NASA-96-cr4749

Computation of Thermally Perfect Propertiesof Oblique Shock Waves

Kenneth E. TatumLockheed Martin Engineering & Sciences l Hampton, Virginia

NASA Contractor Report 4749

National Aeronautics and Space AdministrationLangley Research Center l Hampton, Virginia 23681-0001

August 1996

Computation of Thermally Perfect Properties of Oblique Shock Waves

1

Summary

A set of compressible flow relations describ-ing flow properties across oblique shock waves,derived for a thermally perfect, caloricallyimperfect gas, is applied within the existingthermally perfect gas (TPG) computer code. Therelations are based upon a value of cp expressedas a polynomial function of temperature. Theupdated code produces tables of compressibleflow properties of oblique shock waves, as well asthe original properties of normal shock wavesand basic isentropic flow, in a format similar tothe tables for normal shock waves found inNACA Rep. 1135. The code results are validatedin both the calorically perfect and the caloricallyimperfect, thermally perfect temperatureregimes through comparisons with the theoreti-cal methods of NACA Rep. 1135, and with astate-of-the-art computational fluid dynamicscode. The advantages of the TPG code foroblique shock wave calculations, as well as forthe properties of isentropic flow and normalshock waves, are its ease of use, and its applica-bility to any type of gas (monatomic, diatomic,triatomic, polyatomic, or any specified mixturethereof).

The TPG code computes properties ofoblique shock waves for given shock wave angles,or for given flow deflection (wedge) angles,including both strong and weak shock waves,including the special case of a normal shock.The code also allows calculation of a sequence ofshocks where each successive shock wave isdependent upon the flow conditions from thepreceding one. Both tabular output and outputfor plotting and post-processing are available.The user effort required is minimal, consisting ofsimple answers to 14 interactive questions (orless) per table generated.

Introduction

The traditional computation of one-dimen-sional (1-D) isentropic compressible flow proper-ties, and properties across normal and obliqueshock waves, has been performed with calori-cally perfect gas equations such as those foundin NACA Rep. 1135 (ref. 1). When the gas of

interest is air and all shock waves present arenormal to the flow direction, the tables of com-pressible flow values provided in NACA Rep.1135 are often used. For shock waves in air thatare oblique to the flow direction, NACA Rep.1135 contains charts of certain flow properties.These tables and charts were generated from thecalorically perfect gas equations with a value of1.40 for the ratio of specific heats . The applica-tion of these equations, tables, and charts is lim-ited to that range of temperatures for which thecalorically perfect gas assumption is valid. How-ever, many aeronautical engineering calcula-tions extend beyond the temperature limits ofthe calorically perfect gas assumption, and theapplication of the tables or equations of NACARep. 1135 can result in significant errors. Theseerrors can be greatly reduced by the assumptionof a thermally perfect, calorically imperfect gasin the development of the compressible flow rela-tions. (For simplicity within this paper, the termthermally perfect will be used to denote a ther-mally perfect, calorically imperfect gas.) Previ-ous papers (references 2 and 3) described acomputer code, and the underlying mathemati-cal formulation, which implements 1-D isen-tropic compressible flow and normal shock waverelations derived for a thermally perfect gas.The current paper presents an enhanced com-puter code, and the corresponding mathematicalderivation, for the computation of the obliqueshock wave relations based upon the assumptionof a thermally perfect gas.

A calorically perfect gas is by definition agas for which the values of specific heat at con-stant pressure cp and specific heat at constantvolume cv are constants. Reference 1, as well asmany compressible flow textbooks, derive andsummarize calorically perfect compressible flowrelations based upon this definition. The accu-racy of these equations is only as good as theassumption of a constant cp (and therefore a con-stant ). For any non-monatomic gas, the valueof cp actually varies with temperature and canbe approximated as a constant for only a rela-tively narrow temperature range. As the tem-perature increases, the cp value begins toincrease appreciably due to the excitation of the


2

vibrational energy of the molecules. For air, thisphenomenon begins around 450 to 500 K. Thevariation of cp with temperature (and only withtemperature) continues up to temperatures atwhich dissociation begins to occur, approxi-mately 1500 K for air. Thus, air is thermallyperfect over the range of 450 to 1500 K andapplication of the calorically perfect relationscan result in substantial errors. A similar rangecan be defined for other gases over which the gasis calorically imperfect, but still thermally per-fect. At yet higher temperatures cp becomes afunction of both temperature and pressure, andthe gas is no longer considered thermally per-fect.

