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900MAN2253A DISTRIBUTION 21.2.3

VOLUME 6

Special Report

Optical Signatures Code

Volume 6

DIPREF Theory Manual

June 2016

SPONSORED BY THE MISSILE DEFENSE AGENCY

NOTICE – The views, opinions, and/or findings contained in this document are those of the author(s) and should not be construed as an official Department of Defense position, policy, or decision unless designated by other official documentation.

DISTRIBUTION STATEMENT C – Distribution authorized to U.S. Government Agencies and their contractors; Critical Technology, Export Controlled; 30 November 2015. Other requests for this document shall be referred to MDA/ESP (Attn: Mark Underwood or active program COTR), 5224 Martin Road, Redstone Arsenal, AL 35898 or higher DoD authority.

WARNING – This document contains technical data whose export is restricted by the Arms Export Control Act (Title 22, U.S.C. sec 2751 et seq.) or the Export Administration Act of 1979, as amended, Title 50, U.S.C., App. 2401 et seq.

Violations of these export laws are subject to severe criminal penalties.

Disseminate in accordance with the provisions of DoD Directive 5230.25.

DESTRUCTION NOTICE – For classified documents follow the procedures in DoD 5200.22-M, Industrial Security Manual, Section 11-19 ORD or 5200.1-R, Information Security Program Regulation Chapter IX. For unclassified, limited documents destroy by any method that will prevent disclosure of contents or reconstruction of the document.

NOTICE – This document contains information that may be exempt from mandatory disclosure under the Freedom of Information Act.

i

NOTICE

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Per Department of Defense (DoD) Manual 5200.01, Volume 4, February 24, 2012, Objective

Simulation Framework (OSF) documents, including this document, are considered to be

Unclassified but For Official Use Only (FOUO) and exempt from the Freedom of Information

Act (FOIA) as a result of compilation. The OSF software and documentation are considered export controlled. Though no discrete portion of this document is considered to be export controlled, the information that may be extracted from this document as a whole is considered to be export controlled and therefore may be exempt from FOIA and is considered FOUO.

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Optical Signatures Code Volume 6 – DIPREF Theory Manual OSF Contract No.:

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900MAN2253A

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Controlled; 30 November 2015. Other requests for this document shall be referred to MDA/ESP (Attn: Mark Underwood or active program COTR), 5224

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13. ABSTRACT (Maximum 200 words)

DIPREF (Direct Procedure for ReEntry Flow) generates the aerothermal environment suitable for hardbody thermal response and optical signature calculations from ascent to impact of missiles. DIPREF is accessed through the OSC (Optical Signatures

Code) endoatmospheric thermal response model, RVTEMP or within the OPTIcal Signature In-Line Generator (OPTISIG).

This manual provides users with a technical discussion of DIPREF’s capabilities.

14. SUBJECT TERMS 15 NUMBER OF PAGES

aerothermal, ascent, flowfield, reentry, optical signatures, OSC, OPTISIG, DIPREF 59

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SPECIAL REPORT

900MAN2253A

OPTICAL SIGNATURES CODE

Volume 6

DIPREF Theory Manual

June 2016

NOTICE – The views, opinions, and/or findings contained in this document are those of the author(s) and should not be construed as an official Department of Defense position, policy, or decision unless designated by other official documentation.

DISTRIBUTION STATEMENT C – Distribution authorized to U.S. Government Agencies and their contractors; Critical Technology, Export Controlled; 30 November 2015. Other requests for this document shall be referred to MDA/ESP (Attn: Mark Underwood or active program COTR), 5224 Martin Road, Redstone Arsenal, AL 35898 or higher DoD authority.

WARNING – This document contains technical data whose export is restricted by the Arms Export Control Act (Title 22, U.S.C. sec 2751 et seq.) or the Export Administration Act of 1979, as amended, Title 50, U.S.C., App. 2401 et seq.

Violations of these export laws are subject to severe criminal penalties.

Disseminate in accordance with the provisions of DoD Directive 5230.25.

DESTRUCTION NOTICE – For classified documents follow the procedures in DoD 5200.22-M, Industrial Security Manual, Section 11-19 ORD or 5200.1-R, Information Security Program Regulation Chapter IX. For unclassified, limited documents destroy by any method that will prevent disclosure of contents or reconstruction of the document.

NOTICE – This document contains information that may be exempt from mandatory disclosure under the Freedom of Information Act.

Prepared For

Missile Defense Agency

Redstone Arsenal, Alabama

Under Contract Number

HQ0147-11-D-0015, Task Order 9

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This document was prepared and released in accordance with CDRL A039.

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Updated distribution markings, task order number, sponsor, text on page 4-1, Table 4-2, Table 4-4, and references 1 and 37.

v

ABSTRACT

DIPREF (Direct Procedure for ReEntry Flow) generates the aerothermal environment suitable for hardbody thermal response and optical signature calculations from ascent to impact of missiles. DIPREF is accessed through the OSC (Optical Signatures Code) endoatmospheric thermal response model, RVTEMP, or within the OPTIcal Signature In-line Generator

(OPTISIG). This manual provides users with a technical discussion of DIPREF’s capabilities.

Portions of this manual that describe input options are not necessarily updated with each

OSC or OPTISIG release. Users are advised to review the latest OSC Basic Option Reference

Manual (Volume 1) and OSC Basic Option Example Case Manual (Volume 2) for a complete set of input options and example cases.

APPROVED BY:

vi

FOREWORD

The hardbody flowfield model DIPREF, discussed herein, is part of the Optical Signatures

Code (OSC) and OPTIcal Signature In-line Generator (OPTISIG). OSC and OPTISIG are under the technical direction of Dr. Mark Underwood (MDA/ESP) for the Missile Defense Agency.

