Progress in Aerospace Sciences 42 (2006) 469530

the current hypersonic experimental database. A total of 18 one- and two-equation turbulence models are reviewed, and

model sensitivities. In addition, model corrections (e.g., compressibility corrections) should be carefully examined for their

ARTICLE IN PRESS

www.elsevier.com/locate/paerosci0376-0421/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.paerosci.2006.12.002

Corresponding author. Tel.: +1334 844 5187; fax: +1 334 844 6803.E-mail address: cjroy@eng.auburn.edu (C.J. Roy).results of turbulence model assessments for the six models that have been extensively applied to the hypersonic validation

database are compiled and presented in graphical form. While some of the turbulence models do provide reasonable

predictions for the surface pressure, the predictions for surface heat ux are generally poor, and often in error by a factor

of four or more. In the vast majority of the turbulence model validation studies we review, the authors fail to adequately

address the numerical accuracy of the simulations (i.e., discretization and iterative error) and the sensitivities of the model

predictions to freestream turbulence quantities or near-wall y mesh spacing. We recommend new hypersonic experimentsbe conducted which (1) measure not only surface quantities but also mean and uctuating quantities in the interaction

region and (2) provide careful estimates of both random experimental uncertainties and correlated bias errors for the

measured quantities and freestream conditions. For the turbulence models, we recommend that a wide-range of turbulence

models (including newer models) be re-examined on the current hypersonic experimental database, including the more

recent experiments. Any future turbulence model validation efforts should carefully assess the numerical accuracy andReview and assessment of turbulence models forhypersonic ows

Christopher J. Roya,, Frederick G. Blottnerb

aAerospace Engineering Department, Auburn University, 211 Aerospace Engineering Building, Auburn, AL 36849-5338, USAbConsultant, Sandia National Laboratories, Albuquerque, NM, USA

Abstract

Accurate aerodynamic prediction is critical for the design and optimization of hypersonic vehicles. Turbulence modeling

remains a major source of uncertainty in the computational prediction of aerodynamic forces and heating for these

systems. The rst goal of this article is to update the previous comprehensive review of hypersonic shock/turbulent

boundary-layer interaction experiments published in 1991 by Settles and Dodson (Hypersonic shock/boundary-layer

interaction database. NASA CR 177577, 1991). In their review, Settles and Dodson developed a methodology for assessing

experiments appropriate for turbulence model validation and critically surveyed the existing hypersonic experiments. We

limit the scope of our current effort by considering only two-dimensional (2D)/axisymmetric ows in the hypersonic ow

regime where calorically perfect gas models are appropriate. We extend the prior database of recommended hypersonic

experiments (on four 2D and two 3D shockinteraction geometries) by adding three new geometries. The rst two

geometries, the at plate/cylinder and the sharp cone, are canonical, zero-pressure gradient ows which are amenable to

theory-based correlations, and these correlations are discussed in detail. The third geometry added is the 2D shock

impinging on a turbulent at plate boundary layer. The current 2D hypersonic database for shockinteraction ows thus

consists of nine experiments on ve different geometries. The second goal of this study is to review and assess the validation

usage of various turbulence models on the existing experimental database. Here we limit the scope to one- and two-

equation turbulence models where integration to the wall is used (i.e., we omit studies involving wall functions).

A methodology for validating turbulence models is given, followed by an extensive evaluation of the turbulence models on

effects on a

become ava

mo

r

Keywords: Co

Contents

1.3.

2. Turbu

2.3.

2.4.

3.4.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

4.2. Validation of turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 4695304704.2.2. Two-equation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

4.2.3. Physical freestream turbulence quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

4.3. Turbulence model application to the hypersonic validation database . . . . . . . . . . . . . . . . . . . . . . . . . 498

4.4. Previous ow geometries with adverse pressure gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

4.4.1. Case 1: 2D compression corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

4.4.2. Case 2: cylinder with conical are . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

4.4.3. Case 3: cone with conical are . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5034.2.1. One-equation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4964.1.1. Flat plate/cylinder. . . . . . . . . . . . . .

4.1.2. Sharp circular cone . . . . . . . . . . . . .3.5. Updated hypersonic turbulence model validation database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

3.5.1. Previous ow geometries with adverse pressure gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

3.5.2. New ow geometries with and without pressure gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

3.6. Conclusion and recommendation of adequacy of experimental database . . . . . . . . . . . . . . . . . . . . . . 490

4. Usage of the hypersonic validation database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

4.1. Validation of theory-based correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.3.2. Correlations for the sharp cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Direct numerical simulation database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4793.2.3. ERCOFTAC database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

3.2.4. Holden database. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

3.2.5. Other limited reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

3.3. Theory-based correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

3.3.1. Correlations for the at plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4783.2.1. AGARD experimental review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

3.2.2. Experimental reviews by Settles and Dodson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4762.5. Turbulence model sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

2.6. Turbulence model validation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

3. Turbulence validation database for 2D/axisymmetric hypersonic ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

3.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

3.2. Previous experimental databases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476Turbulence models examined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

Model implementation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

Efforts to establish numerical accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

2.4.1. Grid convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

2.4.2. Iterative convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4752.1. Cases examined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

2.2.1.4.2. Compressibility effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

lence model validation methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4741.4. Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

1.4.1. Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473Scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Molecular transport for hypersonic ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4731. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

1.2.mputational Fluid Dynamics (CFD); Turbulence model; Hypersonic ow; Boundary layer; Shock wave; Compressible owdels and thus increase their predictive capability.

2007 Elsevier Ltd. All rights reserved.standard, low-speed validation database. Finally, as new experiments or direct numerical simulation data

ilable with information on mean and uctuating quantities, they should be used to improve the turbulence

. .

re gr

. . .

. . .

. . .

ce m

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

sform

ansp

. . .

these tests generally provide only small amountsof data, usually with large experimental uncertain-ties. There are many more ground-based wind

the associaablation, e

Settles and1990s. Diff

assessing turbulence models as applied to theexisting hypersonic experimental database.

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530 471Dodson [14] conducted in the early forced to limit the scope of this article. Prediction ofThe current effort builds on the reviews by available experimental data are enormous, so we arey on computational uid dynamics andted models for turbulence, chemistry,tc.

The validation of turbulence models shouldnecessarily include a wide range of ows. However,the extremely wide range of turbulent ows andtunnel tests on simplied geometries in hypersonicow. These ground tests generally provide muchmore data than the ight tests, and usually withsmaller experimental uncertainties. However, due tothe extremely high velocities found in hypersonicow, the hypersonic ground tests generally do notmatch the same high total enthalpy and lowfreestream turbulence levels typical of hypersonicight. The validation of turbulence models withwind tunnel data thus generally involves signicantextrapolation to ight enthalpies. Because of thesedifculties in obtaining validation data for turbu-lent, hypersonic ows, designers are forced torely heavil

The current article can be considered both as anupdate to the Settles and Dodson work, as well asan extension which includes the steps of validatingthe turbulence models. Finally, we soften the Settlesand Dodson requirement that the upstream bound-ary layer be fully characterized by the experiment incases where the predictive capabilities of theturbulence model are judged to be sufciently good,i.e., at plates or cylinders with natural transition.

1.2. Scope4.4.4. Case 4: axisymmetric impinging shock

4.5. New ow geometries with and without pressu

4.5.1. Case 5: 2D impinging shock . . . . . .

4.5.2. Case 6: at plate/cylinder . . . . . . . .

4.5.3. Case 7: sharp circular cone . . . . . . .

4.6. Conclusion and recommendation on turbulen

5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6. Recommendations . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . .

Appendix A. Compressibility corrections . . . . . . . .

A.1. Two-equation k2 turbulence models . . . . . .Appendix B. Turbulent at plate correlations. . . . .

B.1. Correlation of skin-friction data . . . . . . . . . .

B.2. Correlation of heat transfer . . . . . . . . . . . . .

B.3. Mean temperature proles . . . . . . . . . . . . . .

B.4. Mean velocity proles . . . . . . . . . . . . . . . . .