NACA Rep. 1135 does present one methodfor computation of the 1-D compressible flowproperties of a thermally perfect gas. In thisapproach, the variation of heat capacity due tothe contribution from the vibrational energymode of the molecule is determined from quan-tum mechanical considerations by the assump-tion of a simple harmonic vibrator model of adiatomic molecule (ref. 4). With this assumptionthe vibrational contribution to the heat capacityof a diatomic gas takes the form of an exponen-tial equation in terms of static temperature anda single constant . The complete set of ther-mally perfect compressible flow relations basedupon this model is listed in NACA Rep. 1135Imperfect Gas Effects. Tables of these ther-mally perfect gas properties are not providedbecause each value of total temperature Ttwould yield a unique table of gas properties.Instead, NACA Rep. 1135 provides charts of the1-D thermally perfect air isentropic and normalshock properties normalized by the caloricallyperfect air values and plotted versus Mach num-ber for select values of total temperature. Foroblique shock waves, an even more limited set ofcharts is presented for shock wave angle ,downstream Mach number M2, and pressurecoefficient as functions of deflection angle forfour static temperatures at two total tempera-tures. This approach can only be applied todiatomic gases (e.g., N2, O2, and H2) because thefunctional form which describes the variation ofheat capacity with temperature accounts onlyfor the harmonic contribution.

Because the imperfect gas method of NACARep. 1135 is applicable only to diatomic gases, adifferent method of computing the 1-D isentropicand normal shock flow properties of a thermallyperfect gas was developed and described in refer-ences 2 and 3. The current paper extends thatmethodology to the computation of flow proper-ties across oblique shock waves. The method uti-lizes a polynomial curve fit of cp versustemperature to describe the variation of heatcapacity for a gas. The data required to generatethis curve fit for a given gas can be found in tab-ulated form in published sources such as theNBS Tables of Thermal Properties of Gases(ref. 5) and the JANAF Thermochemical Tables(ref. 6). Actual coefficients for specific types ofpolynomial curve fits are also published in NASPTM 1107 (ref. 7), NASA SP-3001 (ref. 8), andNASA TP-3287 (ref. 9). Use of these curve fitsbased upon tables of standard thermodynamicproperties of gases enables the application ofthis method to any type of gas: monatomic,diatomic, and polyatomic (e.g., H2O, CO2, andCF4) gases or mixtures thereof, provided a dataset of cp versus temperature exists for the indi-vidual gas(es) of interest.

In this report, a set of thermally perfect gasequations, derived for the specific heat as a poly-nomial function of temperature, is applied to thecalculation of flow properties across obliqueshock waves. This set of equations was codedinto a previously developed computer programreferred to as the Thermally Perfect Gas (TPG)code. The new oblique shock wave capabilitywas added as an optional output to the isen-tropic flow and normal shock wave tables of theoriginal code. All of the output tables of the TPGcode are structured to resemble the tables ofcompressible flow properties that appear inNACA Rep. 1135, but can be computed for arbi-trary gases at arbitrary Mach numbers and/orstatic temperatures, for given total tempera-tures. All properties are output in tabular form,thus eliminating the need for graphical interpo-lation from charts. As in the original TPG code,the biggest advantage is the validity in the ther-mally perfect temperature regime as well as inthe calorically perfect regime, and its applicabil-ity to any type of gas (monatomic, diatomic, tri-


3

atomic, polyatomic, or any specified mixturethereof). The code serves the function of thetables and charts of NACA Rep. 1135 for any gasspecies or mixture of species, and significantlyincreases the range of valid temperature appli-cation due to its thermally perfect analysis.

