, such as Mach and

Reynolds numbers and shock strength, for 2D and Axisymmetric Hypersonic Shock Wave Turbulent Boundary Layer interactions obtained by

Navier-Stokes methods using the SST turbulence model. Baseline parametric sensitivity response is provided in part by comparison with vetted

experiments, and in part through updated correlations based on free interaction theory concepts. A recent database compilation of hypersonic

2D shock-wave/turbulent boundary layer experiments extensively used

in a prior related uncertainty analysis provides the foundation for this

updated correlation approach, as well as for more conventional validation. The primary CFD method for this work is DPLR, one of NASA’s

real-gas aerothermodynamic production RANS codes. Comparisons are

also made with CFL3D, one of NASA’s mature perfect-gas RANS codes.

Deficiencies in predicted separation response of RANS/SST solutions to

parametric variations of test conditions are summarized, along with recommendations as to future turbulence approach.

1 Introduction

Design optimization for aerodynamic and aeroheating of hypersonic and

reentry vehicles rely on advanced computational methods, such as RANS

(Reynolds-Averaged Navier-Stokes) CFD codes with advanced turbulence models, validated for correct response over a sufficient range for

input relevant design parameters. Conventionally, this validation is done

by comparison of CFD results with a select group of experiments at

discrete fluid dynamic conditions. However, credibility of optimization

results rely not just on conventional CFD validation for discrete experimental conditions at sparse points located in parameter space, but also

by ensuring the proper response trends of the underlying analysis procedures throughout the entire input parameter space considered.

Multi-Discipline Optimization (MDO) procedures, embedded with

RANS methods and reliance on advanced turbulence models, when applied to the design of modern Hypersonic and Atmospheric ReEntry

vehicles (Garcia, et al., [1]) can result in aerothermodynamic performance gains combined with weight savings so as to exhibit potential for

improved mission operations and costs with consequent enhanced economic viability of the evolving nascent US Space industry. The success

of these MDO Optimization procedures depends not just on their algorithmic efficiency, but also on the physical accuracy of the modeling of

the turbulent RANS analysis procedures over a large range of possible

vehicle and fluid dynamic performance parameters. The validation of

turbulent models in RANS procedures, however, often depends on com1
parisons of turbulent RANS analysis results with but a small number

of vetted available experiments covering only a small range of physical

parameters.

Examination of the sensitivity of these turbulent RANS analysis

methods to fluid dynamic parametric variations relative to established

correlations in conjunction to these vetted experiments help establish the

degree and range of credibility for optimized design results of MDO procedures with embedded turbulent RANS aerothermodynamic analysis.

One such recent uncertainty analysis was accomplished by the present

author (Brown, Refs. [2,3]) at the discrete conditions of a vetted database

of hypersonic experiments (Marvin, Brown and Gnoffo, Ref. [4]. However, realized during the course of that study was that by considering

only a limited number of fluid conditions as described by a small select group of experiments, no matter how well-behaved the experiments,

leads to an incomplete picture of the analysis validity; rather, there is

need to ensure that the proper physical trends are obeyed. Essentially,

it is only when both the point-wise accuracy and the trend sensitivity

of the turbulent RANS methods embedded in MDO methods are physically correct that optimized design results from MDO procedures can

be trusted as being valid. By point-wise accuracy, we mean accuracy of

design relevant quantities such as separation extent, separation bubble

pressure, peak pressure and peak heating at the discrete flow parameters

of the considered experiments, while by ‘trend sensitivity’ we mean the

partial derivative or variation of these design relevant quantities with respect to small variations of the flow parameters (typically Mach number,

Reynolds number, shock strength, wall cooling, etc.) about the discrete

conditions of these considered experiments.

The present study attempts to examine the ability of current turbulent RANS methods applied to hypersonic Shock Wave Turbulent Boundary Layer Interactions by use of the recently released database of vetted

hypersonic experiments (Marvin, et al., [4]), supplemented by the previous database of Settles and Dodson, Refs. [6]- [9]. Comparison of CFD

results (such as separation extent, separation bubble pressure and peak

heating) with such experiments provides a basis for credibility at the

discrete fluid dynamic conditions of these experiments. But it does not

provide confidence in the ability of these CFD methods to properly follow

the correct trends (e.g., rate of change in {separation extent, separation

bubble pressure or peak heating} with respect to incremental change

in {Mach no, Reynolds number, wall cooling}) when the fluid dynamic

conditions change over a large range as might be required to accomplish

comprehensive MDO optimization. To accomplish examination of parametric trend sensitivity of these methods, we make use of correlations

related to the long-established concept of free-interaction theory (Chapman, et al., [23], and Erdos and Pallone, [25]), updated, where needed,

to include the latest hypersonic database. Other accepted correlations

are to be considered as well. Only by identifying these deficiencies, can

2
an attempt then be made to advance CFD turbulence modeling so as to

improve both the accuracy and the parametric trend sensitivity of the

overall CFD turbulent analysis procedure.

A primary purpose of this paper is to evaluate and to identify deficiencies of current turbulent RANS CFD solution methods as applied

to 2D Hypersonic Shock Wave Turbulent Boundary Layer Interactions,

giving particular attention to the success of their results in conforming

to proper trends with relevant fluid dynamic parametric variations.

In the following sections, we first provide a brief overview of two

databases providing the hypersonic experimental data of primary use in

this current report, next summarize the approach and results of the recent uncertainty analysis accomplished in the recent works of Brown [2,3]

and of Gnoffo [19,20]. We then describe free interaction theory concepts

and additional correlations that establish the proper physical parametric trends which computations should follow. The application of free

interaction theory is reformulated slightly so as include and compare

with these updated databases and also to better align the application

of free interaction correlations to current CFD methodology. A comparison is then provided of turbulent CFD results with both the discrete

experimental data of the databases and the parametric trends described

by the updated free interaction related correlations. We finally identify

defects in SST turbulence model response, so as to provide a basis for

improved performance of the model relative to the database experiments

and to free interaction theory. Primary emphasis is made of the Mach 8.2

Compression Corner experimental series of Holden and the Mach 2.85

Compression Corner experiments of Settles as these are two of but few

hypersonic/supersonic datasets that includes variations in both shock

strength and Reynolds number.

1.1 Hypersonic Shock-Wave Turbulent Boundary Layer

Interaction Experiment Database

Much of the development of ‘Free Interaction’ SWBLI theory occurred

based on early supersonic experiments. The primary source of experimental data used for the current study is obtained from two databases

that include more recent hypersonic data and in a form that facilitates

the use of that data. These are the supersonic/hypersonic database of

Settles and Dodson [6], and the hypersonic database of Marvin, Brown

and Gnoffo [4]. The databases themselves do not present new experimental data, but review the available experiments for the accuracy of the

data, for complete definition of the geometry of the test and suitability

and completeness of the stated test conditions for use with present-day

Navier-Stokes solvers, and also seek permission from the experimentor

to satisfy possible copyright issues. The databases then compile the

data tabulated in easily read ASCII text files. More information can

be obtained from the reports describing these databases, however the

3
experiments specifically used in this current study include both hypersonic and supersonic compression corner and impinging shock cases from

Mach 3 to Mach 11, see Tables 1-4.

The primary parametric variation of interest to this study, that ideally would prove to be available in a single experiment having an upstream boundary layer of fixed properties, is a variation in shock strength

from attached to incipient separation to several fully separated cases.

Additional parameters of interest, likely to be found among the collection

of experimental cases, and therefore having upstream boundary layers of

differing properties, are variations in Reynolds number, Mach number

and degree of wall cooling (Tw/Taw). Of the experiments compiled in

these two databases, the primary focus therefore will be on the Holden

Mach 8 cold wall compression corner cases, the Settles Mach 3 adiabatic

compression corner cases, and the Sch ̈ulein Mach 5 cold wall impinging

shock cases. The remaining experiments in these databases, as tabulated in Tables 1-4, then provide supplemental information to, at least

partially, provide indication of the effects of Mach number, Reynolds

number, degree of wall cooling and primary shock strength.

1.2 Computational Methods

Computational methods used for this study include both the EddyBl

boundary layer code of Wilcox [13] to provide definition of the test

flat plate (or axisymmetric cylinder) boundary layer, and DPLR, one of

NASA’s production Real-Gas Reynolds Averaged Navier-Stokes codes.

The Wilcox ‘EddyBL’ boundary layer code with the Cebeci-Smith turbulence model option is used as the standard basis by which to evaluate

the upstream flat plate boundary layer properties. However, it is the

DPLR RANS code, using the SST turbulence model, and its’ ability to

properly analyze hypersonic shock wave turbulent boundary layer interaction behavior, that is of primary interest to this study. As all the

experimental cases considered prove to be ideal gas, additional ideal-gas

RANS solutions using the well-tested CFL3D code of Rumsey provide

supplemental support to the DPLR/SST results.

These two codes, DPLR [14,15] and CFL3D [16], although providing

the same nominal SST turbulence model of Menter [33]- [36], differ in

details of the implementation of that viscous modeling, as well as their

inviscid solver algorithm.

1.3 Uncertainty Analysis

NASA researchers provided, in a collection of papers, a recent uncertainty assessment of various aspects of hypersonic aerothermodynamics prediction capability, see Refs [17, 18]. This included an assessment

of two primary types of shock wave/turbulent boundary layer interactions: that of the compression corner, see Gnoffo [19, 20]; and, of the

4
impinging shock wave boundary layer, see Brown [2, 3]. A goal of these

last two papers was, based on rigorous statistical basis, to provide a

confidence interval estimate of the performance of NASA production

real-gas Navier-Stokes solvers in predicting selected design parameters

(peak heating, peak pressure, separation extent, etc.) relevant to possible NASA mission and program needs for turbulent SWTBLI. This

extended the conventional validation approach of simply examining and

comparing, experiment by experiment, the performance of select turbulence models and CFD codes performance against each of the selected

experiments, but also provided a collective CFD assessment over a range

of conditions by statistical analysis of physics quantities of interest. The

resultant uncertainty intervals should prove helpful in providing a rigorous statistical basis of risk assessment of NASA missions and programs

interested in using CFD methods. Although an improvement over simple validation approaches used previously, one limitation is the small

number of vetted experiments suitable for such a statistical analysis,

but also that it only makes use of a point analysis from each experiment test condition, although collectively a wide range of test conditions

are covered. What is not covered by such an approach is that about

each experimental test condition(e.g. Mach number, Reynolds Number,

Shock strength, etc), there exists a response in test results (separation

extent, separation bubble pressure,peak wall pressure, peak wall heating,

etc) covered by conventional validation, but also there exists a trend response of these same physical quantities of interest to small perturbations

in test condition parameters, (e.g., ∂separation extent/∂Mach Number,

∂separation extent/∂Reynolds Number, etc). It is this latter ‘parametric trend response’ that we wish to explore more thoroughly, and to add

to the validation process, in this present paper.

As to the Uncertainty Analysis conducted by both Gnoffo and by

Brown, it was found that although separation extent and bubble pressure contributed to the overall uncertainty of current status of CFD

solvers in prediction of SWTBLI, it is the post-reattachment wall heating

that is the largest contributor to uncertainty, giving as much as ≈ 50%

overshoot. Such a large overshoot in wall heating prediction can have

significant impact on NASA’s risk assessment in the design and operation of space vehicles, leading to unnecessarily large thermal protection

systems, or limiting allowable mission performance. As such, there is a

compelling need to reduce this uncertainty with improvements in CFD

predictive methods, including advances in hypersonic turbulence models.

1.4 The ‘Free Interaction’ and λ-Shock Separation

A 2D or axisymmetric, supersonic or hypersonic boundary layer subjected to the sudden pressure rise of a primary shock system of sufficient

strength will undergo separation. There are at least two distinct viscousinviscid interactions that may be invoked for such a shock-separated re5
gion: the viscous-inviscid interaction controlling the separation process;

and, the viscous-inviscid interaction controlling reattachment. It has

been observed by experiment and by theoretical analysis, that the behavior of the separation region preceding the primary shock system is

most directly defined by the properties of the boundary layer just upstream of separation regardless of the geometry of the primary shock

system, whether by compression corner or by impinging shock, or forward facing step or obstacle or by means of a supersonic flow on the

walls of an over expanded nozzle.

The overall shock-wave/boundary layer separation process is dominated by a strongly coupled viscous-inviscid interaction between the

viscous separating boundary layer and the inviscid effects of the λ-shock

that forms in response in the supersonic (or hypersonic) free stream immediately adjacent. The viscous and inviscid effects of this strong coupling are mutually induced. The separating boundary layer displacement

thickness undergoes rapid growth induced by the inviscid pressure rise

imposed on the boundary layer; the rapid displacement surface growth

induces the formation of a leading λ-shock immediately above the separating boundary layer, and which projects upstream of the primary

shock system. In turn, the λ-shock formation leads to a pressure rise imposed on the separating boundary layer. This shock-separation interaction has been found to be independent of, or free of primary dependence

on, the details of the primary shock system; it has become known as

a ‘Free Interaction’. The λ-shock separation-for supersonic and hypersonic flows-process has proven to be nearly independent of the generating

mode of the downstream primary shock system, whether that mode is of

compression corner, of impinging shock, or of forward-facing step type.

Rather, the properties of the ‘Free Interaction’ separation bubble (e.g.

pressure level and shape of the pressure distribution within the separation bubble region) seem to be established by the properties of the

upstream boundary layer: the primary controlling factors are observed

to be the wall shear stress, wall pressure, boundary layer thickness (or

displacement thickness), edge Mach number, and whether the boundary

layer is turbulent or laminar.

In contrast, the process of reattachment of the separated boundary

layer is subsequently accomplished in the vicinity of -and dependent

on- the primary shock system, and not considered a ‘Free Interaction’.

Naturally, should the primary shock system not be of sufficient strength

to separate the boundary layer, the primary shock system is still present,

but without the leading λ-shock.

First order behavior observed characterizing ‘Free Interaction’ separation is that the shock angle of this leading λ shock and the magnitude

of the associated pressure rise within the separation bubble, will remain

nearly constant -once separation occurs- even if the strength of the primary shock system and its associated overall pressure rise is caused to

increase. An increase in primary shock strength, past that needed to

6
separate the boundary layer, serves mainly to increase the extent or size

of the separation bubble, but without significant further pressure rise

within the separation region nor change in strength of the leading λ

shock and its associated angle. It can be concluded that the ‘viscousinviscid interaction’ of the leading λ shock with the underlying separating

boundary layer is nearly independent -or ‘free’ of direct influence- of the

primary shock system, and it is the ‘free interaction’ of the viscous separating boundary layer with the inviscid leading λ shock that establishes

the properties of the separation bubble; consequently, the properties of

the upstream boundary layer (e.g., the boundary layer wall shear stress,

the boundary layer thickness, the edge Mach number, and whether the

boundary layer is laminar, transitional or turbulent) that dominate in

determining the details of the separation process and the properties of

the leading λ shock.