Recent previous releases were issued under the technical direction of Mr. Cliff Burson

(MDA/ESP) and Ms. Carol Barclay (USASMDC/ARSTRAT).

vii

Table of Contents

1. Introduction .......................................................................................................................... 1-1

2. DIPREF Description .............................................................................................................. 2-1

2.1 DIPREF Overview .......................................................................................................... 2-1

2.2 Input/Output Interfaces with Other OSC Models ....................................................... 2-2

3. DIPREF Theory ..................................................................................................................... 3-1

3.1 Flowfield Analysis ......................................................................................................... 3-1

3.2 Heat Transfer Model .................................................................................................... 3-6

3.2.1 Laminar, Transitional and Turbulent Flow Regions ............................................ 3-7

3.3 Variable Entropy Correction ......................................................................................... 3-8

3.4 High-Altitude Correction ............................................................................................... 3-9

3.5 Base Heating ............................................................................................................. 3-11

3.6 Viscous Interaction Effects at Flares and Flaps ...................................................... 3-12

3.6.1 Plume/Body Interaction ..................................................................................... 3-16

3.7 Subsonic and Low Supersonic Region ..................................................................... 3-16

3.8 Fins ............................................................................................................................. 3-17

3.9 Angle of Attack Effects .............................................................................................. 3-18

3.10 Validation/Comparison ............................................................................................. 3-20

3.11 Discussion .................................................................................................................. 3-23

4. DIPREF Input ........................................................................................................................ 4-1

5. Example Cases ..................................................................................................................... 5-1

6. Diagnostics and Troubleshooting ....................................................................................... 6-1

7. References ........................................................................................................................... 7-1

A. List of Symbols ..................................................................................................................... A-1

B. List of Key Program Variables ............................................................................................. B-1 viii

List of Tables

Table 2-1: DIPREF I/O Summary ................................................................................................ 2-2

Table 2-2: Trajectory File (TRAJ, Unformatted, Unit 7; ATRAJ, Unit 1) ..................................... 2-3

Table 2-3: File for Communication between DIPREF and F678 (FORF678, Unit 20) ............ 2-4

Table 4-1: How to Access DIPREF .............................................................................................. 4-2

Table 4-2: RVTEMP Program Control ......................................................................................... 4-2

Table 4-3: Input for SHINE .......................................................................................................... 4-2

Table 4-4: Namelist NAME Input for DIPREF ............................................................................. 4-3 ix

List of Figures

Figure 1-1: General OSC Program Flow ..................................................................................... 1-1

Figure 1-2: RVTEMP/DIPREF Program Flow .............................................................................. 1-2

Figure 3-1: Sketch of Coordinate System from Reference 2 ................................................... 3-2

Figure 3-2: Sketch of Geometry for Calculating Boundary-Layer Edge Properties ................. 3-9

Figure 3-3: Sketch of Base Geometry from Reference 13 .................................................... 3-12

Figure 3-4: Flow Separation and Formation of Separation Shock, Reattachment, and Dividing

Streamline in Viscous Corner Flow ......................................................................................... 3-13

Figure 3-5: Wall Pressure Distributions for Laminar and Turbulent Flow for Separated Corner

Flow ........................................................................................................................................... 3-13

Figure 3-6: Comparison of Pressure Distribution over Blunt Cones ..................................... 3-20

Figure 3-7: Laminar Heat Transfer Distribution over Spherical Nosecap from Reference 6 ... 3-

Figure 3-8: Comparison with Flight Test Data of Reference 33 ........................................... 3-22

Figure 3-9: Comparison of Pressure and Heat Transfer Distribution for 15° and 30° Flare from Reference 34 ................................................................................................................... 3-22

Figure 4-1: Generic Target Available in DIPREF ........................................................................ 4-6

Figure 5-1: Baseline Target Description .................................................................................... 5-1

Figure 5-2: Baseline Target in WinOSC ...................................................................................... 5-2

Figure 5-3: V2-A4 Target Description ......................................................................................... 5-3

Figure 5-4: Sample WinOSC Output for V2-A4 .......................................................................... 5-4

Figure 5-5: WIND1 Input ............................................................................................................. 5-5

Figure 5-6: WIND1 Sample Output from WinOSC ..................................................................... 5-5

1-1

1. Introduction

The Optical Signatures Code (OSC) consists of a set of simulation components that provide different types of visible through long wavelength infrared (visible - LWIR) optics-related calculations for ballistic missile defense (BMD) applications. The code has various program options, one of which is referred to as BASIC. This option links multiple programs to calculate hardbody trajectories, temperatures, mass ablation rates, optical signatures, optical effects with various atmospheric backgrounds, and sensor performance parameters.

Figure 1-1 illustrates the BASIC option program structure.