Appendix C. Turbulent sharp cone to at plate tran

Appendix D. Perfect gas air model and molecular tr

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction

1.1. Background

Turbulence plays a key role in determining theaerodynamic forces and heating for hypersonicvehicles. However, experimental data for turbulencemodel validation are difcult to obtain. Thereare very few ight tests in the open literature, anderences between the Settles and Dodson3. addresses the steps required for validatingturbulence models, and

4. takes the additional step of reviewing and2.thehas a different scope since only hypersonic owsare considered,includes new experimental data since 1994,1.effo

iews and the current work are that the currentrt:rev. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

adient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

odel validation usage. . . . . . . . . . . . . . . . . . . . . . . 510

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

ort properties for hypersonic ows . . . . . . . . . . . . 524

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525more basic turbulent ows needs to be improved

ARTICLE IN PRESS

Nomenclature

a speed of sound, m/sA incompressible transformation functionB incompressible transformation functioncp specic heat at constant pressure, J/

(kgK)Cf skin-friction coefcientCm turbulence modeling constantf general solution variableF compressible to incompressible transfor-

mation functionF s safety factor in the GCIG Mangler transformation functionhc heat transfer coefcient, W=m2 KH total enthalpy, J/kgk turbulent kinetic energy, m2=s2

l turbulent length scale, mL reference length, mm incompressible transformation function,

at plate to sharp cone scaling exponentM Mach numberp order of accuracy of the numerical

method, pressure, N=m2

Pr Prandtl numberq square root of turbulent kinetic energy,

m/s, heat ux, W=m2

qw wall heat ux, W=m2

r grid renement factor, recovery factorR specic gas constant, J/(kgK)Raf Reynolds analogy factorRn nose radius, mRe Reynolds numberSf at plate to sharp cone scaling factorSt Stanton numbert time, sT temperature, KT dimensionless total temperatureTu turbulence intensityu x-component of velocity, m/su wall-tangent component of velocity in

turbulence coordinatesut turbulence friction velocity ut

tw=rw1=2, m/sv y-component of velocity, m/sV total uid velocity, m/sx spatial coordinate (usually main ow

direction), mX axial distance from the leading edge, my spatial coordinate (usually wall-normal

direction), m

y wall-normal mesh spacing in turbulencecoordinates y uty=v

Greek letters

a incompressible transformation functionb turbulence modeling constant, incom-

pressible transformation functiong ratio of specic heatsd boundary-layer thickness, mDy height of rst cell off the wall, m specic dissipation rate, m2=s3

k Karman constant (typically k 0:41)r density, kg=m3

m absolute molecular viscosity, kg/(m s)mT turbulent eddy viscosity, kg/(m s)n kinematic molecular viscosity, m2=sy momentum thickness, mt shear stress, N=m2

o turbulence frequency, 1/sz enstrophy, 1=s2

Subscripts

1 freestream quantityaw adiabatic wall valuee boundary-layer edge propertyinc incompressible valuek grid level 1 finest grid0 total (stagnation) conditionsRE Richardson extrapolated quantityT turbulence quantityw wall value

Superscript

incompressible valuef denotes Reynolds-average of f~f denotes Favre-average of f

Acronyms

DNS direct numerical simulationERCOFTAC European Research Community

on Flow, Turbulence and CombustionGCI Roaches grid convergence indexNASA National Aeronautics and Space Admin-

istrationRANS Reynolds-averaged NavierStokesRMS root mean square

C.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530472

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530 473and validated before the more complex ows areinvestigated. Furthermore, while we have endea-vored to include all appropriate experimental andcomputational studies, it is inevitable that somequalied studies will be overlooked. We apologize inadvance for such omissions.Herein we consider only two-dimensional (2D)/

axisymmetric hypersonic ows, where the free-stream Mach number is limited to values greaterthan or equal to approximately ve. In addition,only wall-bounded ows are considered, thuseliminating ows such as hypersonic mixing layersand jets. While there are ongoing research efforts inadvanced turbulence models such as Reynolds stressmodels and large Eddy simulation (LES), the mostcomplex models currently employed in designstudies (where a large number of parametric casesmust be considered) are one- and two-equationturbulence models. We therefore limit the currentstudy to these models. We also limit this study tomodels where integration of the governing equa-tions to the wall is performed, thereby eliminatingthe use of wall functions. This choice was primarilydriven by the fact that a majority of the cases ofinterest for hypersonic ows include shock-bound-ary-layer interactions, where the assumptions in-herent in the use of wall functions are difcult tojustify. We further limit our scope to cases where thetransition from laminar to turbulent ow occursnaturally, and where this transition location isspecied in the experimental description. The focushere is not on the prediction of transition, whichitself is a difcult challenge for hypersonic ows.Finally, the effects of surface roughness, ablation,chemical reactions, real gases, and body rotation areall neglected as the existing experimental databasedoes not yet adequately address these phenomena.In most cases, turbulence models are expected to

be valid for a wide range of problems and nottuned for a very limited class of turbulent ows(this latter approach more closely resembles modelcalibration or parameter tting than a true predic-tion). Therefore, the testing of a turbulence modelfor high speed ows should include the evaluationof the model for all speeds and various owgeometries to determine its limitations. Here welimit our study to include only those models whichhave a well-established validation history over awide range of ow conditions including low-speedows. We therefore will not discuss efforts wherethe researchers propose model improvements, but

do not address the effects of these model improve-ments on the prior model validation heritage. Westrongly recommend that future high-speed turbu-lence modelers test their compressible ow modelimprovements on a standard set of incompressibleows as well, or at least give arguments as to whytheir corrections will not impact low-speed ows.See Marvin and Huang [5] for a recommended set ofexternal aerodynamics test cases in the subsonicthrough supersonic regime.

1.3. Molecular transport for hypersonic flows

Due to the difculties of reproducing highenthalpy environments in ground-based facilities,experimental freestream static temperatures areoften quite low, sometimes on the order of 50Kor below. In addition, the most common test gasesare air, nitrogen, and helium. For these reasons,appropriate molecular models for viscosity andthermal conductivity should be used. See AppendixD for recommended low temperature moleculartransport models for both air and nitrogen.

1.4. Turbulence

1.4.1. Physics

The NavierStokes equations contain all of thephysics necessary to simulate turbulent ows.However, due to the wide range of length and timescales associated with simulating turbulence atReynolds numbers typical of ight vehicles, thisdirect simulation approach for turbulence is wellbeyond the capabilities even of todays fastestcomputers. Engineers are thus forced to rely onturbulence models, which account for the effects ofthe turbulence rather than simulate it directly. Thesimplest turbulence modeling approach is Reynolds-Averaged NavierStokes (RANS), where all of theturbulent length and time scales are modeled viatemporal ltering of the NavierStokes equations.For compressible ows, density-weighted (or Favre)averaging is used. See Wilcox [6] for details of theReynolds and Favre averaging procedures.

1.4.2. Compressibility effects

Typically turbulence models have been developedfor incompressible ows and then extended withoutmuch change to compressible ows. This approach inmany cases is not adequate. For complex turbulentows, Coakley et al. [7] have recommended correc-tions to apply to the two-equation k2 and k2o

turbulent eddy viscosity models. In addition, Aupoix

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530474and Viala [8] have proposed corrections to the k2model for compressible ows. The authors have usedat plate ows and mixing layers to assess thecompressible corrections introduced. Signicant ef-forts to assess turbulence models for compressibleows have occurred at NASA Ames ResearchCenter. The results of these investigations have beenpublished by Horstman [9], Horstman [10], Coakleyand Huang [11], Huang and Coakley [12], Coakley etal. [7], Bardina et al. [13], and Bardina et al. [14]. SeeA.1 for additional discussion of the compressibilityeffects for hypersonic ows.

2. Turbulence model validation methodology

The turbulence model validation methodologypresented herein is inuenced heavily by the work ofMarvin [15] and Marvin and Huang [5]. Theproposed validation framework [16] includes guide-lines for documentation, model sensitivities, andmodel validation. In addition, it is recommendedthat a signicant effort be made to estimate thenumerical accuracy of the simulations as part of thevalidation procedure. Listed below are six criteriafor assessing the models. The rst three criteria(2.12.3) focus on the thorough documentation ofthe model evaluation efforts. Details of the ow caseand the models used must be given in enough detailso that the results are reproducible by otherresearchers. The last three criteria (2.42.6) list thespecic standards for evaluating the models. Theturbulence models should be evaluated by rstestablishing the numerical accuracy of the simula-tions, then by examining model sensitivities, andthen nally by validation comparisons to experi-mental data.

2.1. Cases examined

Details of (or references to) the specic owproblem examined should be given including ow-eld geometry and relevant physics (ideal gas versusequilibrium thermochemistry, transport properties,etc.). All required boundary conditions should belisted including inow and outow conditions, wallboundary conditions for temperature, incomingboundary-layer thickness, freestream turbulenceintensities, a measure of the freestream turbulencedissipation rate, etc. One of the difculties encoun-tered in the specication of computational bound-ary conditions is that the level of information

required may not be fully characterized in theexperiment. For example, a large number ofotherwise excellent hypersonic validation data setsfail to report the thickness of the turbulentboundary-layer upstream of the interaction region;this information is especially important when theboundary layer is tripped to force transition toturbulence. It should be clearly stated whether theow is fully turbulent or transitional. Finally, thedata available for model validation should be given(feature location, surface quantities, turbulent eldproles, etc.).

2.2. Turbulence models examined

It should be clearly stated which form of theturbulence model is employed. It is stronglyrecommended that the standard model constantsbe used so as to build on prior turbulence modelvalidation efforts. Where applicable, the form of thelow Reynolds number wall damping functions usedshould be stated. The treatment of the near-wallregions should also be listed (i.e., integration to thewall versus wall functions).