NomenclatureSymbols:

a speed of sound

A cross-sectional area of stream tubeor channel

Aj coefficients of polynomial curve fitfor cp/R of individual species; andfor cp of mixture, equation (3)

cp specific heat at constant pressure

cv specific heat at constant volume,cpR

M Mach number, V/a

p pressure

q dynamic pressure,

R specific gas constant

T temperature

u velocity component normal to theshock wave

V flow velocity

w velocity component tangential to theshock wave

Y mass fraction

z lateral coordinate to upstream flow,measured from wedge leading edge

ratio of specific heats, cp/cv

flow deflection angle;2-D wedge half-angle

molecular vibrational energy con-stant, references 1 and 4

characteristic Mach angle

mass density

shock wave angle relative to theupstream flow direction

Subscripts:

1 upstream flow reference point;e.g., upstream of a shock wave

2 downstream flow reference point;e.g., downstream of a shock wave

i ith component gas species of mixture

j jth coefficient of polynomial curve fitfor cp

jmax maximum j value

lim limiting conditions for oblique shockwaves

max maximum value

mix gas mixture

n total number of gas species thatcomprise a gas mixture

perf calorically perfect

therm thermally perfect

t total (stagnation) conditions

Abbreviations:

1-D one-dimensional

2-D two-dimensional

CFD computational fluid dynamics

CPG calorically perfect gas

cpu (computer) central processing unit

GASP General Aerodynamic SimulationProgram

NACA National Advisory Committee forAeronautics

NASP National Aero-Space Plane

TPG thermally perfect gas

V 22----------


4

Derivation of Oblique ShockRelations

Polynomial Curve Fit for cp

The selection of a suitable curve fit functionfor cp is the starting point for the development ofthermally perfect compressible flow relations.The form chosen for the TPG code was the eight-term, fifth-order polynomial expression givenbelow, in which the value of cp has been nondi-mensionalized by the specific gas constant.

(1)

For a given temperature range, A1 to A8 are thecoefficients of the curve fit. This is the func-tional form for cp of NASP TM-1107 (ref. 7),which provides values of A1 to A8 for 15 differentgas species. NASA SP-3001 (ref. 8) provides sim-ilar cp curve fit coefficient data (A3 through A7)for over 200 individual gas species for a five-term, fourth-order polynomial fit equation.NASA TP 3287 (ref. 9) provides additional cpcurve fit coefficient data for 50 reference ele-ments for both a five-term (A3-A7) and a seven-term (A1-A7) fourth-order polynomial fit equa-tion. Equation (1) is valid for each of these curvefit data sets, with appropriate coefficients set tozero. These definitions are typically multi-seg-mented curve fits defined over contiguous tem-perature ranges so that the cp definition iscontinuous. Thus, for a given species there aretypically two or more sets of coefficients, eachvalid over a given subset of the entire tempera-ture range.

The polynomial curve fit definitions pro-vided in refs. 7, 8, and 9 are defined to be validover specific temperature ranges. Beyond thoseranges the accuracy of the data fit may degraderapidly. All algorithms based upon these datafits should note any calculations at temperaturesoutside of the stated temperature ranges.

Different algebraic expressions for cp/Rcould be exchanged for that of equation (1)within the TPG code. The form of equation (1)

was selected in the current work because of itsease of implementation, and the wealth ofalready available data from references 7, 8, and9. The only requirements are that closed-formsolutions to both and must beknown. These requirements are explained inmore detail in reference 2.

Mixture Properties

The TPG code can be used to compute thethermally perfect gas properties for not onlyindividual gas species but also for mixtures ofindividual gas species (e.g., air). This capabilityis achieved within the code by calculating thevariation of the heat capacity for the specifiedgas mixture. The mixture cp is

(2)

where Yi is the mass fraction of the ith gas spe-

cies. The value of is determined from equa-tion (1) for each component species. This isactually implemented in the code by combininglike terms of the polynomials for each individualgas species; that is

(3)

where

, for j=1, jmax (4)

With a known curve fit expression for cp ofthe gas mixture, the value of for a given tem-perature can be directly computed, as can mix-ture properties of gas constant and molecularweight. See references 2 and 3 for additionaldetails regarding the derivation of the 1-D isen-tropic relations and the normal shock relationsfrom these properties.