Figure 1-1: General OSC Program Flow

This manual discusses DIPREF (Direct Procedure for ReEntry Flow), which is available as the primary endoatmospheric hardbody flowfield model. It is linked into RVTEMP, the endoatmospheric thermal response code shown in Figure 1-2. DIPREF calculates the flowfields and heat fluxes of axisymmetric targets. It was originally focused on mono-, bi-, tri-, quadraconic, and flared targets, but it is now capable of modeling targets with any number of conic frusta. The actual limit is determined by the users’ parameter settings (in the OSC

Backgrounds Low/High Altitude (GFE) Zodiacal Light/Stellar

LOW5/6

EARLIM

RVSNTH TSHA

D 2D Target Model 3D Target Model

BASIK

Executive

ATMOS

Environment

• Sun Position

• Atmosphere

• Earthscene

BALLIS Trajectory

3 DOF, 6 DOF

DIPREF

F678

Ablation, Endo Thermal Response

EXOHT 3 EXOHEAT

1D 3D

Exo Thermal Response

Materials

Databases

SELECT BIDIRC

TPV

Pixel 8, SAO Image

Sort Optical Properties Database

SENSORM

SISMOD

Target Signatures In-band, Spectral Reflection: Lambertian Specular Bidirectional

SPACE

Target Phenomenology Visualizer Resolved Signature Images

Signal-to-Noise

PRIME

Endo Flow

Fields

EZFLOW

SHINE Sun, Earth, Albedo

MBALL

~2D

1-2 modules file, the OSCPref file, or the OPTISIG DIPARM file) and available computer memory.

Aerodynamic fins of conventional planform or grid geometry can also be represented.

Figure 1-2: RVTEMP/DIPREF Program Flow

The flowfield is calculated by a technique in which the program calculates its own grid. The program is self-starting and computes the flowfield around flares and flaps that is only approximated by PRIME (the older endo-environment model). DIPREF generates heat flux values for all speeds of interest from ascent through impact and is used in place of PRIME. It has been interfaced with RVTEMP/F678 (the ablation thermal response model), RVSNTH

(the shape input synthesizer in the BASIC option), EXOHEAT (an exo thermal response model), and SHINE (the endo earthshine/albedo model).

DIPREF has been validated against wind tunnel experiments and recent flight test measurements. DIPREF currently has 0 to 360° angle of attack capability and multiple fins capability. DIPREF is easy to use, as it receives most of its target specification from OSC’s data stream. This manual proceeds as follows: Section 2 describes DIPREF structure;

Section 3 describes the algorithms; Section 4 contains the input specification; Section 5 illustrates some comparisons and validations; and Section 6 describes diagnostics and troubleshooting.

2-1

2. DIPREF Description

DIPREF is currently available as part of the OSC BASIC Option when an endoatmospheric thermal response calculation is required. As shown in Figure 1-2, it is located within RVTEMP and is the preferred alternative to the older PRIME flowfield model. It has been linked with all applicable parts of the BASIC Option:

RVSNTH (shape synthesizer)

EXOHEAT (exo-endo thermal response)

SHINE (endo earthshine/albedo)

RVTEMP/F678 (ablation thermal response)

2.1 DIPREF Overview

The typical RV has a blunt shape to reduce heat transfer around the stagnation region. The shock wave becomes detached from the body, and the flow varies from subsonic to hypersonic Mach numbers over the vehicle. DIPREF solves for the subsonic region and computes the supersonic flow over the rest of the body in subroutine SUPER. The flowfield information is used to compute the heat transfer to the body surface in subroutine HEAT.

Entropy correction to heat transfer is computed in subroutine EDGE. Subroutine LOWDEN applies the correction in the high-endo region due to rarefied gas flow effects. Subroutine

FLARE accounts for the viscous interaction effects at compression corners generated by flares and flaps for attached and detached as well as laminar and turbulent flow conditions.

The results of DIPREF are then passed to RVTEMP/F678 for calculating the ablation effects.

When there is an angle of attack larger than one degree, subroutine INC and the associated subroutines are called to generate the aerothermal environment around the circumference.

DIPREF computes detailed solutions for axisymmetric bodies with, or without fins. A new option, OSC’s EZFLOW, is a hybrid of methods from DIPREF and some approximate solutions. EZFLOW is designed to compute aeroheating on more complex, non-axisymmetric shapes. A subset of EZFLOW, called EZBALL, is invoked for efficient calculation of endoatmospheric heating of spherical objects. Wake flowfields are not computed in DIPREF

– a separate OSC module, WAKEPRO, performs wake flow, chemistry, heat transfer, and signature calculations. Details of WAKEPRO appear in the OSC Basic Option Reference

Manual [1].

2-2

2.2 Input/Output Interfaces with Other OSC Models

In order for DIPREF to calculate the flowfield around a target, it must have:

Target shape (nose radius, cone angles, and lengths)

Trajectory data (altitude, velocity, roll angle, and angle of attack)

Atmospheric conditions (pressure, temperature, and density at the given altitude)

To communicate with RVTEMP/F678 and BIDIRC (signatures), it must also have:

Running lengths/station locations for thermal response histories and signatures

Earthshine, sunshine, and albedo fluxes for each thermal station

Table 2-1 lists the files containing each set of data and the DIPREF routine that obtains it or passes it to RVTEMP/F678.

Table 2-1: DIPREF I/O Summary

DIPREF I/O Summary

Parameters File Unit Number Read/Write Location

Target Shape 8 DOSHAP

Trajectory Data 7 DIPREF

Atmospheric Conditions (Module or Common block) INITAL

Running Lengths 8 DOSHAP

Earth Sun Albedo 23 POINTS

Output for F678 10 POINTS

Reformat Output 20 WRAPUP

Target shape data are either input by the user or sent from RVSNTH. Trajectory data are available from BASIC Option trajectory card input or from BALLIS calculations (see Table 2-2 for the file format). Atmospheric pressure, temperature, and density are obtained via subroutine AIR, which contains a table of atmospheric conditions that was set up by ATMOS.

The DIPREF subroutine POINTS interpolates the DIPREF-generated grid to obtain the values of the parameters required by RVTEMP/F678 and reads SHINE-generated earthshine, sunshine, and albedo fluxes and writes them to Unit 10. Subroutine WRAPUP reformats the data on Unit 10 and writes it to Unit 20 for use in F678. Table 2-3 illustrates the format of

Unit 20.