2.3. Model implementation issues

The form of the governing equations should begiven. For example, different results may be foundwhen employing the full NavierStokes, thin-layerNavierStokes, parabolized NavierStokes, viscousshock layer equations, or boundary-layer equations.The boundary conditions employed in the simula-tion, including both ow properties and turbulencequantities, should be specied. Finally, any limitingof the turbulence quantities should be discussed.For example, limiting of the ratio of production todissipation of turbulent kinetic energy to some ratio(e.g., P=rp5) is often used. In addition, realize-ability constraints on the turbulence variables and/or normal turbulent stresses [17] should also bediscussed.

2.4. Efforts to establish numerical accuracy

The numerical accuracy of the simulations is animportant factor to consider when comparing toexperimental data; for example, if the numericalaccuracy of pressure distributions is estimated to be20%, then agreement with experimental datawithin 5% does not mean the model is accuratewithin 5%. The rst step towards determining the

accuracy of the simulations is code verication, i.e.,

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530 475building condence that the code is solving thegoverning equations correctly. Code verication canbe performed by comparison of the code results toexact solutions to the governing equations, highlyaccurate numerical benchmark solutions, or by themethod of manufactured solutions [1821]. Onceone has condence that the code is veried, then theaccuracy of the individual solutions must beveried. Solution accuracy includes assessing theerrors due to incomplete iterative convergence [16],temporal convergence for unsteady problems, andgrid convergence [18,21]. Methods for estimatingthe grid convergence errors based on systematic gridrenement [22] tend to be the most reliable and areapplicable to any type of discretization includingnite-difference, nite-volume, and nite-element.Grid convergence error estimates for hypersonicows are complicated by the presence of shockwaves, which tend to reduce the spatial order ofaccuracy to rst order on sufciently rened meshes[23,24], regardless of the nominal order of thespatial discretization scheme.

2.4.1. Grid convergence

Discretization error is dened as the differencebetween the exact solution to the discrete equationsand the exact solution to the original partialdifferential equations. Discretization error can beestimated by performing computations on two ormore meshes. The Richardson extrapolation proce-dure [18] can be used to obtain an estimate of theexact solution from the relation

f RE f 1 f 1 f 2

3, (1)

where 1 denotes the ne mesh and 2 the coarsemesh. This relation assumes that the numericalscheme is second-order, that both mesh levels are inthe asymptotic grid convergence range, and that amesh renement factor of two (i.e., grid doubling) isused. A more general expression for the Richardsonextrapolated value is given by

f RE f 1 f 1 f 2rp 1 , (2)

where r is the grid renement factor and p is theorder of accuracy (either formal or observed). Theformal order of accuracy can be found from atruncation error analysis of the discretization

method. If solutions are available on three meshes,then the observed order of accuracy can becalculated from

p lnf 3 f 2=f 2 f 1lnr , (3)

where 2 now denotes the medium mesh and 3 thecoarse mesh. Here it is assumed that the renementfactor between the coarse and medium mesh is equalto that between the medium and ne mesh.The accuracy of the solutions can be estimated

using the exact solution approximated by f RE whichgives the discretization error as

% Error of f k 100%f k f RE

f RE, (4)

where k 1; 2, etc. is the mesh level. Since it isequally possible that the true exact solution is aboveor below this estimate, it is generally recommendedthat some factor of safety be included in the errorestimate. Roache [22] combines the concept of afactor of safety along with absolute values to producean error band rather than an error estimate. Theresulting error (or numerical uncertainty) estimate isreferred to as the Grid Convergence Index, or GCI.The GCI thus produces an error (or uncertainty)band around the ne mesh solution and is given by

GCI Fsrp 1

f 2 f 1f 1

. (5)

When solutions from only two meshes are available,Roache recommends a factor of safety of three. Forthree meshes where the observed order of accuracyagrees with the formal order of accuracy, a much lessconservative value of Fs 1:25 is suggested byRoache.

2.4.2. Iterative convergence

When iterative or relaxation methods are em-ployed, an additional error source arises due toincomplete iterative convergence. The numericalerror due to incomplete iterative convergence isusually assessed by evaluating norms of theresiduals, where the residual is dened by substitut-ing the current numerical solution into the dis-cretized governing equations. For steady-stateows, the residual is calculated with the steady-state terms only, even if the temporal terms areincluded to speed up the convergence process. Theresiduals will approach zero as the steady-statesolution is reached and the current solution satises

the discretized form of the steady equations. These

3.

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530476residuals can generally be driven to zero withinmachine round-off tolerance; however, this extremelevel of iterative convergence is generally notnecessary. Many studies (e.g., [16,20]) suggest thatfor computational uid dynamics simulations, theresidual reduction levels correlate quite well with theactual iterative error in the ow properties.

2.5. Turbulence model sensitivities

Model sensitivity studies should be performedto determine practical guidelines for model use.A systematic study of the effects of the freestreamturbulence levels on the numerical predictionsshould be performed. The normal spacing at thewall y should also be varied in order to testmodel robustness and accuracy for both integrationto the wall and wall functions. In addition toestablishing the solution accuracy, a mesh rene-ment study can also be used to determine a giventurbulence models sensitivity to the mesh density.The sensitivity to the freestream turbulence levels

can manifest in two forms: changes in the locationof transition from laminar to turbulent ow andchanges in the eddy viscosity levels in the turbulentregion. The former may actually be a desirablecharacteristic when bypass transition is beingmodeled, while the latter is generally undesirable.Experimental evidence [25,26] suggests that surfaceproperties (e.g., shear stress) in the fully-developedturbulent region are generally not affected byfreestream turbulence intensity, at least in the caseof low-speed ows.

2.6. Turbulence model validation results

Model validation results should be presented in aquantitative manner rather than qualitatively. Forexample, the percent difference between the predictions(with demonstrated numerical accuracy) and experi-ment should be plotted or explicitly stated. Wheneverpossible, experimental error bounds should be givenfor all measurements used for validation. These errorbounds should include contributions from instrumentuncertainty, experimental run-to-run uncertainty, phy-sical model alignment uncertainty, oweld non-uniformities, etc. Bias errors are generally difcult toquantify, so if possible, multiple measurement techni-ques should be employed and, furthermore, tests inmultiple facilities should be performed. Techniques areavailable for converting some experimental bias errors

into random uncertainties [27].3.1. Overview

The validation of turbulence models must rely onreal-world observations, i.e., experimental data, toestablish model accuracy. The experimental datahave been mainly obtained from wind tunnels, wheredetailed measurements can be performed, rather thanin ight. There is a long history of high speedturbulent wind tunnel ow experiments. Compilationof experimental data for compressible turbulentboundary layers up to approximately 1980 is givenin AGARD reports by Fernholz and Finley [2830].For high speed compressible turbulence, an experi-mental database has been developed by Settles andDodson [14] for two- and three-dimensional shock-wave boundary-layer interaction ows, attachedboundary layers, and free shear ows. The hyperso-nic portions of these databases are described belowalong with other more limited reviews. In addition, adiscussion is provided on the role of both correla-tions and direct numerical simulation (DNS) data inturbulence model validation.

3.2. Previous experimental databases

3.2.1. AGARD experimental review

There has been a signicant effort by Fernholzand Finley [2830] to document available experi-mental data for compressible turbulent ow up toabout 1980. A total of 77 experiments are reviewedwith 59 given in Ref. [28] and 18 given in Ref. [29].A further compilation of compressible boundary-layer data is given in Ref. [30]. The number ofhypersonic experiments is limited. In addition, thesereports do not provide a clear recommendation fora limited list of experiments that should be used forvalidation of turbulence models.

3.2.2. Experimental reviews by Settles and Dodson

A very careful assessment of validation experimentsfor compressible turbulent ow was performed bySettles and Dodson in the early 1990s [14]. Theydeveloped a list of eight necessary criteria for validationexperiments which is given below (in abbreviated form)in the order in which they were applied.

1. Baseline applicability: Supersonic or hypersonicturbulent ow with shock wave/boundary layeraxi

Turbulence validation database for 2D/symmetric hypersonic owsinteraction.

8.

fol

4.

stu

excoat

ex

mime

mixaddof t

layeRefN

son

[4].menaccThe

avaDothe

triepaprecoThe

(1)

ment of Coleman and Stollery [31].

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530 477bou

exxing layer experiments. The authors recom-nded nine experiments as acceptable for attachedndary layers with pressure gradients and threepre

periments of attached boundary layers withssure gradients and 45 supersonic turbulentaccperiments at hypersonic conditions that werensidered as acceptable while seven experimentssupersonic conditions that were considered aseptable. The second study [2] examined 39bouThe initial study [1] examined 105 experimentaldies of shock wave interactions with turbulentndary layers at Mach 3 or higher. There are vethose experiments that reveal ow structure andphysical mechanisms.5.Redundant measurements: Preference is given toexperiments in which redundant data are taken.Flow structure and physics: Preference is given to1. Turbulent data: Turbulent properties (Reynoldsstress, etc.) of the ow eld are given.