1.4.1 Free Interaction Relations

Analyses of and relations describing the ‘Free Interaction’ were provided

in experimental and theoretical studies predominantly in the mid-1950’s

to early-1960’s. Extensions of the ‘Free Interaction’ concept to incorporate non-uniform free stream and real gas effects pertaining to rocket

nozzle applications has occurred more recently. The paper of Chapman,

Kuehn and Larson [23] appear to have first expressed the concept of ‘Free

Interaction’ based on an analysis with integral boundary layer relations

to convey the viscous effects and the Prandtl-Meyer relations to convey

the inviscid effects, but with supporting dimensional analysis and extensive supersonic experimental observations. Chapman’s paper treated

both turbulent and laminar cases, but limited to adiabatic wall boundary layers. A theoretical basis for ‘Free Interaction’ relations had been

explored earlier by Reshotko and Tucker [24], but without clear identification of the ‘Free Interaction’ concept, and the latter thorough development by Erdos and Pallone [25] again using integral boundary layer and

inviscid relations prevalent in that era. In each of these studies, a somewhat differing formulation was arrived at. Lewis, Kubota and Lees [26]

improved the laminar case correlations to include cold-wall boundary layers. Carriere, Sirieix and Solignac [27] extended the cases considered to

include turbulent separation in non-uniform supersonic flow. Additional

experiments by Roshko and Thomke, by Settles, by Holden and others

further enhanced the understanding of and credibility of ‘Free Interaction’ separation for supersonic and hypersonic shock-separated boundary

layers.

Delery and Marvin [29] subsequently provided a comprehensive review, along with a summary of the primary behavior of the ‘Free Interaction’ to parametric variations. Apparent from Delery and Marvin’s

review is that ‘Free Interaction’ theory certainly is not comprehensive in

scope and should not be considered rigorous, as many effects are only

7
approximately inferred from theory, but appears to provide a robust

framework to guide the formation of correlations of experimental data.

The form for the free interaction correlations used in this present

study is that of the free interaction transformations describing the normalized pressure distribution, Pξ(ξ) through the separation region as

given in Equation 3.4 of Delery and Marvin, rearranged as:

Pξ(ξ)=(pw(x) − pw,0)/



2τw,0q0/β, where (1)

ξ = (x − x0)/κξδ∗

0



βτw,0/q0, and (2)

(	sep/δ∗

0) = K(pf − pf,incipient)/pw,0 (3)

Where, the flow variables consisting of the skin friction, τw,0, the

dynamic pressure, q0 = γpw,0M2

e,0/2, and β =



(M2

e,0 − 1) appear in

the relations above. Naturally, for air and N2, γ = 1.4.

The flow variables required for these transformations depend only

on the upstream boundary layer, and so for a given upstream boundary layer, even where the primary shock strength varies with different

experimental case, it is convenient to rewrite these equations as:

Pξ(ξ)=(pw(x) − pw,0)/Aξ, (4)

ξ = (x − x0)/κξBξδ∗

0 (5)

(	sep/δ∗

0) = K(pf − pf,incipient)/pw,0 (6)

Where, Aξ =



2τw,0q0/β, and Bξ = 1/



βτw,0/q0 are obtained using

upstream boundary layer properties, and the variable κξ is fit for each

experiment of the wall pressure distribution through separation region

to the standard curve of [25], but is constant for a particular upstream

boundary layer regardless of whether primary shock strength is allowed

to vary.

For turbulent flow:

Pξ,s = 4.22, at separation, xs, and (7)

Pξ,b = 6.0, for the separation bubble pressure plateau. (8)

Note, that the transformation required to obtain Pξ,s and Pξ,b levels

does not require curve fit, but is determined entirely by the upstream

boundary layer flow parameters, τw,0, q0 and β.

For this current study of the available hypersonic SWTBLI experiments, we make use of the Wilcox EddyBL boundary layer code using the Cebeci-Smith turbulent model to provide the upstream boundary properties for use in these relations (e.g., τwall,0, etc.). Using the

8
EddyBL/Cebeci-Smith approach ensures consistent definition of the upstream boundary layer properties, and also casts these relations as providing a predictive method, rather than depending on the experimental

measured boundary layer properties,

Note, that the growth of the separation bubble, as given by the separation length, 	, is seen to be linear with increase of pf or ‘final’ pressure

of the primary shock system over pf,incipient which is the ‘final’ pressure of the primary shock system for the case of incipient separation, as

might be intuitively expected. The variable K is an experimental curvefit, which may be dependent on Mach number, Reynolds number and

degree of wall cooling.

Erdos and Pallone did not provide a means of estimating the heat

transfer behavior through the free interaction region, but other physics

based discussions indicate that with onset of shock-separation, for the

separation bubble region, the wall heating tends to initially decrease below upstream boundary layer values, even though the pressure rises, but

then rises as reattachment is approached, and then follows the pressurewall-heating correlation described below. However, no trend of wall heating variation in the separation region is clearly established in the various

datasets to date, and so no free interaction theory for wall heating seems

to have gained favor.

1.4.2 Free Interaction Trends

Whether based on free-interaction theory or simply on empirical observation, certain trends can be determined. Generally speaking as the Mach

number increases, the turbulent boundary layer becomes more resistant

to separation and it takes a larger induced pressure rise to initiate separation; and, as the Reynolds number increases (Cf decreases) then the

turbulent boundary layer becomes less resistant to separation, leading to

a lower induced pressure rise to initiate separation. Kuehn [28] based on

extensive experimental data in the range of Mach 1.5-5, albeit mostly of

axisymmetric flare, further states that the effect of heat transfer to the

wall consistently reduces slightly the extent of separation; for axisymmetric cases a thin boundary layer relative to the cylinder diameter tends

to promote separation; and, as might be expected, velocity profiles that

were not fully developed as to turbulence were more prone to separation

than fully developed boundary layers. The effect of bluntness of the flat

plate or nose (entropy layer effect) on separation tended to be accounted

for by using local properties, e.g. δ∗

0 or Cf and Me of the boundary layer

just upstream of the interaction.

Whether such empirically observed trends are indeed correctly incorporated into current CFD methodology is, or should be a concern and is

worthy of future analysis. There does seem to be a need for a comprehensive comparison of these and other semi-empirical correlations with

a modern updated list of hypersonic experiments.

9
The physics based correlations stemming from free-interaction theory, as described in this subsection, provide a basis for behavior trends

that will not be revealed by comparisons of CFD simulations with the

few hypersonics experiment that exist. Also, as the free-interaction work

was mainly accomplished based on supersonic experiments, there does

seem to be a need for a comprehensive comparison of these and other

semi-empirical correlations with a modern updated list of hypersonic

experiments. What may be reasonably inferred from the CFD RANS

hypersonic shock-separated flow simulation results that follow in the

present study is that the understanding reflected in the extensive experimental correlations related to free interaction theory has not been infused

into standard available turbulence models. It is this deficiency in existing

turbulence models that may be a core reason for the high level of statistical variance and uncertainty that is exhibited by CFD simulations of

hypersonic SWTBLI flows.

1.5 SWTBLI Pressure-Heating Correlations

A correlation of the variation of wall heating with wall pressure for

SWTBL interactions was reported by Back and Cuffel [30], and further demonstrated by others, e.g. Holden [31], mainly that throughout

the attached region of a SWTBL interaction-but not within a separation

bubble, if it exists- the heating scales with pressure variation approximately as:

QP85 ≡ Q0 · (p/p0)

0.85, for turbulent 2D SWTBLI (9)

As with the separated region free-interaction relations (for pressure),

the pressure-heating correlation appears to correlate a great deal of experimental data regardless of whether the interaction itself is of the compression corner or impinging shock type.

In that current RANS methods have difficulty predicting wall heating

in the vicinity of reattachment of shock-separated flows, a more accurate

approach for heating analysis of separated SWTBLI may be to postanalyze the CFD simulation pressure rise in the post-reattachment region

using this Q ≈ P0.85 correlation to provide improved post-reattachment

SWTBLI heating predictions as an upper limit on heat transfer rise. The

‘QP85’ correlation is not accurate within the separation bubble itself, but

appears valid from reattachment onwards.

2 Discussion of Results

In the following subsection of this Results Discussion section, we first

update a potentially useful incipient separation correlation for 2D compression corners so as to include the latest datasets included in the hypersonic database of Marvin, et al., [4]. An advantage of correlation ap10
proach in validating CFD, rather than just examining individual cases

is that correlations can best represent collective behavior over a range

of parameters. We then examine several of the 2D SWTBL interactions

for their behavior with respect to Free Interaction Theory, and examine

CFD/SST solutions for their compliance with both experiment and Free

Interaction Theory correlations. Primary observations are made with use

of the Mach 8.2 Compression Corner experimental series of Holden as it is

one of the few hypersonic datasets that includes variations in both shock

strength and Reynolds number. Supplemental observations (both in support of and in contradiction of these observations) of CFD results for the

Mach 2.84 compression corner experimental set of Settles, Bogdanoff and

Vas [10], and of the impinging shock cases of Sch ̈ulein [44], [45], and of

Kussoy [48] are then presented.

2.1 Incipient Separation

Important to methods used in the design and analysis of compressible

flow devices is the determination of the onset of turbulent boundary

layer separation. The onset of turbulent separation can have significant

impact on the aerodynamic and heating performance of such flow devices,

particularly those involving SWTBL interactions.

The shock strength for the onset of separation of 2D and axisymmetric SWTBLI is generally accepted to be primarily a function of Mach

number, but with a weak dependence on Reynolds number. Settles,

Bogdanoff and Vas [10] indicate that for a compression corner at fixed

free stream Mach number, the wedge angle at which incipient separation

is induced will decrease gradually with an increase in Reynolds number,

but at sufficiently high Reynolds number the incipient separation wedge

angle approaches a limit dependent on Mach number alone. Various authors, including Holden, Korkegi, Needham and Stollery, and Babinsky

and Edwards, examining considerable supersonic data, have proposed

correlations for incipient separation, giving either the shock strength,

pinc/p1, or compression corner angle, αinc as a function of Mach number

alone, or a combination of Mach number and Reynolds number. These

correlations should be considered approximate, partly as the experimental determination of incipient separation is itself imprecise. It is useful,

however, to examine the addition of the more recent hypersonic data

appearing in the hypersonic SWTBLI database of Marvin, et al., [4] as

compared against several of these correlations in Figs. 3 through 5. In

these plots, due to the scope of this current paper, only turbulent cases

are considered.

Figure 3 provides a plot of overall pressure rise through the interaction normalized by the initial wall pressure as a function of Mach

number alone for the experiments in the hypersonic database of [4]. The

fully separated cases are solid symbols, and the attached cases are open

symbols. Also shown is a possible correlation line ((pinc − p1)/p1 ≈

11
0.44M exp(−M/2) + 0.13M2.6) which appears to reasonably divide the

compression corner cases into either separated (above the line) or attached(below the line). This correlation line also lies close to the multiequation correlation of Korkegi [37], not shown. Without more data such

a correlation line should be used with considerable caution. The Mach

5 impinging shock cases of Sch ̈ulein, however, is not so easily divided by

this tentative correlation line, with one of the separated cases (10o case)

being below the tentative correlation line. For the impinging shock cases,

the overall pressure rise due to both the incident and reflected shock is

used.

Holden [31] provides, in graphical form, a similar correlation for incipient separation for turbulent boundary layers subjected to 2D and

axisymmetric Shock Wave Boundary Layer interactions. Considerable

data, as of the 1977 date of that publication are also plotted in [31] that

appear to confirm the validity of that correlation. This correlation has,

at most, a weak Reynolds number dependence. Though not given in

Ref. [31], a fit of this correlation line yields approximately the constant

of 0.0165 in the equation:

(pinc − p1)/p1 = M3Cf /0.0165 = (2Mτw/(γp1))/0.0165 (10)

Figure 4 plots this correlation as a dashed line, along with the fully

separated data from Ref. [4] as filled symbols, and the attached data

from this same database as open symbols. Unfortunately, unlike the

data shown by Holden from prior to 1977, the more recent hypersonic

data does not seem to be divided so distinctly by the proposed correlation line. In Figure 4, however, the value for Cf is obtained from

the EddyBL/Cebeci-Smith calculations, whereas Holden either used the

VanDriest II correlation or experimental values.

Figure 5 provides a plot of wedge angle as a function of Mach number

and Reynolds number in the form of α/√

M vs ReL, similar to the plot

of Needham and Stollery [38]. Incipient data from Needham and Stollery

were extracted from their paper and are shown as light grey-filled symbols on this plot, as well as the additional hypersonic data from the

database of [4] with attached (open symbols) and separated cases (filled

symbols) identified separately. A two-part correlation line is also shown

that cleanly separates the attached and separated cases. It is formed

by using essentially the correlation line of Needham and Stollery, which

has α/√

M decreasing in value in the Reynolds number range of 106-107,

but with an extension where there is then a slight increase in α/√

M

in the Reynolds number range above 107. Again, the impinging shock

case of Sch ̈ulein does not conform to this correlation line; the correlation

should only be used with compression corner cases. The correlation line

in Figure 5 (limited to 2D or axisymmetric turbulent compression corner

SWTBLI in the range of ReL between 106 to 150 · 106 and M < 11) is

given analytically as:

12
α/√

M = 10.25/(ReL/107)0.11313; 106 < ReL < 10 · 106, (11)

= 10.25 · (ReL/107)0.0089; 10 · 106 < ReL < 150 · 106 (12)

Data appearing in these figures indicate that these correlations are

imprecise with as much as 2o scatter about the correlation lines indicating

incipient separation and which divide the attached and separation cases.