2-3

Table 2-2: Trajectory File (TRAJ, Unformatted, Unit 7; ATRAJ, Unit 1)

Trajectory File (TRAJ, Unformatted, Unit 7; ATRAJ, Unit 1)

Name Units Description

Record 1

NTRAJ ** Number of trajectory points

REQTR ** meters Equatorial earth radius

RPOLAR ** meters Polar earth radius

IPTEST ** Trajectory signature Δt change flag

TSTART ** seconds Time on trajectory when Δt = H

NOBAND* Number of wavebands (NUMBND maximum)

LWAVE* Number of spectral wavelengths (NMWAVE maximum)

(LLBAND(I),

ULBAND(I))*

microns Lower and upper limit for each waveband

WAVES(I)* microns Spectral wavelengths for each waveband

Records 2 to NTRAJ + 1

TIME ** seconds Time

ALT ** kilofeet Target altitude

ELV ** degrees Elevation angle (with respect to the sensor)

RNG ** kilofeet Range between target and sensor

VEL ** feet/second Target velocity

ATT ** degrees Target angle of attack (= angle between velocity and target axis of symmetry)

ROLL ** degrees Target roll angle facing wind

T(K),K=1,3** meters Target ECR coordinates

O(K),K=1,3** meters Sensor ECR coordinates

E(K),K=1,3** degrees ECR yaw, pitch, and roll

AZIM ** degrees from North Sensor azimuth

ATRU ** degrees True aspect angle (with respect to sensor, = angle between sensor and target axis of symmetry)

AVEL ** degrees Velocity aspect angle (with respect to the sensor. AVEL = ATRU for 3 DOF trajectories)

CDA ** feet Drag area

AMASS(K),K=1,8* pound/second Elemental ablation rates for H, O, C, N, Si, Inert, Element 1, and Element 2 from F678

VDEF* Velocity defect ΔV/V

OBALT* kilofeet Sensor altitude

SIGNA* Watts/steradian Radiant intensity in each waveband

TRANS* Dummy

BCOLOR* Kelvin Color temperature between each waveband pair ((I,J),I=1,NOBAND, J=I+1,NOBAND)

BEMISA* meter Emissivity area between each waveband pair (see BCOLOR)

T1* Watts/steradian/ micron

Spectral signatures

*Added in BIDIRC when WAKMOD = 1.

**Required for the ASCII file ATRAJ. Altitude, velocity, range, and CDA will be in the corresponding metric units if metric units were flagged using TRAJ=ASCIIM instead of

TRAJ=ASCII.

2-4

Table 2-3: File for Communication between DIPREF and F678 (FORF678, Unit 20)

File for Communication between DIPREF and F678 (FORF678, Unit 20)

Name Units Description

Variables Listed as in F678

The following records are written for each station (stagnation point and running lengths)

Record 1

AA (1) = 0 (integer)

AA (2) Case identification (when roll angle body ranges are used, they are added to this number, Real or Double Precision.)

AA (3) Station number (stagnation point = 1, integer)

NV Number of trajectory points

V(J), J=1, NV Velocity defect for each trajectory point

NTOT Number of running lengths

AREA (K), K=1, NTOT meters Surface area associated with each running length

The following records are written for each trajectory point + 1. (The last record contains a negative time to end the data read.)

Records 2 to NTRAJ + 2

TI seconds Time

HSUBRX Btu/pound Recovery enthalpy (function of time)

PDELTX pounds/feet Local pressure (function of time)

ENCZRO Btu/feet /second/ degrees Rankine Heat transfer coefficient (function of time)

VISX pounds·seconds/feet Local viscosity

PFXX pounds/feet Free stream pressure

VELX feet/second Local velocity (function of time)

RDXX pounds/feet Local density

DEFX pounds/feet Free-stream density

ATITX feet Altitude (function of time)

VEFSX feet/second Velocity function of time)

QUNDER Btu/feet /second Heat flux underneath station, used with curved bases (function of time)

TINFO degrees Rankine Free stream temperature, used with curved bases (function of time)

SSHINE Btu/feet /second Sunshine and albedo (function of time)

ESHINE Btu/feet /second Earthshine (function of time)

RETTB 0.0 Laminar Flow

1.0 Turbulent Flow

ABSPT Absorptivity

3-1

3. DIPREF Theory

Endoatmospheric heating in the OSC is currently calculated using the RVTEMP module. The development and expansion of the DIPREF program and its associated upgrades has been motivated by the need for a more versatile endo-model to simulate certain cases of interest, including complex missile and reentry vehicle shapes and vehicles at large angles of attack.

A computer program known as DIPGAF, developed by NASA Langley Research Center [2] was the starting point for DIPREF. DIPGAF was based on the thin-shock layer approximation and generated the detailed flowfield over blunt axisymmetric bodies at zero angle of attack. The flowfield and shock shape generated by DIPGAF compare within a few percent error with more exact methods and take less time than exact methods. DIPREF uses the same core method of computation to generate the flowfield and heat transfer distribution over multiconic axisymmetric RVs, but DIPREF has grown significantly from its original roots.

Since its inception, DIPREF has been upgraded and expanded beyond the initial capabilities of DIPGAF. DIPGAF was limited to supersonic and hypersonic flight at zero angle of attack with targets consisting of up to four conic frusta. DIPREF’s capabilities have been enlarged and extended as part of OSC’s development. DIPREF now computes subsonic, transonic, supersonic, and hypersonic aeroheating at all angles of attack and in all flight regimes from continuum through rarified and free-molecular flow. The code is self-starting and requires minimal user input. (For example, it builds its own computational grids and determines calculation methods.)