2. Realistic test conditions: Flow conditions andboundary conditions typical of actual hypersonicight.

3. Non-intrusive instrumentation: Preference is givento non-intrusive experimental data.eva

lowing desirable criteria were also used in theluation of the experiments:Inthe ow are clearly resolved.

addition to the above necessary criteria, theAdequate spatial resolution of data: Sufcientdata must be presented such that key features ofavailable in tabulated form and capable of beingput into machine-readable form.7.2. Simplicity: Experimental geometries sufcientlysimple that they may be readily modeled by CFDmethods.

3. Specific applicability: Must provide useful experi-mental data for testing turbulence models.

4. Well-defined experimental boundary conditions:Sufcient boundary condition data must besupplied to allow CFD solutions to be performedwithout any assumptions.

5. Well-defined experimental error bounds: Mustprovide an analysis of the accuracy and repeat-ability of the data.

6. Consistency criterion: All data must be self-consistent (i.e., different measurements cannotbe contradictory).Adequate documentation of data: Data must beperiments as acceptable for supersonic turbulent(2) Cylinder with conical flare. For a freestreamMach number of 7, the wall pressure and heatux have been measured and ow eld surveyshave been made in the experiment of Kussoyand Horstman [32].

(3) Cone with conical flare. For a freestream Machnumber of 11 and 13, the wall pressure and heatux have been measured in the experiments ofHolden et al. [33] and Holden [34].

(4) Axisymmetric impinging shock. For a freestreamMach number of 7, the wall pressure, skinfriction and heat ux have been measured andow eld surveys have been made in theexperiment of Kussoy and Horstman [35].

(5) Flat plate with two fins (crossing shocks). For afreestream Mach number of 8.3, the wallpressure and heat ux have been measured,and surveys in the ow eld have been made inthe experiment of Kussoy and Horstman [36].

(6) Flat plate with 3D fin. For a freestream Machnumber of 6, the wall pressure and heat uxhave been measured in the experiment of Law[37]. For a freestream Mach number of 8.2, theTwo-dimensional compression corner. For a free-stream Mach number of 9, the wall pressure andheat ux have been determined in the experi-ares or ow problems will be discussed in thiser, with a total of seven geometries beingmmended for turbulence model validation.Settles and Dodson ow geometries numbersas follows:fouilable in electronic format. The Settles anddson ow geometries have been numbered withrst four being 2D or axisymmetric and the nextr being 3D experiments. Additional ow geome-tabuFor hypersonic ow conditions, seven experi-ts on six ow geometries are classied aseptable for validation of turbulence models.data for all of the acceptable experiments arelated in the Settles and Dodson reports andrefer interaction experiments has been published in. [4].one of the references to the acceptable super-ic experiments are included in the list ofrences of this review, but are available in Ref.of ting layers. The last report [3] has reviewed sevenitional experiments and has corrections to threehe previously reviewed experiments. A summaryhe supersonic and hypersonic shock/boundary-wall pressure and heat ux have been measured

istheFloA c

verdev

A review of the hypersonic experiments that have

3.2.

I

in hrevInshypand

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530478ypersonic ow. There have been several limitediews of hypersonic experiments at the Californiatitute of Technology by Hornung [49] andersonic ow research in Europe by Groenigrevi5. Other limited reviews

t has been nearly 15 years since a comprehensiveew has been performed on the new experimentsTwo-dimensional impinging shock [42,47,48].been performed at Calspan has been made byHolden and Moselle and reported in Ref. [42].There are numerous experiments that have beenperformed that are of interest:

(1) Sharp and blunted cones at Mach 11 withlaminar, transitional and turbulent ow havebeen investigated. This work is documented byHolden [43].

(2) Flow over a cone/are model at Mach 1116 hasbeen investigated. Earlier experiments on thisow geometry is one of the Settles and Dodsonacceptable experiments. This work is documen-ted by Holden [44].

(3) Two-dimensional compression corner [42].(4) Flat plate with 3D n [34,42].(5) Flat plate [42,45,46].(6)3.2.y limited as this effort is in the early stages ofelopment.

4. Holden databaseperfbeing developed on the web [41] and hasname European Research Community on

w, Turbulence and Combustion (ERCOFTAC).omplete review of this database has not yet beenormed, but most of the databases are presentlyA comprehensive database of European work

3.2.3. ERCOFTAC databasehave no available experiments.and surveys in the ow eld have been made inthe experiment of Kussoy and Horstman [38,39].For hypersonic ow at Mach 4.9, the Rodi andDolling experiments [40] are also acceptable.

(7) Cylinder with skewed flare. For hypersonic ow,Settles and Dodson have no available experi-ments.

(8) Three-dimensional compression corner withsweep. For hypersonic ow, Settles and DodsonOlivier [50].3.3. Theory-based correlations

Theory-based correlations exist for two of thesimpler geometries discussed herein: the at plateand the sharp cone. In some respects, theory-basedcorrelations can be considered superior to any singleexperimental data set since they mitigate theexperimental bias errors that vary from facility tofacility as well as bias errors associated with a givenmeasurement technique.

3.3.1. Correlations for the flat plate

The turbulence properties of interest are thewall skin friction, heat transfer, and proles ofvelocity and temperature across the boundary layer.While a summary of the at plate correlations ispresented here, a detailed discussion can be found inAppendix B.

Correlation of skin-friction data: The basic ap-proach which transform the experimental compres-sible local skin friction and momentum thicknessReynolds number to incompressible values for a atplate is the Van Driest II theory [51]. Squire [52]estimates that the accuracy of the Van Driest IIcorrelation is within 3% for the at plate. Basedon the sometimes erratic agreement between experi-ments and the correlation (e.g., see [53,45]), we feelthat this error estimate is somewhat optimistic andshould be increased to at least 5% for hypersonicows.

Correlation of heat transfer: Reynolds analogy isused to predict the wall heat ux, which is expressedas the compressible Stanton number St qw=reueHw Haw. Reynolds analogy is written interms of the compressible local skin friction2St=Cf Raf , where Raf is the Reynolds analogyfactor. Experiments indicate that 0:9oRafo1:3.While there are insufcient reliable experimentaldata to establish the Reynolds analogy factor, forhypersonic ows a reasonable choice at presentappears to be Raf 1. Additional work is needed toestablish the appropriate value for the Reynoldsanalogy factor for hypersonic ow.

Mean velocity profiles: In the log-law region,similarity of the Favre-averaged velocity ~u can beobtained with the Van Driest velocity transforma-tion (see Appendix B). Huang et al. [54] have alsoobtained the transformed velocity from the wall tothe edge of the boundary by taking into account theviscous sublayer and by including a wake function.This procedure gives the skin friction, velocity, and

temperature proles as a function of the Reynolds

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530 479number. It has been developed as a seven stepprocedure with iteration of the solution untilconverged (see Appendix B for details).

Mean temperature profiles: The general form ofthe mean temperature across the zero-pressuregradient turbulent compressible boundary layer asa function of the mean turbulent velocity andturbulent kinetic energy is

T Tw a ~u b ~u2 gTk. (6)

Huang, Bradshaw, and Coakley (HBC) [54] havedeveloped the temperature equation by neglectingthe convective terms in the momentum and energyequations. Their analysis (see Appendix B) yieldsthe following relations for the variables in Eq. (6):

a PrT=cpqw=tw; b PrT=2cp,

gT PrT=cp. (7)

3.3.2. Correlations for the sharp cone

One of the problems with the sharp cone is thelack of a theoretical correlation of the experi-mental data to use as a benchmark solution. Forlaminar ow, the skin friction and heat transferfor a at plate are multiplied by

3

pto obtain

the cone values. There does not appear to be awell established approach to transform the turbu-lent at plate results to the cone. Van Driest [55]has suggested an approximate approach that hasbeen developed further in the book by White [56]using the von Karman momentum integral relation(see Appendix C for more details). The atplate skin friction and wall heat ux are multipliedby a scale factor G that gives the Cone Rule asfollows:

Cf cone GCf plate; qwcone Gqwplate, (8)

where G 3

p 1:732 for laminar ow and G

1:13 for turbulent ow.A correlation of the heat transfer on axisym-

metric ight vehicles with at plate relations hasbeen investigated by Zoby and Sullivan [57] and anadditional correlation including ground data hasbeen investigated by Zoby and Graves [58]. Theformer includes six references for experimental dataon sharp cones where the Mach number varies from2.0 to 4.2. An assessment of the theoreticalcorrelations for sharp cones was given in Roy and

Blottner [59].3.4. Direct numerical simulation database

The numerical solution of the unsteady NavierStokes equations with rened grids as formulated inthe DNS and LES methods has the potential ofproviding an accurate numerical simulation data-base with limited or no turbulence modelingassumptions. However, the increased delity ofthese approaches requires additional temporal andspatial information for the specication of the initialand boundary conditions. As computer speed andmemory size have increased, the accuracy andcapabilities of these computational uid dynamicapproaches have increased and will increase in thefuture. Next a brief indication is given of theturbulent at plate and sharp cone ow problemsthat are being solved with the DNS and LESmethods.