2.2 Holden Mach 8.2 Compression Corner Shock/

Boundary Layer Interaction

In this subsection, reported are comparisons of our CFD solutions using

both DPLR/SST and CFL3D/SST with the hypersonic compression corner shock/boundary layer data obtained by Holden at a nominal Mach

8.2 in the CUBRC shock tunnel facility. This data is reported by Holden

in Ref [32] and [31], and also in the database of Marvin, Brown and

Gnoffo, Ref [4]. Test data are reported for wedge angles of 27o, 30o,

33o and 36o, providing 2D flat-plate/wedge interactions ranging from

attached to incipient separation to fully separated. While the reported

data in Ref [32] and Ref [4] are at a single unit Reynolds number of

ReU ≈ 137 106/m, Ref [31] provides additional data at two additional

unit Reynolds numbers of ReU ≈ 105 106/m and ReU ≈ 193 106/m,

thereby allowing an examination of Reynolds number effects. An additional compression corner experiment of Holden at Mach 11.3 is also

reported in Ref [32] and [4], giving some insight into the effects of an increase in nominal Mach number. The compression corner for the Mach

8.2 cases was located at 0.99568 meters from the leading edge of the

flat plate, while for the Mach 11.3 case, the 36o compression corner was

located at 1.02362 meters from the flat plate leading edge.

The nominal test conditions for the Holden Mach 8.2 compression

corner cases are given in Tables 2-4, and are used for all calculations

in this report. As the facility used was a shock tunnel, variations occur during and between each run. The reported run-to-run variations

amounted to about 1.5% in unit Reynolds number and free stream Mach

number, and ≈ 4% in free stream temperature. These variations are not

considered significant to the conclusions of this report. Shock relations,

based on NACA 1135 relation, Ref [21], supplemented by the algorithm

of Wolf [22], provide the interaction shock strength, p2/p1, based on the

wedge deflection angle, and are given in Table 5 for each wedge angle.

The boundaries of the grids used for the CFD solutions for the 27o

through 36o Holden Mach 8.2 cases are shown in Figure 6. To reduce

the impact of grid variations on the solutions, the grids for each of these

cases are not independently obtained, but have common origination, in

that they are generated by means of a simple rotational and stretching transformation of the Mach 11.3, 36o grid. All solutions presented

13
made use of grids with 513 grid points in the stream wise direction, with

189 points in the wall-normal, however checks using a higher resolution

grid (1025x377) proved to demonstrate satisfactory grid convergence of

solutions. The number of grid points in the span wise direction varies

with the particular code used: CFL3D using 2; DPLR2D using 1; and

DPLR3D using 4. As the case is 2D, the extra span wise points are

obtained by addition of simple X −Y planes shifted in Z. The 1st point

off the wall is located typically at y+

1 ≈ 0.1. Transition is numerically

specified to occur over the range 0.05-0.2 meters from the leading edge

so as to match experimental heat transfer results.

Incipient separation for this Holden Mach 8.2 case, as evaluated, by

the correlation shown in Fig. 5 should occur for a wedge angle in the

range of 30o ± 1o, but see below for the discussion on sensitivity to

Reynolds number.

2.2.1 Separation Sensitivity to Interaction Strength

Figure 7 depicts the experimental wall pressure distribution through the

separation region in terms of the Free Interaction Variables, Pξ vs ξ for

the four Holden M8.2 wedge angle cases. To accomplish this plot, the

Free Interaction Theory is applied as two independent transformations:

1. The experimental wall pressure data, pw is transformed into the

normalized Pξ variable by means of Eq. 4; and,

2. The corresponding x-locations of these pressure measurements are

transformed into the normalized ξ variable by means of Eq. 5.

To accomplish the transformation of Eq. 4, the upstream boundary

layer wall shear stress, τw,0 as obtained from the Wilcox EddyBL boundary layer code using the Cebeci-Smith turbulence model is used, rather

than experimental data. Similarly, the transformation of Eq. 5, makes

use of the displacement thickness, δ∗

0, from the Wilcox EddyBL/CebeciSmith calculations since measured skin friction and displacement thickness are typically not available for many of the experiments considered

in this report. By using the Wilcox EddyBL/Cebeci-Smith results to

describe the upstream boundary layer, the Free Interaction Theory approach as used within this report is truly predictive (relying on analysis

without direct reference to experiment), rather than postdictive (relying

on post-experiment analysis). The same nominal 2D boundary layer appears for each of these 4 wedge angles, and the EddyBL/Cebeci-Smith

model is quite accurate for flat plate results with complete description

of the boundary layer for these conditions. Also, by consistently using

the EddyBL/Cebeci-Smith code results for the pw → Pξ Free Interaction Theory transformations for all the cases reported in this study, the

impact of experimental uncertainty of boundary layer measurements,

or a lack of sufficient upstream boundary layer experimental measurements(e.g., τw and δ∗), on evaluation of the Free Interaction correlations

is eliminated. A variation from the x → ξ transformation of Eq. 5,

14
is that the location of upstream influence, x0, from the largest wedge

angle, 36o, is used when evaluating the transformation for each wedge

angle, thus giving an indication of the growth of the separation extent.

A curve fit of κξ = 0.30 is found to be required for use in Eq. 5 for this

Holden Mach 8.2 case to best fit the experimental separation pressure

distributions of the 33o and 36o wedge angle cases, to the standard Free

Interaction Pξ vs ξ curve, obtained from Ref [25] also shown in Figure 7.

Values of Aξ = 5726.62, Bξ = 14.55, pw,0 = 10.188kPa, δ∗

0 = 0.6299mm

and X0 = −0.05842m from the 36o case are used for the transformations

leading to Figure 7. These values required for the Free Interaction transformations are accumulated in Table 6 and 7, along with those quantities

for the rest of the experimental cases. Also given in Table 7 is the separation wall pressure, pw,s = pw,0 + 4.22Aξ, and the separation bubble

pressure plateau, pw,b = pw,0 + 6.0Aξ, obtainable by means of the Free

Interaction Relations, Eqs. 4, 7 and 8.

It is seen in Figure 7, that the normalized pressure level of Pξ = 6

is achieved experimentally for both of the fully separated 33o and 36o

wedge angle cases, with the leading edge of these normalized pressure

distributions having shapes quite similar to the ‘standard’ Free Interaction Theory normalized pressure distribution shape of Erdos and Pallone

(shown as a solid line). A similar but slight upward trend in normalized

pressure is also observed for both these experimental cases with increasing ξ though the remainder of the separated region. The experimental

scatter in pressure in the separation bubble region is fairly high at about

20%. Apparent is that the experimental 27o wedge angle case is fully

attached, while the 30o wedge case is at or near incipient separation.

Figures 8-11 provide comparisons, in dimensional form, of wall pressure from RANS CFD with experimental data for the four M = 8.2

wedge cases of Holden. Figures 8 and 9 present results from DPLR/SST,

while Figures 10 and 11 present results from CFL3D/SST. Figure 9 is

an expanded scale version of Figure 8 emphasizing the separation region, while Figure 11 is an expanded scale version of Figure 10, also

emphasizing the separation region. Apparent in these figures are that

the solutions, in comparison to the experiment:

a. The pressure solutions through the separation bubble region, for

both DPLR/SST and CFL3D/SST, agree well with both the normalized Free Interaction theory distribution and level, as well as with the

experimentally observed pressure separation levels.

b. The DPLR/SST solutions consistently indicate early separation

(by ≈ 3o or 10% in wedge angle), then follows with overprediction in

separation extent for each separated wedge angle, but with approximate

agreement with experiment as to observed growth of separation extent

with further increase in wedge angle.

c. The CFL3D/SST solutions, in contrast, agree quite well with the

experimental data as to separation extent for the same wedge angle case

and subsequent separation extent growth with increase in wedge angle.

15
d. Both DPLR/SST and CFL3D/SST indicate separation for the 30o

case, whereas the experiment indicates this case is at or close to incipient

separation.

e. Both DPLR/SST and CFL3D/SST post-reattachment constant

pressure region levels agree quite well with the experimental data for the

particular case, and also with the p2/p1 ratio obtained by means of the

NACA 1135 relations for the primary shock system for that particular

case.

2.2.2 Separation Sensitivity to Reynolds number

To examine the Reynolds variation effect additional M8.2 cases at differing Reynolds numbers for the 33o wedge angle case as presented in [31]

were also calculated with DPLR/SST and CFL3D/SST. These experimental results are shown, along with experimental measurements where

available, in Figures 12 through 16. In these figures, the experimental

separation extent is estimated for each case by use of the pressure distribution, with separation location estimated by Pξ ≈ 4.22. Figure 12 and

13 depict, at two Reynolds numbers, the variation of the extent of experimental separation observed with change in wedge angle or equivalently

primary shock strength. Projection of the separation extent to zero so

as to obtain incipient separation indicates that it occurred at approximately 29o regardless of Reynolds number in the experiment. This is

in excellent agreement with the incipient separation correlation estimate

given earlier.

Figure 14 also shows, for the 33o wedge angle, the significant decrease

in experimental separation extent with an increase in Reynolds number.

Specifically the experimental separation location, as given by Pξ ≈ 4.22,

was approximately 39.37 mm ahead of the compression corner at a unit

Reynolds number of 105.6 · 106/m, and then decreased to 30.48 mm and

20.3 mm ahead of the compression corner at unit Reynolds numbers of

137.1 · 106/m and 193.2 · 106/m, respectively. This represents about a

36% decrease in separation extent for a 37% increase in Reynolds number. Although the separation extent might be expected to scale with upstream boundary layer thickness (or displacement thickness, or momentum thickness), using results from Wilcox/Cebeci-Smith calculations,

the upstream boundary layer thickness only decreases by ≈ 12% over

this Reynolds number range. Thus, the experimental Reynolds number

effect on separation extent exceeds that which could be accounted for by

the change in upstream boundary layer condition.

In contrast, Figure 15 shows that for the 33o case, although the

DPLR/SST solutions gave larger extent of separation for these cases,

the sensitivity to variation of unit Reynolds number of separation extent

predicted by the DPLR/SST solutions was about an order of magnitude

less, being 53.7, 51.2 and 50.7 mm ahead of the compression corner for

these same unit Reynolds numbers of 105.6 · 106/m, 137.1 · 106/m and

16
193.2 · 106/m, shown in the figure as solid blue, black and red lines respectively. In Figure 16, the CFL3D/SST solutions, although in better

agreement with the magnitude of extent of separation, also showed virtually no sensitivity to a change in unit Reynolds number, for the specified

experimental unit Reynolds numbers. Unlike the Wilcox/Cebeci-Smith

calculations for the upstream boundary layer at these Reynolds number

levels, however, both the DPLR/SST and CFL3D/SST solutions show

almost no effect on the upstream displacement thickness and boundary

layer thicknesses, and only a small effect on the momentum thickness.

Figure 17 provides velocity and temperature profiles for the upstream

boundary layer at two of the Reynolds number levels, as obtained from

the DPLR/SST solutions, and indicate that the boundary layer thickness

undergoes little change with the increase in Reynolds number; rather, the

effect of change in Reynolds number mostly affects the inner portion of

the upstream boundary layer for the RANS solutions. It is anticipated

that deficiencies in turbulence modeling that primarily affect the inner

boundary layer will be most felt in surface heating and shear stress,

whereas changes in modeling so as to improve predictions of the extent

of separation would require addressing the entire boundary layer.

To explore further the reduced sensitivity by RANS/SST of separation extent to changes in Reynolds number for this Mach number and

wedge angle, two additional cases were constructed, similar to the experimental Reynolds number variation but with a wider change in Reynolds

number. These additional ‘constructed case’ results are also shown in

Figures 15 and 16 for DPLR/SST and CFL3D/SST calculations at these

two additional Reynolds numbers. In Figure 15 and 16, these additional

cases are shown as the dashed blue line which represents a unit Reynolds

number of 137.1 · 106/

√10/m (or 13.2 · 106/ft), and the dashed red line

representing a unit Reynolds number of 137.1·106·

√10/m (or 136·106/ft),

thereby giving an order of magnitude overall change in Reynolds number, centered about the middle experimental Reynolds number. This

larger Reynolds number variation represents about a 5x larger variation than was explored in the experiment, but seen is that the increase

in separation extent over this larger range approximated that seen in

the experiment. Thus with the wide spacing in Reynolds number, the

RANS/SST solutions provide only 20% the sensitivity of separation extent to Reynolds number than that seen in the experiment. This low

sensitivity of RANS/SST calculated separation extent to Reynolds number could prove problematic in optimization design procedures of such

devices as either hypersonic control surfaces or scramjet inlets, both devices making use of compression corners. Of some further concern to

hypersonic design methods use of embedded RANS/SST analysis is that

with the more narrow spacing in Reynolds number spacing (at the widest

experimental values of Reynolds number), the sensitivity of separation

extent to Reynolds number for the RANS/SST solutions is even smaller,

≈ 10% of the experimental sensitivity. Thus, the RANS/SST sensitiv17
ity of separation extent to Reynolds number is not even consistent over

the Reynolds number range observed. Effectively this is a form of ‘digital noise’ as can sometimes be generated by numerical differentiation of

closely spaced inputs, and which to some MDO optimization methods

could prove unstable. Whether experiment or CFD, certain quantities of

interest such as separation extent or wall heating can be subject to runto-run variation, and place a lower practical limit on how closely spaced

input test conditions should be set before running the risk of the desired

observation of parametric trends sinking into the background level of run

noise.

The DPLR/SST solutions indicate early separation (by ≈ 3o or 10%

in wedge angle) with reduced Reynolds number sensitivity, but with approximately the correct separation extent growth per degree of wedge angle. The CFL3D/SST solutions also exhibit low response of separation

extent to change in Reynolds number. In contrast to the DPLR/SST

solutions, the CFL3D/SST solutions do indicate good agreement with

experiment as to the separation extent along with the correct growth

of separation extent per degree of wedge angle. This differing behavior

between the two codes might suggest that the implementation of the

SST model in these two codes differ in some significant way. However,

options available to each code were exercised to ensure quite similar set

of assumptions (e.g., Vorticity-based production vs. ‘Exact’ production,

1st vs 2nd order accuracy, etc.), with each of these options having insufficient effect to explain the offset in separation onset or insufficient

Reynolds number parametric trend. We observe that for strong viscous/inviscid interaction flows such as these compression corner shock

cases, however, the physics effects of both viscous and inviscid effects

are present and competing for dominance. Thus, it is best to consider

not just the viscous implementation but that differences between these

two codes in numerical modeling exist for both the viscous and inviscid

physics and may prove of equal importance.

2.2.3 Mach Number Sensitivity

Holden also provided a 36o Mach 11.3 fully separated compression corner related to the Mach 8.2 cases. Only the single fully separated case is

available experimentally. As a consequence, Incipient separation is not

determined experimentally for this case, but a comparison of DPLR/SST

and CFL3D/SST with experimental results can prove of value in considering sensitivity of the turbulence model accuracy to Mach number

variations. A comparison of Figures 18 and 19 for wall pressure, indicates that, as with the Mach 8.2 Holden cases, the DPLR/SST solution

overpredicts the separation extent by ≈ 100%, while the CFL3D/SST

solution is in excellent agreement with experiment. Also due to there

being only a single experimental case, no evaluation of sensitivity of the

extent of separation to change in shock strength, compression corner an18
gle, nor to Reynolds number effects can be made. The utility of adding

additional runs for such effects, accomplished at little extra effort or cost,

should be obvious to future experimenters.