3.1 Flowfield Analysis

In this subsection, the theoretical formulation and method of flowfield calculation are presented.

The coordinate system used is a shock-oriented orthogonal curvilinear system with the x-coordinate measured along the shock and y-coordinate measured along the inward normal toward the body shown in Figure 3-1. The equations of motion for inviscid flow over blunt axisymmetric bodies traveling at hypersonic speeds with thin shock layers are given by the following:

y vr x ur

(3-1) u x v u y uv

R p ys

(3-2)

3-2 x v v y u

R ys

2 1

(3-3) s x v

(3-4)

Figure 3-1: Sketch of Coordinate System from Reference 2

These equations are transformed to streamline coordinates (x,) using the Von Mises transformation, and the resulting equations are as follows:

u

R y

R r s s

(3-5)

H h u vt 2 2

(3-6)

S S (3-7)

For a thin shock layer, y/Rs << 1, and, since the streamlines are nearly parallel to the shock wave, Maslen [3] suggested neglecting v/x in Equation (3-5 and replacing u and r by us and rs, respectively. Integrating Equation 3-5 from the shock to the body gives

3-3 p p r R s s s s s

(3-8)

Equation (3-8 was improved by Maslen for axisymmetric flow by including an estimate of the v/x term as follows:

r R v r R r s s s s s s s s s tan cos1

2 2

. (3-9)

The normal velocity component, v, is calculated from the following equation:

v v yt y

R

1 1.

. (3-10)

The flowfield geometry gives r r ys s cos

, (3-11) and, using the definition of the stream function, the y-coordinate of a point in the shock layer is obtained from the following equation:

r r u s s s cos cos d

1 1

. (3-12)

The distance measured along the axis of symmetry from the shock wave is given by:

Z Z ys s sin

. (3-13)

These equations are solved iteratively to obtain the flowfield, subject to the boundary conditions of v = 0 on the body and Rankine-Hugoniot conditions immediately behind the shock.

The flowfield is divided into a subsonic region and a supersonic region, and the method of solution is different for each region. In the subsonic region, the iterative procedure is started by representing the shock wave shape by a Taylor series approximation given by:

R x R s st st

2 (3-14)

3-4 and the calculation is continued up to a point on the shock where the Mach number is 1.3 for perfect gas flow and 1.8 for real equilibrium air flow. The local flow properties and geometry are calculated using Equations (3-9 through (3-13. The computed body is scaled to the desired body, and the scaling factor is used to calculate a new shock shape. This process is repeated until the computed and desired body agree within a specified error of

10-4. The entire subsonic region is calculated in this process, and the converged results are stored and printed if desired. The shock and body data are available at this stage at six equally spaced shock arc lengths, and the flow properties and grid geometry are available at

15 points along each ray normal to the shock at each of the six shock arc lengths.

To proceed into the supersonic region, a small increment for the last converged values of the shock radius of curvature and shock angle is given, and the resulting shock geometry and flowfield properties are calculated. An error value is computed for the body radius, and the shock radius of curvature is adjusted until satisfactory convergence is obtained.

Calculations continue until the entire body is generated. The converged supersonic results and the grid generated are available for subsequent calculations of heat transfer distribution and viscous interaction effects at compression corners.

Equilibrium-air thermodynamic correlations developed by Zoby [4] are used in the flowfield computation for free-stream velocities from 1.829 to 9.15 km/sec (6,000 to 30,000 ft/sec) and for free-stream pressures from 2.0 x 10-5 to 0.1 atm., or altitudes from 250 to 20 kft.

The correlations for enthalpy and temperature are given by:

h

RT

C p p o h o

0 97

0 98

(3-15) or

T

C p p o

T

0 7

0 68

. (3-16)

The parameters Ch and CT are given by:

C a Uh i

(3-17)(a) and

3-5

C b UT i i*

. (b)

The coefficients ai and bi are incorporated into the program and are not listed here. Uh and

UT are given by:

U A h s sin .4384

.4384

U

U T s sin

0 3048

A exp ln

2 10

15 0

C C BCT T * exp

B ln

2 10

50 0

C

U UT T

12 0

12 0

20 0

18 0

0 1

By combining the shock-wave relations with the enthalpy correlation equation, the following relation is obtained for the density ratio, , across the shock wave:

p M

M h m n s m n

2 2

2 2 2

1 1 sin

. (3-18)

The equation is solved iteratively for the density ratio, and the pressure and enthalpy behind the shock are computed from the following equations:

Ms

1 12 2 sin

(3-19)

Ms

1 2 2 2

. (3-20)

3-6

3.2 Heat Transfer Model

Stagnation point heat transfer rate is modeled using the experimental correlation given by

Detra, Kemp, and Riddell [5]:

q R u H H

H H s

N SL

s w

17 600

26 000

0 5 3 15

Btu ft sec2

. (3-21)

The heat transfer distribution around the spherical nosecap is computed using Eckert’s method [6] as follows:

ds q q u r H H du ds u r H H s e e e o aw w e e s e e e o

0 . (3-22)

The convective heat transfer over frusta away from the stagnation region is modeled using the relations developed by Zoby, Moss, and Sutton [7]. Both the laminar and turbulent heating relations make use of skin friction relations based on momentum thickness

Reynolds number and a form of Reynolds analogy to relate skin friction and heat transfer, and the compressibility effects are accounted for by Eckert’s reference enthalpy method.