Martin [60]: Martin [60] has started to develop aDNS database of hypersonic turbulent boundary-layer ows over a at plate. She provides a review ofprevious DNS solutions that have been obtained forhigh speed compressible ows. The list includes areview of other work as well as her previous paperswith coworkers. Martin has obtained DNS solu-tions for perfect gas and reacting air ows over a atplate. Martin presented DNS solutions for perfectgas ow with the gas viscosity modeled with apower law dependence on temperature. The simula-tions use freestream conditions corresponding to analtitude of 20 km and the Mach number varies from3 to 8. The wall temperature is specied to be nearlythe adiabatic temperature. At Mach 8, the walltemperature is 2713K and would result in signi-cant dissociation of the oxygen in air. The perfectgas model is not adequate to simulate these physicalow conditions. From the simulation solutionsobtained, the mean ow velocity across the bound-ary layer has been determined, then transformedwith the Van Driest transformation to incompres-sible form, and presented in gures for the casessimulated. No information is presented on the wallskin friction and heat transfer. This work isimportant as it is starting to provide useful DNSsolutions at hypersonic ow conditions. However,there is a need to extend this work by obtainingsimulations with a gas model that are moreappropriate for ight conditions or modify the owto wind tunnel conditions.

Yan et al. [61]: Yan et al. [61] have obtained LESsolutions with the Monotonically Integrated LES

(MILES) technique for at plate ow at Mach

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530480number 2.88 and 4. Both adiabatic and isothermalwall boundary conditions are used. The authorshave provided a list of researchers that have studiedcompressible LES with no work performed athypersonic ow conditions. The authors velocityprole predictions are compared to the law of thewall analysis and experimental data of Zheltovodovat Mach 2.9 and 3.74, although the computationsare at a lower Reynolds number than the experi-ment. LES temperature proles are also comparedto experimental data. The comparisons for velocityand temperature at Mach 2.9 are good but thecomparison at Mach 4 is poor. The interestingresults of this investigation are concerned with heattransfer and Reynolds analogy. The authors in-dicate that Reynolds analogy factor is Raf 2St=Cf 1=Prtm 1:124 where the mean Prandtlnumber Prtm 0:89. At Mach 4, the LES solutiongives Raf 1:23 while the experimental value isRaf 1:12. The Reynolds analogy factors differ by10%. The simulations obtained in this articleindicate the potential of the LES technique to helpvalidate turbulence models; however, the subgridscale model required in LES can affect the resultsand therefore is a limitation of this approach.

Pruett and Chang [62]: The Pruett and Chang [62]investigations are concerned with DNS of hyperso-nic boundary-layer ows on sharp cones and cone-are models. The initial work in 1995 is anapproximate simulation of the geometry and owconditions in the wind tunnel experiment of Stetsonet al. [63]. A 7 half-angle cone in a Mach 8 ow issimulated. The inviscid ow at the edge of theboundary layer is specied, and the wall tempera-ture is specied as the laminar adiabatic walltemperature, which is given as 611K. The free-stream properties are estimated from the Sims tables[142], which give the total temperature as 733K,static temperature as 53K, and the unit Reynoldsnumber as 3:407 106=m. Pruett and Chang [64] in1998 published an investigation of DNS of hyper-sonic boundary-layer ow on a ared cone. TheDNS solution is for the quiet (low freestreamturbulence) wind tunnel experiment of Lachowiczet al. [65], where the freestream turbulence has beenreduced signicantly. The axial length of the cone-are model is 0.51m and the sharp cone axial lengthis 0.254m. The freestream ow conditions for thesimulation are specied as Mach number 8, statictemperature 55K, total temperature 450K, and unitReynolds number 8:85 106=m. In both of the

above DNS the air viscosity is determined withSutherland viscosity law, and a perfect gas model isused. Although the Pruett and Chang DNScomputations do not provide useful informationon fully developed turbulent ow on the conicalpart of the models, the numerical simulationsindicate the future potential for providing valuabledata for validation of compressible turbulencemodels.

Comments on DNS database: For ows withoutchemical reactions and for typical ight conditions,the wall temperature needs to be sufciently low.The maximum gas temperature occurs in theboundary layer due to viscous dissipation and canbe sufciently high to produce vibrational excita-tion. Complete simulation without chemical reac-tions requires a vibrational non-equilibrium model.The solutions from the complete model can bebounded by using perfect gas and thermally perfectgas models, which makes DNS solutions with thesemodels valuable. The gas models need a moreappropriate viscosity model than Sutherland visc-osity law. For hypersonic wind tunnel conditions,the stagnation temperature is sufciently high tohave vibrational excitation while the freestreamtemperature in the test section is low. The modelrequirements are the same as for ight conditions asvibrational non-equilibrium effects can be impor-tant. Keyes viscosity model should be used due tothe low gas temperatures. The desired databaseshould include a matrix of accurate solutions whichdepend on Mach number, boundary-layer momen-tum thickness Reynolds number, and wall tempera-ture. A series of solutions should be obtained withonly one of the variables varying and with the othertwo variables held constant. These solutions wouldprovide a database that can be used to validate theVan Driest transformation approach to correlatecompressible turbulent skin friction and heattransfer (Stanton number) and to determine theReynolds factor in the Reynolds analogy. Inaddition, the DNS method should be extended toow over sharp cones. The database would helpto determine the Mangler transformation requiredto transform compressible turbulent ow for theaxisymmetric case to the 2D case.

3.5. Updated hypersonic turbulence model validation

database

In this section we present our recommendationsfor the current 2D/axisymmetric experimental data-

base for validating turbulence models. We adhere to

uncertainties on the surface quantities given inRefs. [31,67], nor are any uncertainties presented inthe Settles and Dodson reviews [1,4]. Holden [48]points out that the interaction region in the Cole-man and Stollery/Elfstrom experiments may be tooclose to the transition zone, resulting in a differenttrend of separation zone size versus Reynoldsnumber than seen in equilibrium turbulent bound-ary layers, which require a distance of approxi-mately 50100 boundary-layer thicknesses betweenthe transition point and the interaction region. Oncethe oncoming turbulent boundary layer has reachedan equilibrium state, the trend should be a decreaseof separation zone size (and incipient separationpoint) with increasing Reynolds number, while these

ARTICLE IN PRESS

5 2D impinging shock Kussoy and Horstman [38]

Schulein et al. [7678]

6 Flat plate/cylinder Van Driest (VDII)a [51]

Huang et al. (HBC)a [54]

Aupoix et al. (AVC) [8]

Hopkins and Keener [7981]

Horstman and Owen [8284]

Coleman et al. [85]

Kussoy and Horstman [36,38]

Hopkins et al. [86,87]

Holden et al. [42,45,46,48,88]

7 Sharp circular cone Van Driest (VDII)a[51] and

Whitea [56]

Kimmel [89,90]

Rumsey et al. [91,92]

Chien [93]

Hillier et al. [9497]

Holden et al. [4244,46,48,98]

aDenotes a data correlation.

C.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530 481Settles and Dodsons necessary criteria for accep-tance of experiments into the validation database[14] with one exception. Here we relax criteria #4(well-dened experimental boundary conditions)and allow cases where the boundary layer is notcharacterized upstream of the interaction region aslong as this upstream boundary layer can beadequately predicted by turbulence models orcorrelations (i.e., for at plates, cylinders, or sharpcones where an equilibrium boundary layer hasbeen established). Note that in the original Settlesand Dodson validation database, they also appearto have relaxed this requirement for the Colemanand Stollery/Elfstrom [31,66,67] compression cornerexperiment.

3.5.1. Previous flow geometries with adverse pressure

gradient

This section describes the four 2D/axisymmetrichypersonic ow geometries that were assessed in theSettles and Dodson database [14]. For each ofthese ow geometries, we present both the experi-ments that were included in the Settles and Dodsondatabase, as well as new experiments conductedsince 1994 which we recommend for inclusion in a2D/axisymmetric hypersonic validation database.The experiments discussed in this section areincluded as Cases 14 in Table 1.

3.5.1.1. Case 1: 2D compression corner. There aretwo hypersonic experiments for the 2D compressioncorner which are deemed acceptable with somecaveats. The rst experiment was conducted in theMach 9 nitrogen gun tunnel and includes heattransfer measurements (see [31,66]) and surfacepressures from a separate experiment (see [67]).The second experiment was conducted at theCalspan 48 in and 96 in shock tunnels at Mach 8by Holden [48].

Coleman and Stollery/Elfstrom experiment

[31,66,67]: Elfstrom [67] reported surface pressuremeasurements and Coleman and Stollery [31] andColeman [66] reported surface heat transfer mea-surements for a Mach 9.22 ow over a 2D wedge/compression corner. The wedge angle was variedbetween 15 and 38, and includes ows which arenominally attached, at incipient separation, andfully separated. It is not fully clear whether or notthe pressure [67] and heat transfer [31] measure-ments had the same upstream length for the atplate. While this experiment is considered accepta-

ble by Settles and Dodson, there are no numericalTable 1

Turbulence validation database for 2D/axisymmetric hypersonic

ows

Case

no.