2.2.4 Wall Heating

Figures 20 through 22 depict comparisons, for the Holden Mach 8.2 27o36o wedge angle cases at the unit Reynolds number of 137.1 · 106/m condition, of wall heating distributions from DPLR/SST computations with

experimental measurements and the (Q/Q0) ≈ (p/p0)0.85 correlation of

Back and Cuffel [30]. The ‘QP85 heat transfer’ distributions shown in

these figures, are based on the Back and Cuffel correlation and rely on the

wall pressure distributions of Figure 8. The overprediction of separation

by the DPLR/SST computations is evident in these figures, but evident

also is the nearly 50% overprediction or spike in the wall heating past

reattachment for those cases where separation occurs. For the attached

case of 27o wedge angle, the RANS solution prediction of wall heating

agrees with the heat transfer derived from the QP85 correlation from

the experimental pressure measurements, while being consistently high

relative to the experimental wall heat transfer measurements. Either

the QP85 correlation-in which there has proven to be some confidence

outside the separation region- is proving to be too high for these cases

by ≈ 25%, or it is the experimental heat transfer measurements that are

too low. But for the fully separated cases of 33o and 36o wedge angle,

the wall heating is underpredicted through the separation region along

with the ≈ 50% overprediction spike in wall heating peak. Once the

post-reattachment constant pressure region is reached, a discrepancy in

wall heating, between experimental measurements and RANS solution,

is limited to ≈ 20%.

2.2.5 Comments on Observed Turbulence Model Deficiencies

From the above observations of SST calculations for the Holden SWTBL

interaction cases, and from observations previously reported by Brown

[2], several deficiencies in SST model applied to hypersonic SWTBLI

seem clear:

1. Overprediction by the standard SST model for the Wall Heating

spike past reattachment;

2. Overprediction by the standard SST model for the Separation

extent for SWTBLI cases;

3. Lack of proper sensitivity to Reynolds number as to separation

extent;

4. Wall Heating for ‘acreage’/upstream flat plate boundary layer

regions is ≈ 10% high for SST model relative to experiment; and

5. Grid Sensitivity of the standard SST model (at larger y+ values).

19
To explore a possible basis for some of these turbulence model deficiencies, Figures 23 through 25 are provided for the Holden Mach 8.2 36o

case of the DPLR/SST solutions. Figure 23 depicts a streamline array

(in red lines) originating in the upstream boundary layer in the vicinity of separation. Note this upstream boundary layer streamline array

lifts over the recirculation region and reattaches past the compression

corner. Plots of the turbulent kinetic energy development and of the

eddy viscosity along the individual streamlines (as a function of X) of

this streamline array are provided in Figures 24 and 25. The individual streamlines are labeled with their initial upstream boundary layer

distance from the wall in millimeters. In both Figures 24 and 25, the

response of the turbulent field in progressing from the upstream boundary layer (marked as BL in Figure 23) to the initial λ-shock (marked

as λ) seems slow, particularly compared to the later growth while the

viscous layer develops as an adverse pressure wake region located above

the recirculation region, and then followed by a rapid further growth of

the turbulent field to the compression or reattachment shock (marked as

C in Figure 23). The eddy viscosity does rise gradually from X=-0.1 m

to -0.05 m, but the reattachment shock in the vicinity of X=0.05 gives

rise to a substantial rise in eddy viscosity as reattachment is approached.

It is clear that although the SST turbulence model may well mitigate

the buildup of the turbulent field in the vicinity of reattachment (relative

to other two-equation models), as is suggested by Menter and Esch [35],

the problem of high eddy viscosity levels at reattachment remains for

these hypersonic turbulent separating/reattaching SWTBLI cases.

To address the model deficiencies enumerated above might require

a more substantial development of eddy viscosity in the early phases of

separation to promote resistance to separation, while near reattachment

the effect of the high levels of eddy viscosity on heat transfer would need

to be moderated. The overprediction of heat transfer in the vicinity of

reattachment, where eddy viscosity is high suggest the strong linking of

turbulent momentum and energy transport implied by the use of constant turbulent Prandtl number in the current form of the SST model

be replaced by an alternate approach where either separate modeling for

the turbulent heat transfer, or of a variable turbulent Prandtl number

be introduced. Essentially, the turbulent eddy diffusivities for momentum and heat transfer appear to no longer be closely linked through the

SWTBLI separation and reattachment processes and are inadequately

represented in the non-equilibrium regions by a simple constant turbulent Prandtl number. Efforts along these lines have been conducted by

Sommer, So and Zhang [41], by Xiao, Edwards and Hassan [42] and

by others. Additionally, implementation of realizability constraints and

Explicit Algebraic Reynolds Stress modeling may well promote a more

accurate development of the turbulent field at both separation and reattachment.

20
2.3 Settles Mach 2.84 Compression Corner Shock/

Boundary Layer Interaction

The Settles experiment provides cases with a variation in shock strength

for an adiabatic wall 2D compression corner at low supersonic Mach

number of 2.84, which we examine to contrast with observations we have

made from the hypersonic Mach 8.2 results for Holden. As with Holden’s

experiment, the parametric variation provided for by Settles [8, 11] included a variation in shock strength from attached through incipient separation to fully separated; but, was conducted at only a single reported

Reynolds number, and a single reported Mach number. An earlier publication of Settles [10], however, describes a more extensive dataset-mostly

provided in non-dimensional analyzed form-that does include a considerable variation in Reynolds number. The lower supersonic Mach number

of this case, relative to Holden, allows for the potential of a more accurate inviscid contribution to a CFD solution since numerical inaccuracies

associated with inviscid total Pressure loss through the shock structure

will likely have reduced impact on the numerical accuracy of the overall

CFD solutions. Furthermore, as it is an adiabatic wall supersonic case,

the accuracy of heat transfer modeling should play a reduced role in

overall accuracy of CFD solutions.

Navier-Stokes solutions were accomplished for the Settles configuration and test conditions using both DPLR and Cfl3d with the SST twoequation turbulence model. The grids were a single block of 1025x289

grid points and are based on a simple geometric transformation of the

same basic compression corner grid shown in Figure 6 and mentioned previously in the Holden section above, but adjusted for test plate length,

grid spacing at the wall and wedge angle to conform to requirements for

the Settles cases. The grid spacing at the wall was y+

1 ≈ 0.05.

As the Settles test was run with a wedge mounted on a wind tunnel

wall rather than on a flat plate suspended in the wind tunnel, it was necessary to adjust the flat plate run length for the CFD grid so as to result

in a best match the measured upstream boundary layer properties. As

a consequence, the CFD grid was set at L = 1.8m (similar to the 1.98m

nozzle to wedge run length in the experiment) with EddyBL/CebeciSmith computations giving τw = 145 Pa, δo = 23.7 mm, δ∗

o = 6 mm

and θo = 1.15 mm compared to the experimental measured τw = 141.7

Pa, δo = 23.0 mm, δ∗

o = 6.1 mm and θo = 1.2 mm for the 20o case

(measurements for the other wedge angles differed slightly).

First, discussing aspects of the Settles experiment, we consider the interaction strength for onset of separation. In Settles experiment, the 16o

case is reported to be ‘nearly incipient separation’, while the 20o and 24o

cases are reported as fully separated. By the correlation shown Figure 5,

the incipient separation demarcation would occur at αi ≈ 10.25√Me ≈

17.3o, within range of accuracy of that correlation. DPLR/SST solutions were obtained at various wedge angle cases for the Settles exper21
iment and intervening wedge angles as well, with the onset of separation occurring at 14o (not shown), approximately 2 − 3o early, similar

to the performance of DPLR/SST solution for the Holden experiment.

The role of inviscid modeling inaccuracies relative to viscous modeling

is likely reduced for the Settles case compared to the Holden; hence, it

appears likely the nearly constant 3o of early incipient separation seen

in DPLR/SST across this wide Mach number range points to the SST

model or its implementation.

Next, the Free Interaction transformed pressure distribution is presented in Figure 26, for Settles experimental measurements of the fully

separated 20o and 24o wedge angles. As can be observed, the experimental wall pressures in the separation bubble region conform quite well

to the pressure levels and leading edge shape of transformed pressure

distribution. In terms of the transformed variables, the growth of the

separation extent over the range of this 4o increase in wedge angle corresponds to Δξ ≈ 0.5, giving ∂ξ0/∂αi ≈ 0.125, which considering the low

accuracy of the curve fit of the ξ transformation, is within range of the

∂ξ0/∂αi ≈ 0.3 seen in Figure 7 for the Holden cases.

Figure 27 compares the DPLR/SST and experimental wall pressure

distributions for the Settles 16o, 20o and 24o cases. Indicated is that, as

with the Holden Mach 8.2 case, DPLR/SST is predicting early separation

by about 3o, or ≈ 10% in wedge angle. Others, e.g. Ref. [50], also show

the SST model as over-predicting the separation extent for the 24o Settles

case (note, in Ref [50] their separation extent for their calculations and

reference to the experiment appears to be in terms of x/δ, with δ = 2.3cm

from the paper by Settles [11]). However, Figure 28 demonstrates that

this early separation leads to a quite similar overprediction of separation

for the fully separated 20o case for both the DPLR/SST and Cfl3d/SST

solvers. This is in contrast to the results for the Holden Mach 8.2 case

for DPLR/SST vs Cfl3d/SST results shown in Figure 9 vs Figure 11, in

which is indicated that although the DPLR/SST overpredicts separation

comparable to an increase in wedge angle by 3o, the Cfl3d/SST solver

was quite accurate as to separation extent. Thus, the over-prediction in

separation extent shown by DPLR/SST is consistent across these two

cases, while the Cfl3d/SST prediction varies in its accuracy. Among the

possible reasons for this is the acknowledged experience that the SST

model tends to overpredict separation, but this may in part by offset

by different inviscid modeling of these two codes in their ability to deal

with the weaker vs stronger shocks of Settles vs Holden cases. The

DPLR/SST results are more consistent in overprediction regardless of

the shock strength, while the Cfl3d/SST results seem to actually improve

with increase in shock strength. For these strong interaction flows, it is

both the inviscid numerical implementation and the viscous numerical

implementation which must interact together with accurate modeling of

all physical features of the SWTBL interaction.

In Figures 29, 30 and 31 is explored the sensitivity of the separa22
tion extent to Reynolds number variation. Figure 29 presents, in nondimensionalized form, separation extent data and correlation extracted

from Figure 17 of Ref. [10] and which incorporates both Reynolds number

and wedge angle effects for the Settles Mach 3 compression corner experiments. This correlation should still apply to the later data of Ref [11].

Figures 30 and 31 provides a comparison of the measured wall pressure

distribution in the vicinity of separation of the 20o case from Ref. [11]

with computed wall pressure distributions obtained with DPLR/SST

and Cfl3d/SST, for the single Reynolds number of that dataset, but

supplemented by additional Reynolds number cases which should be addressable by means of the Settles correlation.

Figure 29 describes Settles 1975 data’s non-dimensionalized upstream

influence extent dependence on both Reynolds number and wedge angle in one plot, along with a correlation for these variables specific to

Mach ≈ 2.84 Settles experiments. The range of Reynolds numbers runs

from Reδ0 = 0.52 106 to 7.64 106 (with Rex up to ≈ 109), and wedge

angles from 10o to 20o. ‘Upstream influence’ extent is as obtained from

measured wall pressure distributions as the location in front of the compression corner at which the interaction pressure rise begins (see Ref. [10]

for details), and is presented in non-dimensional form as ΔX/δ0. The

B variable in this figure characterizes the wedge angle and Reynolds

number of the particular case through the definition:

B ≡ α + 20. − 1.26 · log Reδo (13)

The correlation that Settles obtained, was:

ΔXu/δo ≡ 0.03 + 0.0016 · e0.288·B (14)

Emphasis must be made that this correlation should only be considered as verified for use with Settles M ≈ 3 data. We should be able to

make use of this correlation, however, to ‘extend’ the single Reynolds

number for the 200 separated Settles case as considered in Figures 30

and 31 to cover a range of Reynolds numbers (where the single experimental Reynolds number is scaled by 0.316x, 0.562x, 1.778x and 3.16x

for ‘constructed cases’ as well as the actual single measured Reynolds

number as used from Ref. [11] and labeled as Re1.000x in these figures).

Considering, then Figures 30 and 31, computed wall pressure distributions from both DPLR/SST and Cfl3d/SST for the 20o case agree

with each other for the measured Re1.000x case and as to the additional ‘constructed’ Reynolds, in that they both overpredict the extent

of separation in the measured Re1.000x case by effectively ≈ 3o in wedge

angle, and also as to the effect that the constructed Reynolds number

cases influence their extent of separation observed in the wall pressure

distributions. This is in contrast to the Holden Mach 8.2 case considered previously(contrast Figs 30 and 31 with Figs 15 and 16), where the

23
experimental Reynolds number sensitivity of separation extent was not

observed by RANS/SST.

To determine whether the magnitude of Reynolds number sensitivity

is consistent with the Settles correlation, we apply the Settles correlation

so as to determine the anticipated upstream influence extent for these

cases. Relative to the Reynolds number of the measured 20o case of

Settles [11], the ‘constructed cases’ computed in Figs 30 and 31 have

Reynolds numbers of Re = 0.3162x, 0.562x, 1.000x, 1.778x and 3.162x

or a variation in Reynolds number of ± 1/2 order of magnitude. By

the correlation of Settles, the relative extent of the separated region for

these cases should be ΔX/ΔXRe1.000x ≈ 1.634, 1.277, 1.000, 0.7721 and

0.6071, respectively. In Figure 30 for DPLR/SST and Figure 31 for

Cfl3d/SST, the computed separation extent is slightly larger than these

values by a mere ≈ 10%. Thus, at Mach 2.84, the computed separation

extent is slightly more sensitive to Reynolds number variation, whereas,

in contrast, for the Mach 8.2 Holden case the computed separation extent

was inaccurate and quite insensitive to Reynolds number variation.