Oblique Shock Relations

Figure 1 illustrates the geometric relation-ships associated with a shock wave at an arbi-trary shock angle relative to the upstream flowdirection. These relationships can be expressedas

cpR----- A1

1T 2------ A2 1T--- A3 A4 T( ) A5 T 2( )+ + + +=

A6 T3( ) A7 T

4( ) A8 T5( )+ + +

cp Td cp T( ) Td

cpmixY icpi

i 1=

n

=

cpi

cpmixA jmixT

j 3

j 1=

jmax=

A jmix Y iRiA jii 1=

n

=


5

(5)

and

(6)

where u1 and u2 are the upstream and down-stream velocity components normal to the shock,and w=w1=w2 is the tangential velocity compo-nent which is conserved across the shock. Sinceonly the normal component of velocity changesacross the shock, the flow is turned through anangle . Solving equations (5) and (6) for w, asingle equation can be written:

(7)

The normal velocity components can beexpressed as

(8)

and

(9)

Thus, equation (7) can be rewritten as

, (10)

a function of shock angle , deflection angle ,and the upstream and downstream velocities V1and V2.

The continuity and momentum equationsfor 1-D flow across a shock wave in a shock-fixedcoordinate system (a stationary shock) are givenas follows

(11)

(12)

Dividing the momentum equation by thecontinuity equation gives

(13)

Using the ideal gas law ( ), equation (13)can be written in terms of only temperature andnormal velocity as

(14)

Substituting for u1 and u2 with equations (8)and (9), equation (14) becomes

, (15)

a function of shock angle , deflection angle ,the upstream and downstream velocities V1 andV2, and the upstream and downstream statictemperatures T1 and T2. However, as derived inreference 2, the velocity may be expressed as afunction of temperature

(16)

at any point in the flow field, whether upstreamor downstream of the shock. Thus, the depen-dencies on velocity in equations (10) and (15) canactually be expressed as dependencies on tem-perature.

Assuming all upstream flow conditions areknown (state 1), and substituting equation (16)for V1 and V2, equations (10) and (15) representa set of two nonlinear equations for threeunknowns: , , and T2. Given any one of thesevariables, the other two may be found numeri-cally by means of Newton iteration. (Appendix Aprovides additional details regarding implemen-tation of this Newton algorithm within the TPGcode.) If the shock angle is known, the tangen-tial velocity component is

(17)

Thus, the downstream normal velocity u2 can bedefined as

, (18)

a function of T2 only, instead of both T2 and asin equation (9). Thus, instead of equation (15),equation (14) reduces to a function of T2 onlyand can be solved independently of

tanu1w-----=

( )tan u2w-----=

( )tantan

-------------------------

u2u1-----=

u1 V 1 sin=

u2 V 2 ( )sin=

( )tantan

-------------------------

V 2 ( )sinV 1 sin

--------------------------------=

1u1 2u2=

p1 1u12

+ p2 2u22

+=

p11u1----------- u1+

p22u2----------- u2+=

p RT=

RT 1u1

---------- u1+RT 2u2

---------- u2+=

RT 1V 1 sin------------------ V 1 sin+

RT 2V 2 ( )sin-------------------------------- V 2 ( )sin+=

V 2 2 cp TdTT t=

w w1 w2 V 1 cos= = =

u22 V 2

2w

2 2 cp T w

2d

T 2

T t= =


6

equation (10). For the case of known deflectionangle , equations (10) and (15) must be solvedsimultaneously for T2 and .

Once the primary variables of temperatureT2, , and are known, all other flow quantitiesmay be computed.

(19)

(20)

(21)

where V2 is known from equation (16).

Pressure and density downstream of theshock wave can be calculated by the normalshock wave relations derived in reference 2, not-ing that all velocity terms (including Mach num-ber) are defined normal to the shock. That is,

(22)

(23)

The values of total pressure loss pt,2/pt,1 and theratio of upstream-static to downstream-totalpressure p1/pt,2 can then be calculated fromcombinations of static and total pressure ratios.See reference 2 for further detail.