Flow geometry Experiments

1 2D compression

corner

Coleman and Stollery/Elfstrom

[31,66,67]

Holden [48]

2 Cylinder with

conical are

Kussoy and Horstman [32]

Babinsky and Edwards [68,69]

3 Cone with conical

are

Holden [44]

4 Axisymmetric

impinging shock

Kussoy and Horstman

[35,70,71]

Hillier et al. [7275]experiments showed the opposite trend.

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530482Holden experiment [48]: Due to the concernsregarding the equilibrium nature of the upstreamboundary layer, the compression corner experi-ments conducted by Holden [48] are also includedhere, although they too fail to report uncertaintieson the surface measurements. While no upstreamboundary-layer prole is measured in this experi-ment, the surface quantities compared well to theVan Driest II correlations [99]; furthermore, thetransition location can be easily determined fromthe surface skin friction and heat transfer measure-ments made on a at plate and reported in the samereference. The freestream Mach number for thiscase is approximately 8, and the ratio of walltemperature to the freestream stagnation tempera-ture was 0.3. The Reynolds number based on theboundary-layer thickness at the interaction locationwas varied between 100,000 and 10 million.Measurements are reported for surface pressure,skin friction, and heat transfer in the interactionregion for wedge angles of 27, 30, 33, and 36 andalong the at plate in a conguration without thewedge. Span effects were also investigated andshown to be negligible. This was the rst experimentto show the reversal of the separation zone size andincipient separation with Reynolds number withinthe same experiment. To our knowledge, thisexperiment has not been employed for validatingturbulence models.

3.5.1.2. Case 2: cylinder with conical flare. Thereare two experiments which meet the Settles andDodson criteria for the axisymmetric cylinder-aregeometry. The rst was included in the Settles andDodson review and was performed by Kussoy andHorstman [32] at NASA-Ames Research Center.The second is a more recent experiment performedin the HSST supersonic blow-down wind tunnel andis detailed by Babinsky [69] and Babinsky andEdwards [68].

Kussoy and Horstman Experiment [32]: Kussoyand Horstman [32] studied the ow over axisym-metric ogive-cylinder-ares at a freestream Machnumber of 7 for are angles between 20 and 35.Data include surface pressure and surface heattransfer both upstream of the shock/boundary-layerinteraction (see Case 6: at plate/cylinder ow) andin the interaction region. Pitot-probe surveysthrough the boundary layer are presented at variousaxial locations for the 20 are case only, with theboundary-layer surveys upstream of the interaction

conrming a fully-developed turbulent boundarylayer. Information is given on freestream RMSvalues for stagnation temperature and mass ux aswell as temperature for the water-cooled modelsurface. The data set includes experimental uncer-tainty estimates for each measured quantity. De-rived boundary-layer quantities (displacementthickness, momentum thickness, etc.) are alsoreported upstream of the interaction. Surface datawere taken at 90 locations to conrm that the owwas axisymmetric, and multiple runs were con-ducted to reduce run-to-run uncertainty. A rela-tively long model was employed to allow for natural(non-tripped) transition to occur at approximately0.4 to 0.8m from the tip, which is at least 0.6mupstream of the interaction region.

Babinsky and Edwards Experiment [68,69]: Ba-binsky and Edwards [68] conducted careful experi-mental studies of cylinder-are ows at Mach 5.1for are angles between 3 and 20, with additionaldetails presented by Babinsky [69] for are angles of15 and 20. The experiments were conducted in thesupersonic blow-down wind tunnel (HSST) at DRAFort Halstead, Great Britain using a Mach 5 nozzlethat included a cylindrical centerbody which ex-tended upstream of the test section to the nozzlethroat. This centerbody was deemed necessary toallow for the formation of a fully-developedturbulent boundary layer without the use of ow-intrusive boundary-layer tripping mechanisms.However, the use of the centerbody led to thepresence of non-negligible axial gradients of pitotpressure (10% variation) and Mach number (3%variation) in the test section. Data were presentedfor surface pressure, surface heat transfer (via high-resolution liquid crystal thermography), and pitotpressure through the boundary layer at variousaxial stations. Detailed experimental uncertaintieswere also provided for each of the measuredquantities. Derived quantities presented includevelocity proles, skin friction, boundary-layer thick-ness, and displacement thickness. Conventionaltheory suggests that for are angles of 20 andbelow, no separation will occur. However, investi-gations using shear stress sensitive liquid crystalsshowed a small separated region (possibly in thelaminar sublayer) for both the 15 and 20 arecases. The authors suggest that the presence ofthis small separation zone destroys the similaritybetween pressure and heat transfer.

3.5.1.3. Case 3: cone with conical flare. There is

only one experiment for the cone/conical are case

reference a 1984 AIAA Paper [34], a 1986 CUBRCinternal report [33], and a 1988 AFOSR technical

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530 483report [100], all by Holden and coworkers. How-ever, the initial reporting of these cone/conical areexperiments was not until 1991 in Ref. [44], and thisis conrmed by examining Refs. [42,101] which arereviews of the experimental hypersonic programconducted at Calspan. In any case, the datapresented by Settles and Dodson [1] does appearto be the same data given in Refs. [44,101]. It shouldbe noted that there is also some question regardingthe are angle for this case. Ref. [44] does not makeit clear whether the are angle is measured from thesymmetry axis or from the initial cone angle of 6,while Ref. [101] clearly shows that the are angleshould be measured from the symmetry axis.However, Settles and Dodson [1] state that the areangles should be measured from the 6 forecone,and crude angles measured from Schlieren photo-graphs in Refs. [44,101] seem to support thisconclusion. Subsequent communications with theauthor of Ref. [44] conrmed that the flare angleshould be measured from the 6 cone, not thesymmetry axis [102].

Holden experiment [44]: Holden [44] studied theow over 6 (half angle) cones with conical ares atfreestream Mach numbers of 11 and 13 for areangles of 30 and 36 as measured from the forecone(36 and 42 from the symmetry axis). The smallerare angle represents an incipient separated owcase, while the larger angle a fully separated ow.Data include surface pressure and heat transfer aswell as pitot pressure and total temperature withinthe interaction region. Experimental uncertaintiesare given for the freestream conditions as well asheat transfer coefcient 5% and pressure coef-cient 3%.

3.5.1.4. Case 4: axisymmetric impinging shock. Thereare two different axisymmetric impinging shockexperiments which are deemed acceptable for turbu-lence model validation. The rst is a series ofexperiments conducted by Kussoy et al. at a Machnumber of 7 on a cone-ogive-cylinder modelthat is appropriate for turbulence model validation.Holden [44] performed experiments in Calspans 96in shock tunnel at Mach numbers of 11 and 13. Thisis one of Settles and Dodsons accepted hypersonicexperiments; however, there is some discrepancyregarding the references. Settles and Dodson [1,4][35,70,71]. The second is a more recent experimentalinvestigation by Hillier et al. at Mach 9 on a hollowcylinder model [7275].

Kussoy and Horstman Experiment [35,70,71]:Kussoy and Horstman studied the ow over anaxisymmetric cone-ogive-cylinder model at a free-stream Mach number of 6.9. Axisymmetric cowls of7:5 and 15 were used to impinge axisymmetricshock waves onto the turbulent boundary layer onthe cylinder. Surface data include surface pressure,heat transfer, and skin friction [35,70]. Pitotpressure, static pressure, and total temperature weresurveyed throughout the interaction region [35,70].Ref. [71] also presents turbulence intensity andReynolds stress proles for these two cases. Thecowl length is relatively short, thus the leadingshock wave and subsequent expansion fan mergebefore the shock impinges on the surface. It istherefore strongly recommended that future com-putations of this experiment also include the viscousboundary layer on the outer cowl itself. The lengthof the model cylinder is 3.3m, thus suggesting afully-developed turbulent boundary layer is formedwell ahead of the interaction region. The modelsurface is water-cooled to maintain a temperature of300K. The data set includes experimental uncer-tainty estimates for each measured quantity, withthe exception of the turbulence measurements ofRef. [71]. Derived boundary-layer quantities (dis-placement thickness, momentum thickness, etc.) arealso reported upstream of the interaction. Settlesand Dodson [1] give the experimental data for the15 shock generator case in tabular form.

Hillier et al. Experiment [7275]: Hillier et al. [72]studied the ow over an axisymmetric cone-ogive-cylinder model at a freestream Mach number of 8.9and Reynolds number of 52 106=m. An axisym-metric cowl with a quadratic expression for theshock-generating surface is used to impinge anaxisymmetric shock wave onto the turbulentboundary layer formed on the cylinder. Surfacedata include surface pressure and heat transfer.Transition of the boundary layer on the cylinderbegins at 80mm and ends at 170mm, with the shockinteraction occurring at roughly 520mm. Due to thecurved nature of the shock-generating cowl, it isrecommended that computations of this experimentalso include the viscous boundary layer on the outercowl itself. Although not reported in the experi-mental description, the model surface temperaturewas 300K [103]. The data set includes experimentaluncertainty estimates for the surface pressure and

heat transfer. An additional discussion of the

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530484experimental uncertainties is given in Ref. [73]which quotes uncertainties of 4% and 7% forsurface pressure and heat transfer, respectively.More recent data for shock generator angles of4:7 (attached ow) and 10 (separated ow) aregiven by Murray and Hillier [74,75] for a hollowcylinder forebody. Although no experimental un-certainties were quoted in these last two references,the same experimental techniques that wereused in Refs. [72,73] were employed. Personalcommunication with one of the authors conrmedthe uncertainty levels given above [103]. Forthe 10 shock generator case, the length of theshock-generating outer cowl had to be reduced toprevent choking of the shock system. This recentexperiment has not yet been used in the validationof one- or two-equation turbulence models but isrecommended.