Another observation available from Figures 27-31, is that the magnitude of the DPLR/SST and Cfl3d/SST computed wall pressures in the

separation region agree quite well with the corresponding experimentally

measured wall Pressure magnitudes, and also the pressure levels (either

pw,b ≈ 46.476kPa or equivalently p/p0 ≈ 1.97) that should occur from

Free Interaction Theory for the Settles experimental conditions as indicated in Table 7. Overall the Settles Mach 2.84 cases appear to be

reasonably well predicted by both the DPLR/SST and Cfl3d/SST methods, except for early incipient separation, followed by a nearly constant

overprediction in separation extent.

However, the troubling change in Reynolds number sensitivity behavior, from well-behaved at Mach 2.84, to inaccurate at Mach 8.2 is significant to potential use of so-called high-fidelity Physics based methods

(RANS) in design and MDO methods. This Reynolds number sensitivity inconsistency with Mach number is not fully understood at this time,

but none of the possible origins can be eliminated, whether traceable to

viscous or inviscid modeling applied to the interaction or the upstream

boundary layer computations, or even grid requirements. DPLR/SST

exhibits a consistent early incipient separation by ≈ 3o in wedge angle,

whereas there is an actual improvement in Cfl3d/SST where at Mach

2.84 early separation by ≈ 3o exists, but which does not appear at Mach

8.2.

2.4 Sch ̈ulein Mach 5 Impinging Shock/Boundary Layer

Interaction

The Sch ̈ulein impinging shock/boundary layer interaction flow topology

is shown in Figure 2. This experiment (reported in Refs. [44] and [45],

also in the database [4]) was conducted at a fixed unit Reynolds number

24
of 38.8 106/m, and Mach number 5. A 2D boundary layer developed on

the test surface consisting of a flat plate with a shock generator positioned above the flat plate and inclined so as to generate a shock that

impinged on the flat plate boundary layer at approximately x = 0.35m.

Various shock angles of 6o, 10o and 14o were generated so that impinging

SWTBL interactions occurred on the flat plate ranging from attached

to incipient separation to fully separated. Nominal run conditions for

the Sch ̈ulein experiment are presented within the Tables 1-3. The several cases primarily differed in shock strength only, with no deliberate

variation in Reynolds number nor Mach number.

Extensive examination of DPLR/SST solution uncertainty for this

flow was examined in the prior publication by this author [2, 3], as well

as by others, e.g., Fedorova, et al., [46] and Zhang, et al., [50]. (Note that

Zhang, et al., use of ‘x’ appears to be actually the non-dimensionalized

x/δ0). In this subsection we provide additional examination of the experiment with respect to the Free Interaction Theory formulation, and

of comparisons with both DPLR/SST and CFL3D/SST solutions.

The DPLR/SST and Cfl3d/SST solutions were accomplished with

the grids reported and available from Ref. [4]. These 3 block grids,

being 1056x256 cells for the 6o fine-grid case, 528x128 cells for the 6o

medium-grid 6o and 10o case, 1056x256 cells for the 14o medium-grid

case, and 2112x512 cells for the 14o fine-grid case. The number of cells

span wise depended on grid requirements of each particular code for

these essentially 2D cases.

Figure 32 presents a comparison of the Free Interaction transformed

experimental wall pressure distribution, Pξ vs ξ, for the fully separated

14o impinging shock case with the nominal theoretical distribution. Note

the experimental Pξ distribution does demonstrate a continued slight rise

rather than a strictly constant level past separation (Pξ ≈ 6, See also

Fig. 26; however, within the accuracy of the fit, and of the experimental

wall pressure accuracy, the experimental wall pressure nearly follows the

‘ideal’ Free Interaction wall pressure distribution shape and magnitude

within the separation region. This helps to substantiate the accuracy

of the experimental separation region wall pressure measurements, inasmuch as the computed DPLR/SST and Cfl3d/SST separation wall pressure levels are observed from Figs 33 and 34 to both be significantly lower

(≈ 1/2−2/3) relative to the experiment. This behavior of low computed

separation bubble pressure is seen to be consistent with virtually every

other Navier-Stokes/SST computation for this case, e.g. Steelant [51]

and Kovar [52], except that of Fedorova, et al., [46, 47]. Figure 33 shows

the DPLR/SST wall pressure distribution comparison from DPLR/SST

solutions with experiment for all three of the impinging shock generator

cases, 6o, 10o and 14o. Figure 34 provides a comparison of DPLR/SST

and Cfl3d/SST wall pressure distribution with experiment for the 14o

case. Seen is that the only significant region of disagreement in wall

pressure distribution between CFD and experiment is for the separation

25
region. In the separation region, DPLR/SST and Cfl3d/SST both underpredict the pressure level by approximately 35% − 50%, and, unlike

the compression corner cases discussed above, both slightly underpredict

the separation extent.

All three experimental cases, whether attached or separation, exhibit

a rise in pressure through the interaction region to an extended postreattachment region of constant pressure level within excellent agreement of the inviscid level predicted by simple inviscid analysis of the

2-shock (impinging and reflecting shocks) problem, see Brown [3] for

further description of the double-shock solution. Using the p2/p1 analysis entries from Table 5 for each of these cases, this post-reattachment

region pressure level should be by inviscid analysis: 16.19kPa for the

6o case, 32.82kPa for the 10o case, and 58.68kPa for the 14o case. In

Figure 33, experimental and CFD wall pressures within this extended

post-reattachment constant pressure region are in excellent agreement

amongst each other and as to the inviscid double-shock problem analysis.

This extended post-reattachment constant pressure region is terminated

(with decreasing wall pressure) by the expansion fan emanating off the

shock generator trailing edge. Thus, the ‘footprint’ of the region between

where the impinging shock impinges the boundary layer (x ≈ 0.35m for

all three cases) and the earliest impingement of the expansion fan (for

the 14o case at x ≈ 0.45m) is about a duration of 0.1 meters minimum,

or about Linteraction/δo ≈ 21, using δo = 4.813mm from Table 4. This

appears to be sufficient stream wise distance required for the interaction

region to express full development of the various flow structures-including

separation and an extended post-reattachment constant pressure region.

Hence, we refer to this case of Sch ̈ulein as being a ‘long’ or ‘fully developed’ impinging SWTBLI. We will briefly show an example of a ‘short’

or ‘early termination’ impinging SWTBLI in the next subsection.

Figures 35-37 depict comparisons of wall heat transfer distributions

for the three experimental cases. Figure 35 compares the DPLR/SST

computed wall heat transfer distributions with the experimental measurements. Figure 36 compares the wall heat transfer as computed from

DPLR/SST wall pressure results based on the QP85 correlation with

experimental measurements. And, finally, in Figure 37, a comparison

of QP85 correlation based wall heat transfer is presented based on both

the DPLR/SST and measured experimental wall pressure distributions.

What is clear from these three figures is that, the measured wall heat

transfer throughout most of the interaction is better predicted-based

on DPLR/SST pressures-by a post-analysis using the QP85 correlation

than by direct use of wall heat transfer results from the DPLR/SST solutions, see also Brown [3]. The direct wall heat transfer predictions by

DPLR/SST is seen to produce an overshoot by nearly 50% past reattachment for the separated case, before following a downward trend-yet remaining well above-the measured wall heat transfer throughout the postreattachment constant pressure region. The wall heat transfer within the

26
separation region is not predicted by either the direct DPLR/SST wall

heat transfer results, nor the wall heat transfer obtained by the QP85

correlation whether from the DPLR/SST solution or as an indirect inference from measured wall pressures. The QP85 correlation should not be

used within the separation region, but post-reattachment seems to perform better than the current state-of-art in CFD turbulence modeling

for wall heating. We do observe in Figures 35 and 36 that the measured

wall heat transfer, particularly for the 14o case undergoes a gradual decline in magnitude throughout the post-reattachment constant pressure

region. This is likely due to a natural stream wise development of an

otherwise nearly equilibrium boundary layer-with increase of boundary

layer thickness and consequent reduction of wall heating- occurring for

this constant pressure region.

To explore the possible physical basis for the low separation pressure,

as well as for the post-reattachment wall heating overshoot, we consider

the contour plots in the region of the Sch ̈ulein 14o interaction from both

DPLR/SST and Cfl3d/SST solutions presented in Figures 38 and 39.

These two plots are Pressure contour plots (flooded contours in color,

with solid black contour lines), supplemented by streamlines in red. Figure 38 shows an expanded scale of the DPLR/SST separated region,

while Figure 39 shows (to the same expanded scale) of the Cfl3d/SST

separated region. The Cfl3d/SST pressure contour plot is shifted slightly

in X by 4mm to a lower X value to account for the Cfl3d/SST separated

region being located slightly earlier than is the DPLR/SST separated

region. The size of the two separated regions are actually quite similar.

The shift to lower X of the Cfl3d/SST separated region is due to the

inviscid solver of Cfl3d generating a somewhat steeper shock than does

the inviscid solver of DPLR.

The flow topology of these two solutions is generally quite similar, but

differ in detail. Of particular note in both of these contour plots are the

4 streamlines in red that are placed at the same location in the upstream

boundary layer: Y0 = 1, 2, 3 and 4 mm away from the wall, starting at

X = 0.316m. In both these contour plots, these streamlines demonstrate

the upstream boundary layer lifting away from the wall at separation,

being turned by the λ-shock so as to become a wake-like viscous layer

bounding the recirculating region from above, encountering the shock

triple point, then followed with an expansion wave just past the peak

height of the recirculating region, where the wake-like viscous layer is

turned sharply back to the surface to close the recirculating region at the

reattachment point. The reattached boundary layer past reattachment

is significantly reduced in thickness (along with the increase in density)

compared to the upstream boundary layer, and encounters an adverse

pressure gradient as it continues past reattachment.

Correct computation of this sequence of events places a considerable challenge on both the viscous and inviscid numerical modeling. It

can be seen from the pressure color flooded contours past the expan27
sion wave, that the separation bubble pressure appears to be consistent

with and possibly established by the pressure level (seen in ‘light blue’

in these color flooded contour plots) generated ‘above’ and to the ‘backside’ of the recirculating region as the separated viscous wake traverses

between the expansion wave and the reattachment point. As to the inviscid solvers, DPLR and Cfl3d, more detail and a higher peak pressure

seems to be generated by Cfl3d just past the expansion wave and above

the recirculation region, as experienced by the 4 streamlines in red being discussed. These two codes differ as to inviscid flux function, DPLR

being a modified Steger-Warming, while Cfl3d solutions used a mixed

Roe-Van Leer formulation.

While the inviscid formulation of these two solvers accounts for some

of the differences between their solutions in the separated region, another

apparent possible source of discrepancy between the CFD solutions and

the experiment likely to arise is from the extensive turning occurring

with the streamlines in the vicinity of the pressure peak at the triple

point-expansion wave region located at the peak of the recirculation region. This extensive turning by this viscous layer suggests the need for

curvature terms to be incorporated into the turbulence modeling, not

present in either the DPLR or Cfl3d SST formulations. Whether incorporation of curvature terms into the turbulence model would improve

the separation bubble pressure prediction is yet to be explored.

Another concern is the overshoot in wall heating past the reattachment location (at about x = 0.36m). Figure 40 attempts to explore that

issue, supplementing information observable in Figure 38. Again we refer to the 4 streamlines located in the upstream boundary layer visible

in Figure 38, and located initially at Y0 = 1, 2, 3 and 4 mm away from

the wall, starting at X = 0.316m. Note the curve associated with the

streamline closest to the wall (labeled Y0 = 1mm) is colored black in

Figure 40, while the rest are colored red, to facilitate deciphering the

plot. Figure 40 follows the development of DPLR/SST turbulent kinetic

energy along each of these streamlines through the interaction region

from x = 0.3m (before separation at x ≈ 0.32m) to past reattachment

(at x ≈ 0.35m and past the wall heat transfer overshoot (at x ≈ 0.36m).

Seen is a rapid growth in turbulent kinetic energy as the viscous layer

first separates then converts into a wake-like viscous layer experiencing

first the λ-shock followed by a slight adverse pressure gradient. A slight

decrease in TKE then occurs at about x = 0.34, believed to be associated with favorable pressure gradient experienced at the upper boundary

of the recirculating region, followed by a rapid increase in TKE as the

adverse pressure gradient is encountered by these streamlines as they

approach the reattachment location.

An unfavorable feature of two-equation eddy viscosity models is their

over-prediction of turbulent energy production in encounters with decelerations involving predominately normal strains such as shocks, stagnation and reattachment points. A primary contributing factor to this

28
overproduction is that two-equation ‘K’-based models, of necessity, assume the normal stresses are isotropic, which when simply multiplied

by the divergence of the velocity field numerically produces a high turbulence production. In contrast, a realistic turbulence field -anistropic

and subject to realizabilty constraints- subject to pure normal stresses

will produce a considerably lower turbulence production. Coupled with

the oversimplification of the connection of eddy diffusivity for momentum transfer to heat transfer invoked by the use of a constant turbulent

Prandtl number, it is not surprising that the overshoot in wall heating

past reattachment occurs.

Additionally, Simpson [53] established the important role that the

difference in normal stresses, (u2 − v2), and not just the shear stress

uv, can play in both the momentum and turbulent equations for the

two-dimensional separation process. Conventional two-equation models,

of course, will incorrectly yield zero for such normal stress formulations.

By these observations and physical arguments, it seems unlikely that

simple modifications to two-equation eddy viscosity models, such as for

even the SST model, making use of constant turbulent Prandtl number

will lead to a robust physically accurate prediction of wall heating near

reattachment points. At present the simplest advanced anisotropic modeling of turbulent fields appears to be EARSm models such as that of

Wallin and Johansson [54], or Hellsten [55, 56]. To decouple the eddy

diffusivity for momentum and heat transfer in a manner more advanced

than a simple constant turbulent Prandtl number has been examined by,

for example, Xiao, et al., [42, 43].

2.5 Kussoy Mach 8.2 Impinging Shock/Boundary Layer

Interaction

In this subsection, we briefly examine the Kussoy Mach 8.2 Impinging

Shock/Boundary Layer Interaction as an example of a ‘short’ or ‘premature termination’ Impinging SWTBLI, so as to contrast with the Sch ̈ulein

‘long’ or ‘fully developed’ Impinging SWTBLI.