The resulting laminar heat transfer rate equation is as follows:

.q R u P H Hw L e e e e r aw wL w

0 22

0 6

(3-23)

L e s e e u r

0 644 ds* *

. (3-24)

The turbulent heat transfer rate is computed using the following relation:

q C R u P H Hw T j e e j e e r aw wT w, 0 4

(3-25) where

N R Re e 12 67 6 5 121

. . log . log, , 3-7 j

N

C C N

N N j j

1 5 1 2

C j C2 11

C j3 1

C C4 3

(3-26)

C N5 2 24333 0 93 . .

j C s C e e

C u r

* * ds

3.2.1 Laminar, Transitional and Turbulent Flow Regions

The transition from laminar to turbulent flow is influenced by several factors. Test data indicates a trend of increasing transition Reynolds number with rising Mach number, and the boundary layer edge Mach number and Reynolds number are generally recognized as being important parameters. The default transition is based on the following empirical correlation between momentum thickness Reynolds number and edge Mach number:

R Me T e 275 0134exp .

. (3-27)

User inputs for LRNCOF (default = 275) and LRNEXP (default = 0.134) for ascent and reentry allow users to adjust the intercept and slope of the correlation. Because of the uncertainty of transition, users have the option to control transition based on altitude by setting the flag ITURB = 1 or the local Reynolds number using the flag KSAND = 0. Several other transition methods/criteria are discussed in the OSC Basic Option Reference Manual

[1].

In the transition region, the heat transfer rate distribution is computed from the weighting function of Dhawan and Narasimha [8] as follows:

, , ,q q w q qtr w L f w T w L (3-28) w zf tr 1 0 2exp .412

(3-29)

3-8 z t t t t tr beg end beg

(3-30) and t stands for local momentum thickness Reynolds number for LTRAN = 1 or for local

Reynolds number for LTRAN = 2.

If the boundary layer transition occurs near the stagnation point, Van Driest [9] pointed out the possibility of the maximum turbulent heat transfer rate exceeding the stagnation point heat transfer rate. A modification of the Rose, Probstein, and Adams [10] theory is used to compute the turbulent heat transfer rate:

q P

D u x

D u c T T r e e e e p r w

0 042

0 8 0 8 0 2 0 6

(3-31)

The ratio of turbulent sonic-point to stagnation point heat flux varies from 1.0 to 10 as the

Reynolds number based on nose diameter and conditions behind normal shock wave increases from 105 to 109, as demonstrated by Arthur and Williams [11][12].

3.3 Variable Entropy Correction

The effect of variable entropy conditions at the edge of the boundary layer on the heat transfer to the body is approximated by using local inviscid flow properties located one boundary layer thickness away from the body surface. This method was first suggested by

Ferri [13] and was used by others [14][15]. The boundary layer thickness is calculated from the following relations for laminar and turbulent flow:

L L 5 55. (3-32) w aw r u h

TN

N

N

H

H p w

1 0

1 0 1 0 1 29 0 333

. (3-33)

The following properties at the edge of the boundary layer are calculated using the following geometric relations from Figure 3-2.

x s b sin

(3-34) r rw b

(3-35) z zw b

(3-36)

3-9 y r rs b s cos

. (3-37)

Figure 3-2: Sketch of Geometry for Calculating Boundary-Layer Edge Properties

The stream function at the boundary layer edge is given by:

w w w bu r y y

. (3-38)

The flow properties are interpolated corresponding to the stream function and these values are used in the next iteration to compute corrected flow values at the edge of the boundary layer. The convergence of this procedure is fast, and the final values are obtained in two or three iterations. The converged flow properties have the variable entropy effects incorporated into them, and these values are utilized to compute the entropy corrected heat transfer rate to the body surface.

3.4 High-Altitude Correction

The modification to the continuum heat transfer rates resulting from the rarefied flow at high altitudes is based on a semi-empirical correlation developed by Nomura [16]. The heat transfer rate in the noncontinuum regimes is computed by the following relation:

3-10 q

Kn c q

Kn c

Tr cont FM

(3-39) where the empirical constant, c, is 3.0 and Kn is the Knudsen number based on the boundary layer thickness and is given by:

Kn M

Re

. (3-40)

In the free molecular regime, the local heat transfer rate to a surface element inclined at an angle to the flow is given by the following equation from Oppenheim [17]:

q a

T

T e s s e e w

2 2

2 2 1

2 2 sin sinsin erf sin (3-41) where ae is the accommodation coefficient, is the fraction of specularly reflecting molecules assumed zero for fully diffuse reflection, erf is the error function and s is the speed ratio given by s u (3-42) with

2RT

. (3-43)

The accommodation coefficient can be varied as a function of altitude or held at a fixed value between 0 and 1.0 thereby simulating incomplete thermal accommodation as a function of altitude.

In the noncontinuum regimes, the surface pressure is corrected for low density effects by the correlation developed by Blanchard and Buck [18] from Space Shuttle flight measurements. The correlation is:

p Zc

FM e c

(3-44)

3-11

Z A B Kn exp log

A 0 2553. B 1073. C 185. (3-45)

Kn M

1.491

For Kn 10, representing free molecular flow, Z is set equal to 1.0, and for Kn 0.001, corresponding to continuum flow, Z is set equal to 0.0.

OSC’s exoatmospheric heat transfer models include built-in calculations for free-molecular heating and the models will bridge from free-molecular to continuum heating using approximate relations. This capability, along with the related EZFLOW option, allows users to fly complex, non-axisymmetric shapes on trajectories that result in aeroheating of the target.

These non-DIPREF options are discussed in the OSC Volume 1 Basic Option Reference

Manual [1].