3.5.2. New flow geometries with and without pressure

gradient

In this section we discuss the three new owgeometries which should be added to the 2D/axisymmetric hypersonic turbulence database. Therst is the 2D impinging shock problem and isreferred to as Case 5. Cases 6 and 7 are zero-pressure gradient ows (at plate/cylinder and coneow) where the simplicity of the ow also allows thedevelopment of theoretical correlations.

3.5.2.1. Case 5: 2D impinging shock. A 2D imping-ing shock occurs when an externally generatedoblique shock impinges on a at plate boundarylayer. There are two experimental data sets thatsatisfy the Settles and Dodson criteria: Kussoy andHorstman [38] conducted a careful experimentalstudy of the 2D impinging shock case in the Ames3.5 ft Hypersonic Wind Tunnel at Mach 8.2, andSchulein et al. [7678] studied a similar geometry atMach 5.

Kussoy and Horstman Experiment [38]: Kussoyand Horstman [38] studied a 2D oblique shockimpinging on a turbulent at plate boundary layerat a freestream Mach number of 8.2 for effectivewedge angles of 5, 8, 9, 10, and 11. Datainclude surface pressure and heat transfer bothupstream of the shock/boundary-layer interaction(see Case 6: at plate/cylinder ow) and in theinteraction region. Mean ow surveys through theboundary layer are given for the undisturbedboundary layer (i.e., without the shock generator)

in the vicinity of the shock interaction. Surveys inthe interaction are alluded to in the report butare not presented (these may be included on acomputer disk which is mentioned in the report).The model is water-cooled to maintain a surfacetemperature of 300 5K. The data set includesextensive experimental uncertainty estimates foreach measured quantity, but not for the freestreamconditions. A relatively long 2.2m model wasemployed to allow for natural (non-tripped)transition to occur approximately 0.5 to 1mfrom the leading edge, which is at least 0.5mupstream of the interaction region. The modellength results in a fully developed, equilibriumturbulent boundary layer (conrmed by the ow-eld surveys) that is nearly 4 cm thick near theinteraction region.

Schulein et al. Experiment [7678]: Schulein andcoworkers [7678] have also studied a 2D obliqueshock impinging on a turbulent at plate boundarylayer. The ow is at Mach 5 and wedge angles of 6

to 14 were studied. The at plate was 0.5m longand 0.4m wide, with natural transition occurring0.1m from the leading edge, as judged by peak skin-friction. For all shock generator angles, the inter-action occurs approximately 0.35m from the lead-ing edge. The spanwise extent was chosen to ensure2D ow, which was further conrmed by surfacepressure and pitot survey data taken at variousspanwise locations. The Reynolds number based ondistance from the leading edge at the interactionregion was 13 106, and the boundary-layer thick-ness at this location was approximately 5mm.The wall temperature is 300 5K. The originalreport [76] gives pitot surveys in the interactionregion, as well as wall static pressures with anestimated uncertainty of 5% [78]. Surface skinfriction (obtained via optical means) and heattransfer data are also available in Ref. [77] anduncertainties in the interaction region for both aregiven as 10% [78].

3.5.2.2. Case 6: flat plate/cylinder. The uniformviscous ow over a at plate or cylinder isconsidered, where a laminar to turbulent boundarylayer develops along the surface. The 2D zero-pressure gradient turbulent boundary-layer owproblem is unique. Theoretical analyses of this casefor perfect gas ows have been performed whichresult in correlations of the experimental results.The turbulent properties of interest are the wall skinfriction, heat transfer, and proles of velocity and

temperature across the boundary layer.

A survey has been performed of authors that haveused hypersonic turbulent boundary-layer experi-mental data with zero-pressure gradient to validate

correlations of skin friction and Stanton numberand correlations of velocity and temperature pro-les. Tabulation of the survey is given in Table 2

ARTICLE IN PRESS

Table 2

Model geometries and freestream conditions for hypersonic at plate/cylinder database

Investigator/geometry Date Me Cf St Proles Tabulated data reference

SommerShort [104] (hollow

cylinder, ight range)

1953 5.63, 6.9, 7.0 Yes No No Peterson [105]

Korkegi [106] 1956 5.79 Yes AdW No Peterson [105]

Hill [107] (conical nozzle, nitrogen

gas)

1956 8.39.1 Yes Yes Yes F&F 5901

Hill [108] (conical nozzle, nitrogen

gas)

1959 8.27, 9.07,

10.04

Yes Yes No Hill [108], Peterson [105]

Tendeland [109] (cone-cylinder) 1958 5.04 No Yes No SpaldingChi [110], Author

BrevoortArabian [111]

(downstream inside cylinder)

1958 5.05 No Yes No SpaldingChi [110], Author

WinklerCha [112] 1959 5.14, 5.22,

5.25

Yes Yes Yes F&F 5902

WinklerCha [112] 1961 5.14, 5.22,

5.25

No Yes Authors [112], Peterson [105]

Matting et al. [113] (at wind tunnel

wall, helium gas)

1961 6.7, 9.9 Yes AdW Yes Peterson [105]

Moore [114] 1962 5 No No Yes F&F 6201

Young [115] 1965 5 Yes Yes Yes F&F 6506

WallaceMcLaughlin 1966 7.4, 8.1, 10.7 Yes Yes No CaryBertram [53]

Wallace [117] 1967 7.4, 8.1, 10.7 No Yes No No tabulation

Heronimus [118] 1966 4.6 to 11.7 Yes Yes,

AdW

No Cary [119]

Neal[120] 1966 6.8 Yes Yes No No tabulation

CaryMorrisette [121] (wedge

a 5; 0; 5)1968 6.8, 6.0, 5.3 No Yes No CaryBertram [53]

Hopkins et al. [86] 1969 6.5 Yes AdW Yes Authors [86]

Hopkins et al. [87] 1970 6.5 Yes Yes Yes Authors [87]

Hopkins et al. [122] 1972 6.5 Yes Yes Yes Authors [122]

Cary [123] 1970 6 No Yes No CaryBertram [53]

Weinstein [124] (10 deg wedge) 1970 8, 6.8, 6, 5.2 No Yes No CaryBertram [53]

HopkinsKeener [80] (tunnel wall) 1972 7.4 Yes No Yes F&F 7203

KeenerHopkins [81] (thermally

perfect gas)

1972 6.26.5 Yes No Yes F&F 7204

OwenHorstman [82] (cone-ogive-

cylinder)

1972 7.2 Yes Yes Yes F&F 7205, Authors [82]

HorstmanOwen [83] (cone-ogive-

cylinder)

1972 7.2 Yes Yes Yes Authors [83]

Owen et al. [84] (cone-ogive-

cylinder)

1975 7.0 Yes Yes Yes No tabulation

Holden [45] 1972 6.813 Yes Yes No Author [45]

Holden electronic database [46] 2003 Author [46]

Coleman et al. [85] (at plate

negative angle)

1972 7.159.22 No Yes Yes CaryBertram [53]

Watson et al. [125] (wedge, helium) 1973 910 Yes Yes Yes F&F 7305

Watson [126] (wedge, helium) 1978 1011.6 Yes Yes Yes F&F 7804

LadermanDemetriades[127] (wind

tunnel wall)

1974 9.4 Yes No Yes F&F 7403

Y

Y

C.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530 485KussoyHorstman [38] 1991 8.18

KussoyHorstman [36] 1992 8.28

AdW, adiabatic wall; F&F x, Fernholz and Finley case x.es Yes Yes Authors [38]

es Yes Yes Authors [36]

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530486where the experiments are for at plate ows in airunless indicated otherwise. Since the correlationshave been developed for perfect gas ows and fullydeveloped zero-pressure gradient ow, the experi-mental database is limited to at plates, wedges,wind tunnel walls and cylindrical body ows. Alsothe ow properties should be measured sufcientlyfar downstream where the turbulent ow is fullydeveloped. Upstream history effects can signi-cantly inuence the database as the non-equilibriumturbulent effects sometimes persisting 1000s ofboundary-layer thicknesses downstream [48].For validation of correlation theory and turbu-

lence models, direct measurement of skin friction,heat transfer, velocity proles, and temperatureproles should be made. In addition, the hypersonicexperimental database should be limited to the sameow geometry (at plate) with zero-pressure gra-dient and same freestream perfect gas model. Thesharp wedge and at plate produce the sameturbulent boundary layer as the ow Reynoldsnumber approaches innity. Flow along hollowcylindrical geometries with decreasing transversecurvature effects approaches at plate ow. Whilecylindrical geometries with conical or ogive nosegeometries produce initial disturbances that inu-ence the boundary layer, far downstream a atboundary layer is obtained. Wind tunnel wallboundary layers also have upstream history effectsthat impact the attainment of a at plate ow.Many of the earlier experiments did not measure

both the skin friction and the wall heat transfer andare not as useful. While most experiments areperformed in air, two experiments use helium gasand one uses nitrogen gas. The ideal hypersonicexperimental database for validation of correlationtheories becomes very small if one applies all of therestrictions mentioned above. However, many ofthe experiments in Table 2 have been useful inestablishing the accuracy of the correlation theoriesand determining the validity of turbulence models(see Section 4.5.2). Many of the experiments areuseful in direct comparison between turbulentmodel predictions and experimental results toestablish turbulent model validation. This approachis a signicantly larger computational effort, butwould also help in the evaluation of the importanceof the differences in the experiments.An extensive database of compressible turbulent

ows in a standard form has been compiled byFernholz and Finley [2830] as discussed in 3.2.1.