The experiment of Kussoy and Horstman, Ref [48], is that of a 2D

hypersonic impinging shock generated by a tilted flat plate place above

a flat plate test surface. The test was conducted at a nominal Mach

number of 8.2, with interactions ranging from attached to incipient separation to fully separated provided by variety of shock generator angles:

5o, 8o, 9o, 10o and 11o. For the purposes of this report, we restrict

ourselves to the 10o fully separated case, conducted at test conditions

(given in Tables 1-3 for the B3.Kussoy entry). The test unit Reynolds

number was 4.91 106/m, with the shock impingement occurring at about

x = 0.31m. The upstream wall pressure was p1 = 430Pa, total Pressure was PT = 24.95MPa. Kussoy reports the experimental upstream

wall heat transfer as Q0 = 1.04W/cm2, while the EddyBL/Cebeci-Smith

boundary layer code gives Q0 = 1.128W/cm2. Kussoy also reports the

29
experimental boundary layer thickness, δ0 = 37mm and δ∗

o = 15.9mm,

while the corresponding EddyBL/Cebeci-Smith values (from Table 4) are

δ0 = 39.6mm and δ∗

0 = 17.7mm, and from Figure 41, the DPLR/SST

boundary layer thickness is δ ≈ 21mm. Because the actual test geometry

was being calculated (unlike the Settle test conducted on a tunnel wall),

no closer match in upstream boundary layer properties was attempted.

Figures 42 and 43 (from Ref [2,3], compare wall pressure distributions

and wall heat transfer distribution respectively, for DPLR computations

with several turbulence models, including SST, against the experimental

data obtained by Kussoy. Excellent agreement as to the pressure distribution between computations and experiment is observed, except for details in the separation region. The separation extent for SST is about 2/3

of that obtained in the experiment. Also the SST separation bubble appears to be low relative to the experimental level. Good agreement as to

the wall heat transfer distribution is seen in Figure 43 between the computations and measurements, except the the peak value for SST model

results exceed the peak value from measurements by approximately 30%.

However, in contrast to the the ‘fully developed’ Sch ̈ulein wall pressure

distributions which exhibit an extended post-reattachment region of constant wall pressure corresponding quite accurately to the pressure rise

through both the impinging and reflected shocks, the Kussoy wall pressure distribution, of Figure 42, does not exhibit such behavior, being

more of a bell-shaped curve for both wall pressure and wall heating. For

the Kussoy 10o case, we anticipate the double or impinging/reflection

shock pressure rise to result in a value (from Table 5) of approximately

p2/p1 = 19.155. This rise in pressure ratio level is just reached in Figure

42, but immediately followed by a rapid decline.

The Pressure and Mach contours of Figure 44 and 45 explore why the

anticipated region of constant high pressure does not occur for the Kussoy experiment. Both of these figures concentrate on the flow interaction

about the separation/reattachment region, extending from before where

separation occurs (at x ≈ 0.28m) and the impinging shock encounters

the outside edge of the boundary layer (at x ≈ 0.308m) until just past

where the wall pressure peak is observed (at x ≈ 0.41m).

Note, observable in both Figures 44 and 45, is that the expansion

fan from the trailing edge ([x, y]T E ≈ [0.3, 0.048]m), first encounters the

reattaching boundary layer edge in the immediate vicinity of the wall

pressure peak (at x ≈ 0.41m). The interaction, which extends from

x = 0.28m to x ≈ 0.41m has just sufficient downstream development

distance available to reach the full pressure rise before the expansion fan

‘terminates’ the interaction. Thus, this Kussoy case is not truly ‘prematurely terminated’ by the expansion fan, but rather is at the defining

edge between ‘fully developed’ and ‘prematurely terminated’. Were the

Kussoy 10o shock generator any shorter, there would not likely have been

full development and the full pressure rise would not have occurred. Regardless, there is not sufficient development length available for a post30
reattachment constant pressure region similar to that of Sch ̈ulein case to

occur.

As a further observation, an attempt to fit the experimental wall

pressure distribution in the separation bubble by means of the Free Interaction Theory transform equations similar to the prior discussions with

Holden, Settles and Sch ̈ulein, failed for this ‘short’ impinging shock interaction. Possibly this is because the reflected (or reattachment) shock

observed in Figures 44 and 45 is distorted by the close proximity to the

expansion wave leading to the back-side of the separation bubble to be

bounded by relatively lower pressure (pb ≈ 1kPa) than would otherwise

occur(pw,b ≈ 2.2kPa by Table 7 for Free Interaction Theory). This case

does not seem to fit well with the standard Free Interaction theory due

to the early termination of the interaction by the Trailing edge expansion

fan leading to a distortion of the reflected shock and related flow on the

backside of the separation bubble and the consequent absence of a developed post-reattachment constant pressure region. Note that the distance

along the test surface between the footprint of the impinging shock to

that of the expansion wave is only ≈ 10cm for this ‘thickBL/Short Generator’ (δ0 ≈ 2.1 − 3.7mm upstream boundary layer) interaction, giving

Linteraction/δ0 ≈ 3 − 5, compared to that of δ0/Linteraction ≈ 21 for the

‘thinBL/Long Generator’ Sch ̈ulein case. Oddly, however, most of the

flow-except the separation region- is actually well represented by the

CFD results, possibly since the flow is now more dominated by inviscid

features.

3 Summary

We have examined the performance of independent implementations of

the SST model into two NASA CFD codes as applied to hypersonic 2D

turbulent compression corner and impinging shock interactions.

A detailed examination of the sensitivity of the SST turbulence model

to test parameters when applied to 2D hypersonic shock-wave/turbulent

boundary layer interactions, and as implemented in the DPLR Real-Gas

Navier-Stokes code, has been accomplished. The primary test condition

parameters considered are those of Mach number, Reynolds number,

wedge angle or shock strength. Supplemental calculations were accomplished with the Cfl3d Perfect-Gas Navier-Stokes code so as to isolate

the influence of turbulence model implementation. Emphasis was placed

on computations of the experimental cases of Holden’s Hypersonic Mach

8.2 compression corner, and of Settles Supersonic Mach 2.84 compression corners as each of these provide extensive variation in both Reynolds

number and shock strength at two extremes of Mach number. Two additional cases of impinging hypersonic shock wave turbulent boundary

layer interactions were accomplished so as to provide additional insight

on the performance of the SST turbulence model relevant to scramjet

31
inlet design and analysis.

Supporting information upon which these detailed examinations were

accomplished included engineering based correlations and non-dimensional

pressure distribution transformations based on Free Interaction Theory.

A discussion of Free Interaction Theory leading to a simplified algorithmic formulation for application to these present cases, as well as an

update of an incipient separation correlation based on recent hypersonic

data also facilitated this evaluation of the widely used SST turbulence

model’s ability to accurately predict separated hypersonic SWTBLI.

Overall, the primary observed deficiencies in the computed solutions

were found by comparison of the CFD results with experiment and include:

1. overprediction of separation extent by SST for compression corner

SWTBLI;

2. underprediction of separation bubble pressure by SST for impinging SWTBLI;

3. overprediction of post-reattachment wall heat transfer by SST for

both impinging and compression corner SWTBLI;

4. overprediction of wall-heating for ‘acreage’ boundary layer, along

with inferior post-analysis methodologies of the RANS CFD for upstream

boundary layer thicknesses; and

5. loss of sensitivity to Reynolds number variations of the separation extent at high Mach numbers for the compression corner case. This

contrasted with excellent tracking of Reynolds number variations of separation extent at lower Mach numbers.

The above enumerated behaviors for SWTBL interactions are unlikely to be corrected by simple changes in the SST turbulence model, as

corrections in a simple eddy-viscosity model needed for each of these areas of disagreement are contradictory in their effect. It is suggested that

the most promising avenue of exploration for improvement would be for

incorporation of an Explicit Algebraic Reynolds Stress model, EARSm,

as such a model retains the simplicity of the two-equation models, yet

with the improved constitutive relation providing the full Reynolds stress

tensor, the known deficiencies of two-equation eddy viscosity models in

over-predicting turbulent production for reattaching/stagnation point regions and inability to incorporate known effects of the normal strain

terms in separating flows can be addressed. Furthermore, it is suggested

the turbulent heat transfer modeling be decoupled from turbulent momentum transfer modeling as the use of a constant turbulent Prandtl

number as used with current two-equation models closely couples eddy

diffusivity for heat transfer with that for momentum transfer and prohibits independent correction of the several major model deficiencies

observed (too high heat transfer, but insufficient resistance to separation). Additionally, consideration of the impinging shock flow structures,

specifically as the response of the separated viscous layer at the top of the

recirculation region may play a significant role in the under-prediction

32
of separation bubble pressure, reveals the probable need for inclusion of

curvature correction terms into the turbulence models applied to these

flows.

For compression corner cases, the separation extent tended to be

over-predicted relative to experiment, probably associated with a tendency to predict early incipient separation regardless of Mach number.

DPLR/SST tends to separate too early, by about 10% or 2o-3o in wedge

angle for incipient separation for compression corners. The effect of

this continues with an over-prediction of separation extent for compression corner wedge angles past incipient separation. This contrasts with

CFL3D/SST which consistently agrees with experimental separation extent for all compression corner angles for the Holden Mach 8.2 cases.

A similar result is observed for the Mach 3 Settles experiment, except

both Cfl3d/SST and DPLR/SST tend to separate by about 2o in wedge

angle too early. A change in CFL3D from thin-layer to full-NS resulted

in no change in separation size, while a change in CFL3D turbulent production term from approximate (vorticity-based) to ‘exact’ strain based

turbulent production led to a decrease in separation extent.

At lower Mach number, the DPLR/SST and Cfl3d/SST sensitivity to

Reynolds number variation appeared to track the experiment and related

correlation of Settles, however, at higher Mach number the SST results

became quite insensitive to Reynolds number variation in contrast to

the observed experimental tendency. The predicted separation bubble

pressure was reasonably well-predicted with respect to both experiment

and Free Interaction Theory, for the compression corner cases regardless

of Mach number and Reynolds number.

For the impinging shock cases, two sub-types of such interactions

were observed experimentally: a. ‘thin Boundary Layer/long shock generator’ case, represented by the Mach 5 Sch ̈ulein experiment; and b.

‘thick Boundary Layer/short shock generator’ case, represented by the

Mach 8.2 Kussoy and Horstman experiment. For the Sch ̈ulein case, it

was found that the expansion wave off the trailing edge of the shock generator terminated the interaction only after the interaction had become

‘fully developed’ and followed by an extended post-reattachment region

of constant high pressure level. In contrast, for the Kussoy case, the

expansion wave terminated the interaction soon after reattachment with

the pressure rise only just reaching its’ peak value prior to the decline in

pressure associated with the expansion wave.

For the Sch ̈ulein case, the experimental separation bubble pressure

distribution was found to conform quite well with Free Interaction theory

as to shape and magnitude level. Also, the CFD/SST computed separation extent was in good agreement with the observed experimental measured extent. However, the CFD/SST computed separation bubble was

only about 1/2 of the experimental level (and the level predicted by Free

Interaction Theory). The DPLR/SST and Cfl3d/SST both agreed with

each other as to their basic behavior on the Sch ̈ulein impinging shock

33
experiment. Possible reasons for the observed disagreement between the

numerical and experimental results are considered. It is suggested that

both the influence of the inviscid solver and inadequate representation

of normal stress effects contribute to the under-prediction of separation

bubble pressure for the impinging shock cases.

For the Kussoy case, the experimental separation bubble proved not

to agree with the Free Interaction theory, being at considerably lower

pressure, likely due to the premature termination of the interaction by

the expansion wave distorting the reflection/reattachment shock. An

overshoot in CFD computed wall heat transfer also occurred, although

not as severe as for the compression corner cases.

These observed deficiencies in RANS/SST solutions serve as a cautionary note to those attempting to rely on CFD methods directly embedded as the fluids analysis engine in MDO optimization methods. Such

an approach can well prove problematic when such analysis methods are

applied to hypersonic shock wave turbulent boundary layer interactions,

such as in the design of vehicle control surfaces or scramjet inlets, particularly when separated off-design performance is included in the constraint

list. Rather than pursing the expense and effort of ebedding complex

CFD methods with unvalidated or inaccurate parametric response into

MDO optimization procedures, a more effective analysis engine for MDO

methods may prove to be an intermediate analysis step providing either

interpolation, a neural net or an Engineering-based method making extensive use of engineering correlations (e.g. see Garcia [1] and related

citations), and which are best calibrated by a combination of vetted experiments and sparse CFD solutions. A further cautionary note must be

made by the appearance of what appears to be a form of ‘digital noise’ for

separated SWTBL cases at hypersonic conditions, and may degrade the

performance of MDO design procedures, particularly if MDO algorithms

based on numerical differentiation are employed.

As a final observation, it is clear that over the past 45-50 years, since

the experiments of Settles and of Holden, there has simply been too few

quality 2D hypersonic/supersonic compression corner experiments providing both a range of shock strength and Reynolds number variation

at a variety of Mach numbers. Well-designed hypersonic experiments

are indeed expensive, so to maximize their benefit parametric variation

about nominal test conditions can lend substantial added value at reasonable economic cost. Along with the absence of new experiments, there

is too much scatter to provide adequate confidence in correlations in the

hypersonic regime. In the design of new experiments, it is hoped that

this present study can demonstrate the value of such data obtained at

little additional cost/effort by varying the Reynolds number. The addition of even this extra dimension of parametric variation in hypersonic

experiment design can considerably enhance the utility of such data in

understanding the ability of predictive methods to correctly model general trends as needed for optimization procedures.

34
References

1. Garcia, J. A., Brown, J. L., Kinney, D. J., Bowles, J. V., Huynh,

L. C., Jiang, X. J., Lau, E. and Dupzyk, I. C., “Co-Optimization

of Mid-Lift/Drag Vehicle Concepts for Mars Atmospheric Entry,”

AIAA Paper 2010-5052, 10th AIAA/ASME Joint Thermophysics

and Heat Transfer Conference, Chicago, Illinois, 28 Jun-1 July, 2010.

2. Brown, J. L., “Shock Wave Impingement on Boundary Layers at Hypersonic Speeds: Computational Analysis and Uncertainty,” AIAA

Paper 2011-3143, 42nd AIAA Thermophysics Conference, Honolulu,

Hawaii, 27-30 Jun, 2011.

3. Brown, J. L., “Hypersonic Shock Wave Impingement on Turbulent Boundary Layers: Computational Analysis and Uncertainty,”

J. Spacecraft and Rockets, Vol. 50, No. 1, Jan-Feb 2013.

4. Marvin, J. G, Brown, J. L. and Gnoffo, P. A., “Experimental

Database with Baseline CFD Solutions: 2-D and Axisymmetric

Hypersonic Shock-Wave Turbulent Boundary Layer Interactions,”

NASA TM 2013-216604, Nov. 2013.