Figure 2 illustrates the relationshipsbetween M1, , and for air at a sample Tt. Thefigure shows that, for a given M1 and , two solu-tions exist for , corresponding to what are com-monly called weak and strong shock waves. Amaximum flow deflection angle max exists forany M1, defined as the point at which the weakand strong shock solutions are identical. Strongshock waves always result in subsonic down-stream Mach numbers and have shock waveangles approaching 90. Weak shock waves arethose for which the corresponding wave angle is less than that at which max occurs, and usu-ally result in supersonic downstream Mach num-bers, except as approaches (max).

Oblique shock waves can only exist at Mach

numbers above a limiting case, i.e., M1>M1,lim.For a given , the limiting case occurs at =,where the Mach angle is defined by

(24)

At these conditions the shock strength and reduce to zero. Obviously, is only defined forM1 greater than, or equal to, 1, and thus, obliqueshocks can not exist at subsonic upstream condi-tions. For sonic upstream conditionsM1=M1,lim=1.0, lim=90 and lim=0.

For a given , the limiting case M1,lim corre-sponds to the Mach number for which =max.From a physical perspective, this limit is theminimum Mach number at which an attachedshock wave can occur at the leading edge of awedge. Below M1,lim the flow results in adetached strong shock wave standing a finitedistance upstream of the wedge leading edge,and the flow approaching the point of turning issubsonic, not supersonic.

Oblique Shock Code Description

An interactive FORTRAN computer code,herein referred to as the TPG code, has beenwritten based on the equations described hereinand those described in references 2 and 3. Thiscode delivers a complete table of results withinseconds when run on a computer workstation orpersonal computer. The purpose and primaryoutput of the code is the creation of tables ofcompressible flow properties for a thermally per-fect gas or mixture of gases (styled after thosefound in NACA Rep. 1135). In addition to theisentropic flow and normal shock wave proper-ties of the previous version of TPG, the latestrelease of the TPG code (version 3.1) computesproperties across oblique shock waves, giveneither the shock wave angle or the flow deflec-tion angle . Tabular entries may be based uponconstant increments of Mach number, or con-stant decrements of static temperature from thetotal temperature. Such tables are presented inNACA Rep. 1135 only for isentropic flow andnormal shock waves, and only for calorically per-fect air (=1.40) at constant increments of Machnumber. Only charts (no tables) are presented

2cp2

cp2R

-----------------=

a22 2RT 2=

M2 V 2 a2=

p2p1------ 1 1 M1 sin( )

2 1u2u1----- +=

21-----

u1u2-----=

sin 1M1-------=


7

therein for oblique shock wave properties. As inprevious versions (see references 2 and 3), theutility of the TPG code is its capability to gener-ate tables of compressible flow properties of anygas, or mixture of gases, for any total conditionsover any specified range of Mach numbers orstatic temperatures (T


8

are italicized (and numbered in the left margin).A response of a comma indicates acceptance of aprogram default. Each user response is followedby a carriage return (shown as ).

The first response required [1] is an identi-fier character string (an ID), or symbol name, forinclusion within output files. The ID can be amaximum of 70 characters, entered on a singleline. A default ID (for air) may be specified by acomma or the word DEFAULT (all upper caseor all lower case characters), in which case thenext three questions, [2]-[4], are omitted anddefaults are assumed. A particular database filename may be entered as response [2], the pre-scribed file containing the necessary thermo-chemical data for the gases of interest. The filename can be 80 characters, maximum, includingfile path information. The contents of such adatabase file is described in detail in reference 2,and is summarized in a later subsection. Thecode contains the database necessary to define afour species mixture of air; this default is chosenby entering a comma. The gas mixture defini-tion is completed by responses [3] and [4] inwhich the number of individual species, and thecorresponding mass fractions, are specified.Defaults are shown as part of the program que-ries, and may always be selected; they initiallycorrespond to the standard composition of air.Regardless of the number of fractions, the codechecks to see if the sum of the mass fractions isequal to unity, and if the database contains suffi-cient information for the requested case. If themass fractions do not sum to unity (within asmall tolerance), the code issues an error mes-sage and stops execution. If the sum is withinunity plus-or-minus the tolerance, the last spe-cies fraction is recomputed such that the total isprecisely unity and a warning message is outputdescribing the change. If the database containsinsufficient information an error message is out-put and execution is terminated.