This database is limited to 2D ows where owprole data are available in tabular form. Assess-ment of the data quality or signicance of the data isnot complete. Tabulated data is given for the edgeand wall ow properties and survey propertiesacross the boundary layer. The hypersonic databasegiven in the Fernholz and Finley reports include anumber of experiments where either real gas effectswere an issue, a fully turbulent boundary layer wasnot established after transition process, or the windtunnel side wall was used to generate the boundarylayer (thus bringing the equilibrium nature of theboundary layer into question). As a result, theFernholz and Finley reports are of limited value.Two experiments which will be included from the

Fernholz and Finley database are those of Hop-kinsKeener and HorstmanOwen (discussed indetail below). Additional hypersonic experimentsthat have not been included in earlier databasereviews and are considered useful are discussedbelow.

Hopkins Keener [7981] (Fernholz & Finley Cases6601, 7203 and 7204): The initial work by Hopkinsand Keener [79] was concerned with measuring theproperties of the turbulent boundary layer on theside wall of the NASA Ames 8 7 ft supersonicwind tunnel at Mach 2.43.4. The next investiga-tions were performed in the NASA Ames 3.5 fthypersonic wind tunnel. Hopkins and Keener [80]measured the local skin friction, total-temperatureproles, and pitot-pressure proles on the hyperso-nic wind tunnel wall. Although the pressure gradientis small near the measurement location, thereappears to be signicant upstream history effectsin this experiment. Keener and Hopkins [81]investigated the wind tunnel air ow over a flatplate at Mach 6.26.5. The total temperature ofthe freestream was 7641028K and the temperatureat the edge of the boundary layer was 73K orgreater. The analysis of the air ow propertiesincluded corrections for calorically imperfect gaseffects. The boundary-layer properties were investi-gated with forced and natural transition. Surfaceproperties measured: pressure, temperature, andwall shear stress. Properties measured acrossboundary layer: static pressure, pitot pressure, andtotal temperature.

Horstman Owen [8284] (Fernholz & Finley Case7205): This investigation was performed for air owover an axisymmetric cone-ogive-cylinder at Mach7.2 in the NASA Ames 3.5 ft hypersonic windtunnel. The total temperature of the freestream air

was 667K and the temperature at the edge of the

ARTICLE IN PRESSC.J. Roy, F.G. Blottner / Progress in Aerospace Sciences 42 (2006) 469530 487boundary layer was 59K. Natural transitionoccurred along the body and the boundary layerbecame an equilibrium, constant pressure owdownstream on the body. The transverse curvatureeffects are considered to be negligible for thisgeometry. Fluctuating properties of the ow werealso measured and hu0i=ut 0:17 at y=d 1:1,which gives an indication of the turbulent intensityin the freestream ow. Surface properties measured:pressure, temperature, wall shear stress. Propertiesmeasured across boundary layer: pitot pressure,total temperature.Additional experiments that have not been

included in the Fernholz and Finley database[28,30] follow below.

Coleman Elfstrom Stollery [85] (see also Refs.[31,66,67]): The compressible turbulent boundarylayer on a flat plate was studied at a freestreamMach number of 9 in the Imperial College no. 2 guntunnel. The total temperature of the freestream airwas 1070K. The local boundary-layer edge Machnumber was varied (Mach 3, 5, and 9) by changingthe incidence of the plate from 0 to 26:5. Bothnatural and tripped boundary layer ows wereinvestigated. Theory based on SpaldingChi skin-friction and Reynolds analogy was used to predictthe Stanton number for the three Mach numbers.There was an increased discrepancy between mea-surements of heat transfer and the prediction of thetheory as the Mach number was increased. Surfaceproperties measured: static pressure and heattransfer. Properties measured across boundarylayer: pitot pressure.

Kussoy Horstman [36,38]: The experiments wereconducted with ow over a water-cooled flat plate inthe NASA Ames 3.5 ft hypersonic wind tunnel atMach 8.2. The at plate without a sharp n is thedatabase that is being considered. The plate surfacewas maintained at a constant surface temperature of300 5K. In the rst experiment [38], the proper-ties of the boundary layer 1.87m from the leadingedge were determined. In the second experiment[36], the properties of the boundary layer 1.62mfrom the leading edge were determined. Naturaltransition occurred between 0.5 and 1.0m fromthe leading edge. A fully developed, equilibriumboundary layer was established at the measurementlocation. Tabulated results are presented forboundary-layer surface and edge properties.Tabulated results for the velocity, density andtemperature proles are also given for the measure-

ment location. Surface properties measured: staticpressure and heat transfer. Properties measuredacross boundary layer: pitot pressure, static pres-sure, and total temperature.

Hopkins et al. [86,87,122]: The initial experimentswere performed by Hopkins et al. [86] on simpleshapes for turbulent boundary layers with nearlyzero-pressure gradient in the NASA Ames 3.5 fthypersonic wind tunnel. Local skin friction and heattransfer data were measured on at plates (Mach6.5 and 7.4), cones (edge Mach 4.9, 5.0, and 6.6),and the wind tunnel wall (Mach 7.4). Skin-frictiondata are given in tabulated form. The next experi-ments were performed by Hopkins et al. [87] on asharp leading edge at plate in the same Amesfacility. Flat plate skin-friction was measureddirectly with an edge Mach number of 6.5. Theskin-friction experimental database at various mo-mentum thickness Reynolds numbers and adiabaticwall temperature ratios are given in tabulated form.Hopkins et al. [122] conducted further at plateexperiments in the same Ames facility. This studyprovides additional results to those previouslyreported by Keener and Hopkins [81], but at ahigher Reynolds number. The model was injectedinto the airstream at various angles of attack, whichresulted in local Mach numbers at the measuringstation of 5.9, 6.4, 6.9, 7.4, and 7.8. No boundary-layer trips were used. The model surface tempera-ture was nearly isothermal. Direct measurements ofskin friction and velocity proles were made for thevarious Mach numbers and for Tw=Taw 0:3 and0.5. Real gas corrections as given in Ref. [128] wereused in the analysis of the data. Tabulated results ofthe database are given in the article. Surfaceproperties measured: wall shear stress with a skin-friction balance. Properties measured across bound-ary layer: pitot pressure.

Holden [42,45,46,48,88]: Experiments [48,45] wereconducted on a at plate in the Calspan 48 in and 96in shock tunnels at Mach numbers 6.813. Steadyow was established in these facilities in 1 or 2ms.The investigation measured the wall shear stress andthe heat ux. The wall skin friction and heattransfer results were transformed with the VanDriest, Eckert, and SpaldingChi methods andcompared in gures to the incompressible results.The experimental data is approximately within 30%of the incompressible results, which is more scatterthan expected from experimental results. The paper[45] provides tabulated test conditions, heat trans-fer, and skin-friction. Another experiment [88] was

conducted in the Calspan tunnels on a at plate

with a constant curvature surface downstream witha freestream Mach number of 8. The upstream partof the database on the at plate could be useful, butneeds further evaluation. A brief summary ofexperiments performed on at plates is given byHolden and Moselle [42]. An electronic database[46] of results from the many experiments per-formed by Holden is now available on the internetto qualied users. A further evaluation of theusefulness of the Holden at plate database needsto be performed.

3.5.2.3. Case 7: sharp circular cone. The sharp conemodel is dened by the cone half-angle (angle

p1 r1RT1 p0 1g 12

M21

g=g1.

The nose radius Rn of the sharp cones is onlyavailable for the Kimmel experiment.

Rumsey et al. [91,92]: Rumsey and coworkers[91,92] performed a number of ight tests withvehicles with a sharp-cone nose and at varioussupersonic/hypersonic Mach numbers. The Rumseyand Lee [91] report has data at Mach 5.15 and hasbeen included as a potential part of the presentsharp cone database. The authors present much ofthe needed database information in gures. Thefreestream unit Reynolds number is not consistent

ARTICLE IN PRESS

Model geometries and freestream conditions for cone database

05