3.5 Base Heating

The base heating calculation is limited to flat base geometry and is based on the correlation developed by Bulmer [19] from flight tests. Base pressure is calculated using the following equation for laminar flows:

M

M H

H

R u

B c c c w e c B c

0 51

1 0 2 1 4 02 1 5

1 5

0 4

(3-46) and, for turbulent flow, empirical values are used for the base pressure. The flow is expanded from the pressure, pc, to the base pressure, pB, by Prandtl-Meyer relations, and the base flow properties are determined. The heat transfer rate over the base is calculated from the following relation:

q h u u

Re

Re H HBase c

B B c c

B r

Br w

35 5

1 246

(3-47) where hc is the enthalpy-based heat transfer coefficient on the cone end point, and HBr is the base recovery enthalpy. ReB,r represents the Reynolds number at base property values at a distance, r, from the edge of the cone, as shown in Figure 3-3. The heat transfer rate increases with r and reaches a maximum at the center.

3-12

Figure 3-3: Sketch of Base Geometry from Reference 13

3.6 Viscous Interaction Effects at Flares and Flaps

Flaps and flares on RVs cause an adverse pressure gradient that results usually in boundary layer separation. The occurrence and extent of separation influence inviscid pressure distribution and heat transfer. If separation occurs, the streamlines in the external flowfield will be deflected and the flow geometry is changed such that the supersonic flow will undergo two stages of shock wave compression, which are referred to as separation shock and reattachment shock, as shown in Figure 3-4. The nature of the pressure rise and local flow is shown in Figure 3-5 for laminar and turbulent interaction. It is seen that a pressure plateau exists at the corner, an inflection point in the pressure distribution occurs at both the separation and reattachment points, and there is an upstream influence associated with the interaction. From a thermodynamic point of view, the interaction region is characterized by a reduction of local heat transfer rate at the reattachment. The model in the DIPREF program makes the following determinations:

If flow separation occurs

The geometry of the separated region

The upstream influence and pressure distribution

The heat transfer distribution for laminar and turbulent flow.

3-13

Figure 3-4: Flow Separation and Formation of Separation Shock, Reattachment, and Dividing Streamline in Viscous Corner Flow

Figure 3-5: Wall Pressure Distributions for Laminar and Turbulent Flow for

Separated Corner Flow

The correlations developed by Popinski [20] are used in modeling the pressure coefficients

3-14 and heat transfer coefficients. The separation pressure coefficients for laminar and turbulent flow are given by the following relations:

287.0225.0 1

91.0 oo

Lamps

MRe

(3-48)

Re M ps Turb o o

10 1 2 0 3

(3-49) where Mo is the Mach number at the hinge from inviscid computations, and Reo is the

Reynolds number based on distance along the surface to the hinge line.

The plateau pressure coefficients in laminar and turbulent flow are given by:

pp Lam

2 0

10 25 2 0 373

(3-50) pp Turb

10 1 2 0 205

(3-51)

Using the appropriate plateau pressure, the deflection angle of the dividing streamline is calculated from oblique shock relations. The plateau pressure and flow parameters behind the separation shock are used as upstream conditions to the reattachment shock for an angle equal to the difference between the flare/flap angle and the deflection angle of the dividing streamline. The reattachment pressure is calculated from the following correlation due to Nash:

p p N p pr p f p

(3-52) where N = 0.5, and pf is the inviscid pressure on flare or flap.

The peak heat transfer coefficients at reattachment for laminar and turbulent interaction are given by the following relations:

h Re M pk o Lam p p o o f

5 0 106 0 88

0 5 3 7

(3-53) h Re pk o Turb p p o f

29 5 0 709

0 2

(3-54)

3-15 where ho is the heat transfer coefficient without interaction at the hinge time.

The location of the start of interaction, xo, in turbulent flow is determined from the following equation from Roshko and Thompke [21]:

o f f oc c x * 1 0

(3-55)

1000 0

18 29

2 81

B c Bf

* . . 0 001 1 0 001189 2 o o ox Re 0 37 0 2. .

(3-56)

B = flare angle in degrees

1 = length from stagnation point to the hinge line.

The distance, 1, from the beginning of interaction to the point of separation is correlated by the following relation by Erdos and Pallone [22]:

fo o f c M *

0 5

4 2

0 5

(3-57) where f4 = 3.43 for laminar flow

= 1.4 for turbulent flow

Re fo

0 0574

0 37

0 2

2 584

. log , . 10 10

10 10

7 5

9 7 o o

(3-58)

* . 0125 o .

Using xo and i, the length of separation is calculated using the relation:

3-16

XLS xo i 1 (3-59)

Flow separation is modeled if XLS is positive, or inviscid flare pressure, pf, is greater than the plateau pressure. Otherwise, the flow is treated as attached.

The pressure and heat transfer distributions in the interaction region are modeled using the procedure of Elpstrom [23]. Distributions beyond their peaks in the reattachment region are modeled using the measurements of Coleman and Stollery [24].

3.6.1 Plume/Body Interaction

A relatively new DIPREF feature is the ability to model upstream body flow separation due to the downstream presence of an expanded plume. There are two methods. The internal/native model treats the plume shock front as a virtual frustum that has the potential to create a virtual compression corner on the body. DIPREF estimates the plume shape using the Jarvinen-Hill correlation, and the flow calculations follow the procedure outlined above for compression corners. The model is strictly valid for only zero angle of attack. OSC 21 included an option to input the induced-separation flow properties from an external flow field file, and this capability was expanded in OSC 21.1.