5. Brown, J. L., “Turbulence Model Validation for Hypersonic Flows,”

AIAA Paper 2002-3143, 8th AIAA/ASME Joint Thermophysics and

Heat Transfer Conference, St. Louis, Missouri, 24-26 Jun, 2002.

6. Settles, G. S. and Dodson, L. J., “Hypersonic Shock/BoundaryLayer Interaction Database,” NASA CR 177577, April 1991.

7. Settles, G. S. and Dodson, L. J., “Hypersonic Turbulent BoundaryLayer and Free-Shear Layer Database,” NASA CR 177610, April

1993.

8. Settles, G. S. and Dodson, L. J., “Hypersonic Shock/BoundaryLayer Interaction Database: New and Corrected Data,” NASA CR

177638, April 1994.

9. Settles, G. S. and Dodson, L. J., “Supersonic and Hypersonic

Shock/Boundary-Layer Interaction Database,” AIAA Journal, Vol.

32, No. 7, July 1994, pp. 1377-1383.

10. Settles, G. S., Bogdonoff, S. M. and Vas, I. E., “Incipient Separation of a Supersonic Turbulent Boundary Layer at Moderate to High

Reynolds Number,” AIAA Paper 1975-7, AIAA 13th Aerospace Sciences Meeting, Pasadena, CA, Jan 20-22, 1975.

11. Settles, G. S., Fitzpatric, T. J. and Bogdonoff, S. M., “Detailed study

of Attached and Separated Compression Corner Flowfields in High

Reynolds number Supersonic Flow,” AIAA Journal, Vol. 17, No. 6,

1979, pp.579-585.

35
12. Settles, G. S., Gilbert, R. B. and Bogdonoff, S. M., “Data Compilation For Shock Wave/Turbulent Boundary Layer Interaction Experiments On Two-Dimensional Compression Corners,” Princeton

University Report 1489-MAE, Princeton Univ., 1980.

13. Wilcox, D. C., “Turbulence Modeling for CFD,” 3rd Edition, DCW

Industries, Inc., La Canada, CA, Nov. 2006.

14. Wright, M. J., Candler, G. V. and Bose, D., “Data-Parallel Line Relaxation Method for the Navier-Stokes Equations,” AIAA Journal,

Vol. 36, No. 9, 1998, pp. 1603-1609.

15. Wright, M. J., White, T. and Mangini, N., “Data Parallel Line Relaxation Code User Manual, Acadia-Version 4.01.1,” NASA TM2009-215388, Oct. 2009.

16. Rumsey, C., Biedron, R. and Thomas, J., “CFL3D: Its History and

Some Recent Applications,” NASA TM-112861, May 1997.

17. Bose, D., Brown, J. L., Prabhu, D. K., Gnoffo, P., Johnston, C. J.

and Hollis, B., “Uncertainty Assessment of Hypersonics Aerothermodynamic Prediction Capability,” AIAA Paper 2011-3141, 42nd

AIAA Thermophysics Conference, Honolulu, Hawaii, 27-30 Jun,

2011.

18. Bose, D., Brown, J. L., Prabhu, D. K., Gnoffo, P., Johnston, C. J.

and Hollis, B., “Uncertainty Assessment of Hypersonics Aerothermodynamic Prediction Capability,” J. Spacecraft and Rockets, Vol.

50, No. 1, Jan-Feb 2013.

19. Gnoffo, P. A., Berry, S. A. and Van Norman, J. W., “Uncertainty Assessments in Simulations of 2D and Axisymmetric Hypersonic Shock

Wave-Turbulent Boundary Layer Interactions at Compression Corners,” AIAA Paper 2011-3143, June 27-30, Honolulu, HI, 2011.

20. Gnoffo, P. A., Berry, S. A. and Van Norman, J. W., “Uncertainty Assessments in Simulations of 2D and Axisymmetric Hypersonic Shock

Wave-Turbulent Boundary Layer Interactions at Compression Corners,” J. Spacecraft and Rockets, Vol. 50, No. 1, Jan-Feb 2013.

21. Ames Research Staff, “Equations, Tables, and Charts for Compressible Flow,” NACA Rept. 1135, 1953.

22. Wolf, T., “Comment on Approximate Formulation of Weak Oblique

Shock Wave Angle,” AIAA Journal, Vol. 31, No. 7, 1993, p. 1363.

23. Chapman, D. R., Kuehn, D. M. and Larson, H. K., “Investigation of

Separated Flows in Supersonic and Subsonic Streams with Emphasis

on the Effect of Transition,” NACA Report 1356, 1958.

36
24. Reshotko, E. and Tucker, M., “Effect of a discontinuity on Turbulent

Boundary-Layer-Thickness Parameters with Application to ShockInduced Separation,” NACA TN-3454, May 1955.

25. Erdos, J. and Pallone, A., “Shock-boundary layer interaction and

flow separation,” in: Proc. Heat Transfer and Fluid Mechanics Inst.,

Stanford Univ Press, 1962.

26. Lewis, J. E., Kubota, T. and Lees, L., “Experimental Investigation

of Supersonic Laminar, Two-Dimensional Boundary Layer Separation in a Compression Corner With and Without Cooling,” AIAA

Journal, Vol. 6, No. 1, 1968. p. 7-14.

27. Carriere, P., Sirieix, M. and Solignac, J. L., “Similarity Properties

of the laminar or turbulent separation phenomena in a non-uniform

supersonic flow,” Applied Mechanics, International Union of Theoretical and Applied Mechanics, 1969, p 145-157.

28. Kuehn, D. M., “Turbulent Boundary-Layer Separation induced by

Flares on Cylinders at Zero Angle of Attack,” NASA Technical Report R-117, 1961.

29. Delery, J. and Marvin, J. G., “Shock-Wave Boundary Layer Interactions,” AGARDograph AG-280, NATO, Ed. by Reshotko, E., 1986.

30. Back, L. H. and Cuffel, R. F., “Changes in Heat Transfer from Turbulent Boundary Layers Interacting with Shock Waves and Expansion

Waves,” AIAA Journal, Vol. 8, No. 10, 1970, pp. 1871-1873.

31. Holden, Michael S., “Shock Wave-Turbulent Boundary Layer Interaction in Supersonic Flow,” AIAA Paper 1977-45, AIAA 15th

Aerospace Sciences Meeting, Los Angeles, CA, Jan 1977.

32. Holden, M., MacLean, M., Wadhams, T. and Mundy, E., “Experimental Studies of Shock Wave/Turbulent Boundary Layer Interaction in High Reynolds Number Supersonic and Hypersonic Flow to

Evaluate the Performance of CFD Codes,” AIAA Paper 2010-4468,

40th Fluid Dynamics Conference and Exhibit, Chicago, Illinois, Jun

2010.

33. Menter, F. R., “Improved Two-Equation K-ω Turbulence Models for

Aerodynamic Flows,” NASA TM 103975, Oct. 1992.

34. Menter, F. R., “Two-Equation Eddy Viscosity Turbulence Models

for Engineering Applications,” AIAA Journal, Vol. 32, No. 8, Aug

1994, pp. 1598-1605.

35. Menter, F. R. and Esch, T., “Elements of Industrial Heat Transfer Predictions,” 16th Brazilian Congress of Mechanical Engineering

(COBEM), Uberlandia, Brazil, Nov. 2001.

37
36. Menter, F. R., Kuntz, M. and Langtry, R., “Ten Years of Industrial Experience with the SST Turbulence Model,” in Turbulence,

Heat & Mass Transfer, Vol. 4, Ed. by Hanjalic, K., Nagano, Y. and

Tummers, M., 2003, pp. 625-632.

37. Korkegi, R. H., “Comparison of Shock-Induced Two- and ThreeDimensional Incipient Separation,” AIAA Journal, Vol. 13, No. 4,

April 1975.

38. Needham, D. A. and Stollery, J. L., “Boundary Layer Separation

in Hypersonic Flow,” AIAA Paper 1966-455, AIAA 4th Aerospace

Sciences Meeting, Los Angeles, CA, June, 1966.

39. Babinsky, H. and Edwards, J. A., “On The Incipient Separation Of A

Turbulent Hypersonic Boundary Layer,” The Aeronautical Journal,

June, 1996.

40. Roy, C. J. and Blottner, F. G., “Review and Assessment of Turbulence Models for Hypersonic Flows,” AIAA Paper 2006-713, 2006.

41. Sommer, T. P., So, R. M. C. and Zhang, H. S., “Near-Wall VariablePrandtl-Number Turbulence Model for Compressible Flows,” AIAA

Journal, Vol. 31, No. 1, Jan. 1993.

42. Xiao, X., Edwards, J. R. and Hassan, H. A., “Role of Turbulent

Prandtl Number on Heat Flux at Hypersonic Mach Numbers,” AIAA

Paper 2005-1098, 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 10-13 Jan, 2005.

43. Xiao, X., Hassan, H. A., Edwards, J. R. and Gaffney, R. L., Jr.,

“Role of Turbulent Prandtl Number on Heat Flux at Hypersonic

Mach Numbers,” AIAA Journal, Vol. 45, No. 4, Apr. 2007. pp 806-

813.

44. Sch ̈ulein, E., Krogmann, P. and Stanewsky, E., “Documentation of

Two-Dimensional Impinging Shock/Turbulent Boundary Layer Interaction Flow,” DLR, German Aerospace Center, Rept. IB 223-96

A 49, G ̈ottingen, Germany, Oct, 1996.

45. Sch ̈ulein, E., “Skin-Friction and Heat Flux Measurements in

Shock/Boundary-Layer Interaction Flows,” AIAA Journal, Vol. 44,

No. 8, Aug 2006, pp. 1732-1741, also AIAA Paper 2004-2115, 2004.

46. Fedorova, N. N., Fedorchenko, I. A. and Sch ̈ulein, E., “Experimental

and Numerical Study of Oblique Shock Wave/Turbulent Boundary

Layer Interaction at M = 5,” Computational Fluid Dynamics Journal, Vol. 10, No. 3, Special Issue, 2001, pp. 390-395.

47. Fedorova, N. N. and Fedorchenko, I. A., “Computations of Interaction of an Incident Oblique Shock Wave with a Turbulent Boundary

38
Layer on a Flat Plate,” Journal of Applied Mechanics and Technical

Physics, Vol. 45, No. 3, 2004, pp. 358-366.

48. Kussoy, M. I. and Horstman, K. C., “Documentation of Two- and

Three-Dimensional Shock-Wave/Turbulent Boundary-Layer Interaction Flows at Mach 8.2,” NASA TM 103838, May 1991.

49. Cebeci, T. and Smith, A. M. O., “Analysis of Turbulent Boundary

Layers,” Series in Applied Mathematics and Mechanics, Vol. XV,

Academic Press, Orlando, FL, 1974.

50. Zhang, C., Xu, J., Gao, L. and Gao, G., “Studies of Compression

Corner Flowfields Using THREE Turbulent Models” Procedia Engineering, Vol. 31, International Conference on Advances in Computational Modeling and Simulation, 2012, p. 762-768.

51. Steelant, J., “Effect of a Compressibility Correction on Different Turbulence Models,” Engineering Turbulence Modelling and

Experiments-5, Ed. Rodi, W. and Fueyo, N., Elsevier Science, Ltd.,

2002. pp 207-216.

52. Kovar, A., Hannemann, S. Karl and Sch ̈ulein, E., “About the Assessment of Heat Flux and Skin Friction of the DLR-TAU-code for Turbulent Supersonic Flows,” ODAS, ONERA-DLR Aerospace Symposium, 2007.

53. Simpson, R. L., “Two-Dimensional Turbulent Separated Flows,”

AGARDograph AG-278, Vol. 1, NATO, Ed. by Young, A. D., 1985.

54. Wallin, S. and Johansson, A. V., “An explicit algebraic Reynolds

stress model for incompressible and compressible turbulent flows,”

Journal of Fluid Mechanics, Vol. 403, p. 89-132, Jan. 2000.

55. Hellsten, A. and Laine, S., “Explicit Algebraic Reynolds-Stress Modelling in Decelerating and Separating Flows,” AIAA Paper 2000-

2313, June 2000.

56. Hellsten, A., “New Advanced k − ω Turbulence model for High-Lift

Aerodynamics,” AIAA Journal, Vol. 43, No. 9, 2005.

39
3.1 Tables

Experiment Type Mach No ReU 106/m L, m ShockGenAngle

Wedge

Settles 2D 2.84 67.16 1.98 8, 16, 20, 24o

A2.Holden 2D 8.2 143.74 0.99568 27, 30, 33, 36o

A2.Holden 2D 11.3 36.51 1.02362 36o

A5.Kussoy Axi 7.05 5.8 1.39 20, 30, 32.5, 35o

A6.Williams Axi 8.98 50.3 0.72 36o

Impinging

B1.Sch ̈ulein 2D 5.0 38.8 0.35 6, 10, 14o

B2.Murray Axi 8.99 40.34 0.76 4.7, 10o

B3.Kussoy 2D 8.2 4.91 1.6 5, 8, 9, 10, 11o

B4.Holden 2D 11.4 36.51 0.88392 20o

Table 1. Hypersonic 2D or Axisymmetric SWTBLI Cases. Experimental cases are given in terms of Ref [4] where applicable. Note Settles

experiment was run on a wind tunnel wall, so run length given is from

nozzle to wedge.

Experiment Me Gas PT , MPa To

0 , K To

aw, K To

wall, K

Settles 2.84 Air 0.68 262 244.2 276

A2.Holden 8.2 Air 116.8 1025.8 920.78 296

A2.Holden 11.3 Air 139.0 1619 1447.6 300

A5.Kussoy 7.05 Air 2.495 888.38 799.58 311

A6.Williams 8.98 N2 60.8 1150 1031 293

B1.Sch ̈ulein 5.0 Air 2.2351 410.35 376.264 300

B2.Murray 8.99 N2 60.8 1150 1031 293

B3.Kussoy 8.2 Air 24.95 888.38 799.58 300

B4.Holden 11.4 Air 164.6 1700.5 1520.4 300

Table 2. Total or Stagnation Conditions for SWTBLI Cases.