3.7 Subsonic and Low Supersonic Region

In the subsonic compressible region, potential flow theory is used to calculate the pressure distribution on the axisymmetric multiconic missile body. As the flow progresses, the program checks for the region where the critical Mach number is reached and accounts for the transonic flow from that point over the body. In the case of fins operating in the subsonic region, the pressure coefficient is adjusted using the Karman-Tsien rule given by:

M c p p p incomp incomp

2 1 (3-60) where

1 2 2M .

In the supersonic region, conical pressure fields are modeled using the Linnex-Bailey correlations, and Griffith and Lewis [25] measurements. The heat transfer rates are calculated using the following relations for laminar and turbulent flow given by Eckert [26].

Laminar flow: The heat transfer coefficient is calculated using the following relation:

3-17 h P r

0 332

3. *

(3-61) where the starred quantities are evaluated at the Eckert’s reference enthalpy given by:

H H H H H He w e aw e

* . . 0 5 0 22 (3-62) and the factor is the Mangler transformation factor equal to 3 for conical surfaces and 1 for plates and cylinders.

Turbulent flow: The heat transfer coefficient for turbulent flow is calculated using the following relations:

P u u P Re x x r e e r

0 0296

0 185

0 2

2 584

. log , 5 10 10

5 7

. (3-63)

The starred quantities are evaluated at the Eckert’s reference enthalpy.

3.8 Fins

Swept leading edge heat transfer rate for laminar flow conditions is calculated using the following relation:

. cos q qle L eff 0 75 1 1

(3-64) where sin cos sin eff , is the sweep angle, is the angle of attack, and qL is the heat transfer rate for a sphere with a radius equal to the leading edge radius. For turbulent boundary layer along the leading edge, a modified form of the equation developed by

Beckwith and Gallagher [27] is used to calculate the heat transfer rate:

q h T Tt t r w

(3-65) with ht given by the relation:

3-18 h D

K P u D D u t r

0 031 0 333

0 8

0 6

0 8 0 2

. sin cos cos

. (3-66)

Typical cross-sectional shapes or profiles found on missiles are modeled using the unified supersonic-hypersonic theory and conical wing theory for configurations with subsonic and supersonic leading edge flow conditions.

Body-fin interaction effects on fin heat transfer rate are modeled based on the flight measurements of Coleman and Lemmon [28]. High altitude noncontinuum effects are included for leading edge and fin body based on their respective Knudsen numbers.

The flow properties existing at the location of fins on the body are used as free stream conditions for heat transfer calculations on fins. To capture the body-fin interaction effects on convective heat transfer to fins, the first chordwise division is modeled using the thickness of the body boundary layer, which varies with the altitude and free-stream velocity.

Laminar-turbulent transition on fins is based on the local Reynolds number, which can be adjusted by the users.

3.9 Angle of Attack Effects

The prediction of heat transfer rates over blunt multiconic bodies at angle of attack is a complex problem. There are four major effects on local heat transfer and skin friction due to angle of attack. The first is the streamlines development as they wrap around the body. The second is the variation of the edge properties and local pressure with angle of attack including the effects of entropy gradient. The third is the divergence effect accounted for by the metric factor due to the geometric spreading of the flow. The fourth is the change in the velocity and temperature profiles, the local pressure variation associated with the three-dimensional flow separation caused by the local crossflow within the boundary layer.

A simple solution to the three-dimensional boundary layer problem may be achieved through the axisymmetric analogue approximation. The three-dimensional boundary layer equations are written in streamline coordinates, and the cross-flow velocity is assumed to be negligible. The resulting equations are identical to the axisymmetric equations at zero angle of attack if the distance along the streamline is interpreted as the wetted length, and the metric representing the streamline spreading is set equal to the body radius. This procedure would still require the streamline paths and the pressure distribution around the body. The procedure developed by De Jarnette and Hamilton [29] is used to predict the circumferential distribution of heat transfer making use of the pressure distributions developed by Amick

[30] and Zakkay [31] for large angles of attack. If this procedure fails at large angles of attack or for multiple frusta bodies, a modified form of Lee’s theory is used to calculate the heat transfer distribution. The transition from laminar to turbulent flow for bodies at angle of

3-19 attack is modeled the same as for zero angle of attack. The transitional flow is not modeled, and the input variable LTRAN is not active at present, when there is an angle of attack. Heat transfer to cylinders at angle of attack is modeled using the empirical formulation of

Johnson [32] given by:

. . . sin .

.q q Rees D

0 75 0 01714 0 01235 3 53 100 3

(3-67) where qes is the stagnation point heat flux for a sphere in the free stream with a radius equal to that of the cylinder. The coefficients and the trigonometric function with brackets represent the best curve fit to available experimental data on cylinders. High altitude effects are modeled similarly to those at zero angle of attack.

The angle of attack calculations are carried over to the base heating producing an asymmetric distribution. When the angle of attack exceeds 90, the base becomes the front part and the node region becomes the aft part. The program calculates the base and the nose heat transfer distributions and maps them appropriately. When the angle of attack exceeds 180, the windward and leeward rays are interchanged and the program makes the appropriate transformations to the roll rays around the body.

The expression used for calculating the laminar heat transfer rate is given by Lee’s theory:

q rR r G w stag p p u u o p p u u s e o e e

(3-68)

G M M

2 2

. (3-69)

For turbulent flow the heating rate along the streamline is calculated using the following relation:

q P H H u v h u v h w r e w e e e r o e e r

0 0296

1 05 0 8 0 05

0 6

. (3-70)

3-20

3.10 Validation/Comparison

The most comprehensive validation of DIPREF has been accomplished through comparisons to flight test…