40
Experiment Me U∞, m/s pe, kPa ρe, kg/m3 To

e , K μ∞, kg/m − s

Settles 2.84 570 23.6 0.82014 100.3 7.002 10−6

A2.Holden 8.2 1385 10.188 0.50 71 4.989 10−6

A2.Holden 11.3 1769 1.4437 0.082455 61 4.285 10−6

A5.Kussoy 7.05 1274 0.576 0.0252 81.2 5.700 10−6

A6.Williams 8.98 1500 2.926 0.1467 67.16 4.724 10−6

B1.Sch ̈ulein 5.0 828.4 4.304 0.21815 68.73 4.830 10−6

B2.Murray 8.99 1500 2.898 0.1457 66.97 4.710 10−6

B3.Kussoy 8.2 1446 0.430 0.0187 81.0 5.686 10−6

B4.Holden 11.4 1814 1.611 0.08910 63 4.426 10−6

Table 3. Upstream Boundary Layer properties for SWTBLI Cases.

Keyes Relation for either dry Air or N2 is used for dynamic viscosity.

Experiment Me τ0, Pa Q0, W/cm2 δ0, mm δ∗

0, mm θ0, mm H

Settles 2.84 145.0 −0.946 23.72 6.00 1.15 5.21

A2.Holden 8.2 278.3 14.87 13.3 6.30 0.34 18.17

A2.Holden 11.3 69.28 5.32 21.052 10.506 0.341 30.77

A5.Kussoy 7.05 21.82 0.9887 30.0352 13.647 0.920 14.84

A6.Williams 8.98 113.7 6.8115 13.5289 6.631 0.318 20.83

B1.Sch ̈ulein 5.0 73.48 0.7505 4.813 2.000 0.202 10.94

B2.Murray 8.99 112.046 6.7087 14.028 6.962 0.333 20.87

B3.Kussoy 8.2 18.82 1.128 39.5528 17.6655 0.9868 17.90

B4.Holden 11.4 78.832 6.29 18.715 9.386 0.303 30.97

Table 4. Upstream Boundary Layer properties for SWTBLI Cases, from

Wilcox EddyBL/Cebeci-Smith Turbulence Model.

41
Experiment M1 αo

wedge θoshock M2 p2/p1 T2/T1

Settles 2.84 8. 26.822 2.466 1.7492 1.179

16. 34.571 2.088 2.8631 1.396

20. 39.116 1.888 3.5788 1.525

24. 44.260 1.676 4.4168 1.673

A2.Holden 8.2 27. 35.184 2.947 25.8785 5.279

30. 39.145 2.597 31.0957 6.150

33. 43.313 2.277 36.7477 7.092

36. 47.783 1.976 42.8612 8.112

A2.Holden 11.3 36. 46.783 2.096 78.952 14.129

B1.Sch ̈ulein 5.0 6.0 15.875 : 17.481 4.395 : 3.899 3.762 1.485

10.0 19.376 : 22.237 4.000 : 3.285 7.626 1.899

14.0 23.287 : 27.686 3.603 : 2.736 13.624 2.402

B3.Kussoy 8.2 5.0 10.681 : 11.811 7.012 : 6.116 5.655 1.704

8.0 13.408 : 15.277 6.316 : 5.131 12.418 2.306

9.0 14.384 : 16.516 6.087 : 4.838 15.552 2.543

10.0 15.388 : 17.793 5.861 : 4.561 19.155 2.799

11.0 16.417 : 19.107 5.639 : 4.301 23.232 3.074

B4.Holden 11.4 20.0 25.480 : 31.309 4.363 : 2.746 162.616 10.764

Table 5. Shock Strength for 2D Planar SWTBLI Cases, from NACA

1135, Ref [21] and Wolf, [22]. Impinging shock 2D case values are for

the inviscid 2-shock impinging/reflection shock problem; Hence, 2 shock

angles and Mach numbers are given, while the pressure and temperature

ratios are for the effect of the two shocks combined.

42
Experiment M1 pw,0 Pa τw,o,Pa q0, kPa δ∗

0, mm

Settles 2.84 23600.0 145.00 133.244 6.000

A2.Holden 8.2 10188 278.3 479.529 6.299

A2.Holden 11.3 1443.7 69.28 129.042 10.507

A5.Kussoy 7.05 576.0 21.82 20.040 13.647

A6.Williams 8.98 2926.0 113.7 165.168 6.631

B1.Sch ̈ulein 4.984 4304.0 73.48 74.838 2.730

B2.Murray 8.98 2898.0 112.046 163.587 6.962

B3.Kussoy 8.2 430.0 18.82 20.239 17.665

B4.Holden 11.4 1611.016 78.832 146.557 9.386

Table 6. Free Interaction Transformation Variables for SWTBLI Cases.

Experiment M1 Aξ, Pa Bξ pw,s, kPa pw,b, kPa

Settles 2.84 3812.72 18.59 39.689 46.476

A2.Holden 8.2 5726.62 14.55 34.354 44.548

A2.Holden 11.3 1260.37 12.86 6.762 9.006

A5.Kussoy 7.05 354.00 11.47 2.0690 2.700

A6.Williams 8.98 2051.51 12.76 11.584 15.235

B1.Sch ̈ulein 5.0 1500.84 14.44 10.637 13.309

B2.Murray 8.99 2026.76 12.79 11.450 15.059

B3.Kussoy 8.2 304.66 11.45 1.715 2.257

B4.Holden 11.4 1426.45 12.79 7.630 10.170

Table 7. Free Interaction Variables for SWTBLI Cases.

pw,s and pw,b correspond to Pξ = 4.22, and 6.0, respectively;

where, Pξ = (Pw − P0)/Aξ, and ξ = (X − X0)/(κξBξδ∗

0).

43
3.2 Figures

Figure 1. Hypersonic Compression Corner Separated Shock Boundary

Layer Topology: Holden M∞ = 8.2, 36o case. BL-upstream boundary

layer, P-Primary Shock System, C-compression or reattachment shock,

λ-separation shock.

Figure 2. Hypersonic Impinging Shock-Wave/Boundary-Layer Flow

Topology: Sch ̈ulein Mach 5, 14o case.

44
Figure 3. Incipient Separation: Dependence on Mach Number.

Figure 4. Incipient Separation: Dependence on M3CF .

Figure 5. Incipient Separation: Dependence of Wedge Angle on Mach

and Reynolds Number. Updated Correlation.

45
Figure 6. Holden Mach 8 Compression Corner Grids: 27o, 30o, 33o and

36o Wedge Angle.

Figure 7. Holden Mach 8 Separation Region Wall Pressure Distribution,

Free Interaction Variables: 27o, 30o, 33o and 36o Wedge Angle.

46
Figure 8. Holden M∞ = 8.2

SWBLI, PWall vs X, DPLR/SST vs

Expt.

Figure 9. Holden M∞ = 8.2

SWBLI, PWall vs X, DPLR/SST vs

Expt, Expanded.

Figure 10. Holden M∞ = 8.2

SWBLI, PWall vs X, CFL3D/SST vs

Expt.

Figure 11. Holden M∞ = 8.2

SWBLI, PWall vs X, CFL3D/SST vs

Expt, Expanded.

47
Figure 12. Holden M∞ = 8.2 Separation Extent Dependence on Wedge

Angle.

Figure 13. Holden M∞ = 8.2 Separation Extent Dependence on Primary Shock Strength.

48
Figure 14. Holden M∞ = 8.2, 33o Wedge: Experimental Wall Pressure

Distribution Dependence on Reynolds Number.

Figure 15. Holden M∞ = 8.2, 33o

Wedge: Wall Pressure Distribution

Dependence on Reynolds Number

for DPLR/SST Solns.

Figure 16. Holden Mach 8, 33o

Wedge: Wall Pressure Distribution

Dependence on Reynolds Number

for CFL3D/SST Solns.

49
Figure 17. Holden M∞ = 8.2, 33o Wedge: Upstream Velocity and Temperature Profiles from DPLR/SST Solutions, Reynolds number dependence.

50
Figure 18. Holden M∞ = 11.3, 36o

Compression Corner Wall Pressure

Distribution. Expt vs DPLR/SST

Solutions.

Figure 19. Holden M∞ = 11.3, 36o

Compression Corner Wall Pressure

Distribution. Expt vs CFL3D/SST

Solutions.

51
Figure 20. Holden M∞ = 8.2 SWBLI, Wall Heat Transfer vs X,

DPLR/SST vs Expt.

Figure 21. Holden M∞ = 8.2

SWBLI, Wall Heat Transfer vs X,

QP85 DPLR/SST vs Q Expt.

Figure 22. Holden M∞ = 8.2

SWBLI, Wall Heat Transfer vs X,

QP85 DPLR/SST vs QP85 Expt.

52
Figure 23. Holden M∞ = 8.2 Separated Shock Boundary Layer Topology

and Streamlines, 36o Wedge Angle. DPLR/SST solution.

Figure 24. Development of SST Turbulent Kinetic Energy along Boundary Layer Streamlines through the

Separation and Reattachment regions for the Holden M∞ = 8.2 36o

Wedge Angle SWBLI.

Figure 25. Development of SST

Eddy Viscosity along Boundary

Layer Streamlines through the Separation and Reattachment regions for

the Holden M∞ = 8.2 36o Wedge

Angle SWBLI.

53
Figure 26. Settles M∞ = 2.85 Compression Corner Separation Wall

Pressure Distribution, Free Interaction Variables: 20o and 24o Cases.

Figure 27. Settles Compression

Corner Experimental vs DPLR/SST

Wall Pressure Distribution, M∞ =

2.85: 16o, 20o and 24o Cases.

Figure 28. Settles Compression

Corner Wall Pressure Distribution,

M∞ = 2.85: DPLR/SST and

CFL3D/SST comparison with Experiment, 20o Case.

Figure 29. Settles Correlation for

Upstream Influence Dependence on

Reynolds Number and Wedge Angle

for Turbulent Compression Corner

Wall Pressure Distribution, M∞ =

2.85. Settles Data are open symbols,

Settles Correlation is solid line, Extracted from Ref [10].

54
Figure 30. Settles Compression

Corner Wall Pressure Distribution,

M∞ = 2.85, 20o Case: DPLR/SST

Reynolds Number Sensitivity of

Separation Extent.

Figure 31. Settles Compression

Corner Wall Pressure Distribution,

M∞ = 2.85, 20o Case: Cfl3d/SST

Reynolds Number Sensitivity of

Separation Extent.

55
Figure 32. Sch ̈ulein M∞ = 5

Impinging Shock Separation Region

Wall Pressure Distribution, Free Interaction Variables: 14o Case.

Figure 33. Sch ̈ulein M∞ = 5 Impinging SWTBL Interaction Experimental Wall Pressure Distribution:

6o, 10o and 14o Shock Generator

Cases.

Figure 34. Sch ̈ulein M∞ = 5

Impinging Shock Separation Region Wall Pressure Distribution,

DPLR/SST comparison with

CFL3D/SST: 14o Shock Generator.

Figure 35. Sch ̈ulein M∞ =

5 Impinging SWTBL Interaction

Wall Heat Transfer Distribution,

DPLR/SST vs Experimental Measurements: 6o, 10o and 14o Shock

Generator Cases.

56
Figure 36. Sch ̈ulein M∞ =

5 Impinging SWTBL Interaction

Wall Heat Transfer Distribution,

DPLR/SST QP85 vs Qwall Experimental Measurements: 6o, 10o and

14o Shock Generator Cases.

Figure 37. Sch ̈ulein M∞ = 5 Impinging SWTBL Interaction Wall

Heat Transfer Distribution from

QP85 Correlation, DPLR/SST vs

Expt.: 6o, 10o and 14o Shock Generator Cases.

57
Figure 38. Sch ̈ulein M∞ = 5 Impinging SWTBL Interaction: Pressure

Contours DPLR/SST 14o Shock Generator Case.

Figure 39. Sch ̈ulein M∞ = 5 Impinging SWTBL Interaction: Pressure

Contours Cfl3d/SST 14o Shock Generator Case.

58
Figure 40. Sch ̈ulein M∞ = 5 Impinging SWTBL Interaction, 14o case:

Development of DPLR/SST Turbulent Kinetic Energy along Boundary

Layer Streamlines through the Separation and Reattachment regions.

59
Figure 41. Kussoy M∞ = 8.2 Impinging SWTBL Interaction: Upstream

Boundary Layer Velocity and Temperature Profiles: DPLR/SST compared to Experiment.

Figure 42. Kussoy M∞ = 8.2 Impinging SWTBL Interaction: Wall

Pressure Distribution for DPLR vs

Expt, 10o Shock Generator.

Figure 43. Kussoy M∞ =

8.2 Impinging SWTBL Interaction:

Wall Heat Transfer Distribution for

DPLR vs Expt, 10o Shock Generator.

60
Figure 44. Kussoy M∞ = 8.2 Impinging SWTBL Interaction: Pressure Contours and Contour lines for DPLR/SST, 10o Shock Generator,

Expanded Scale.

Figure 45. Kussoy M∞ = 8.2 Impinging SWTBL Interaction: Mach

Contours and Pressure Contour lines for DPLR/SST, 10o Shock Generator. Expanded Scale.

61
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1. REPORT DATE (DD-MM-YYYY)

01-05-2014

2. REPORT TYPE

Technical Memorandum

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

On Parametric Sensitivity of

Reynolds-Averaged Navier-Stokes SST

Turbulence Model:

2D Hypersonic Shock-Wave Boundary-Layer

Interactions

5a. CONTRACT NUMBER

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6. AUTHOR(S)

James L. Brown

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Ames Research Center

Moffett Field, CA 94035-1000

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REPORT NUMBER

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National Aeronautics and Space Administration

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NASA

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Unclassified-Unlimited

Subject Category 64

Availability: NASA CASI (443) 757-5802

13. SUPPLEMENTARY NOTES

An electronic version can be found at http://ntrs.nasa.gov.

14. ABSTRACT

Examined is sensitivity of separation extent, wall pressure and heating to variation of primary input flow parameters, such as

Mach and Reynolds numbers and shock strength, for 2D and Axisymmetric Hypersonic Shock Wave Turbulent Boundary

Layer interactions obtained by Navier-Stokes methods using the SST turbulence model. Baseline parametric sensitivity

response is provided in part by comparison with vetted experiments, and in part through updated correlations based on free

interaction theory concepts. A recent database compilation of hypersonic 2D shock-wave/turbulent boundary layer

experiments extensively used in a prior related uncertainty analysis provides the foundation for this updated correlation

approach, as well as for more conventional validation. The primary CFD method for this work is DPLR, one of NASA’s

real-gas aerothermodynamic production RANS codes. Comparisons are also made with CFL3D, one of NASA’s mature

perfect-gas RANS codes. Deficiencies in predicted separation response of RANS/SST solutions to parametric variations of

test conditions are summarized, along with recommendations as to future turbulence approach.

15. SUBJECT TERMS

Hypersonic, Shock Wave, Turbulent, Boundary Layer

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