Responses [1] and [2] are sensitive to thecase of the alphabetic characters entered. Thatis, upper and lowercase alphabetic charactershave unique meanings. However, all succeedinginputs are insensitive to case, and either upperor lowercase responses have the same meaning

within the code.

The next inputs define the composition andformat of the table(s). The total temperature,response [5], specifies the upper temperaturelimit (for which the Mach number is zero). Input[6] specifies whether the tables are to be interms of static temperature (T) or Mach (M)number increments. Successive entries in thetable(s) may be given at Mach numbers from aninitial Mach up to an upper Mach number limit,for constant step sizes of Mach, response [7].Alternatively, if the constant static temperatureoption is specified in response [6], the tableentries are of static temperatures from an initialtemperature down to the low temperature limit,for constant step sizes of static temperature, alsoentered as response [7]. The initial values arethe total conditions by default. These inputs areseparated by commas, and any, or all, may bedefaulted simply by entering a comma alone.(See responses [6] and [7], for example.)

Response [8] specifies which properties areto be included in the table(s). Isentropic flowproperties are always included in the table(s),and constitute the default for response [8]. Thebasic table can be expanded to include propertiesacross shock waves through selection of menuoptions 1, 2 or 3. Oblique shock waves can bespecified through either a specific shock waveangle , or as the shock wave required to turnthe flow through a given angle . Normal shocksare a special case of the oblique shock wave with=90. For menu options 2 or 3, response [9] isrequired to specify the appropriate angle indegrees, or , respectively. Response [10]allows the user to choose the desired shock wavetype, either weak or strong, for the given ofmenu option 3, response [8], and is omitted forall other options. Similarly, responses [9] and[10] are both omitted for menu options 0 and 1.

The relative positions of the tables of isen-tropic and shock wave properties are determinedby response [11] regarding the number of tables(either 1 or 2). The shock wave properties andthe isentropic properties may be output in twoseparate tables cross-referenced by Mach num-ber and static temperature, or in a single, wide-format table (156 characters across). The sepa-


9

Table 2: Sample TPG v3.1 interactive inputs

| Thermally Perfect Gas Properties Code | | TPG, Version 3.1.0 | | | | NASA Langley Research Center |

Enter the ID (character string) for the gas mixture: [default = 4-species Air] {enter a comma (,) to accept any default}

[1] Sample Combustion Products Enter the name of the Database File for the Species properties:

[2] CombProd.data Enter the number of species composing the mixture: [default = 4]

[3] 6 Enter the mass fraction, Y, of all 6 component species in the order of the Database File (sum=1.0): [defaults = 0.7553, 0.2314, 0.0129, 0.0004] [defaults = 0.0000, 0.0000,

[4] .1, .6, .2, .0303, .04 Enter the Total Temperature (in degrees K): [default = 400.]

[5] 1500. Output the table entries by incremental Temp (K) or incremental Mach number? (T/M) [default = M]:

[6] , Enter the Initial Mach number, the Mach number Increment, and the Upper Mach number Limit: [defaults = 0., 0.1, 5.0]

[7] , .2, , Enter the appropriate Index from the following Menu: 0 : Isentropic Flow Properties only, ----> [DEFAULT] or additionally include 1 : Normal Shock Wave Properties, 2 : Oblique Shock Wave Properties, Shock angle given, 3 : Oblique Shock Wave Properties, Deflection given,

[8] 3 Enter the Flow Deflection Angle in degrees : [default = 10.000 degrees]

[9] , Weak or Strong Shock solution: w/s [default = Weak]

[10] , Enter the number of Tables to be output, 1 or 2 : [default = 2]

[11] 1 Enter the Tabular output location: screen=6, file=9 [default = 9 : TPG.out]

[12] , Do you want to output a TECPLOT file [TPGpost.dat] ? (y/n): [default = N]

[13] y Continue calculation to a succeeding state? (y/n) [default = N]:

[14] , STOP: NORMAL


10

rate (second) table of shock wave propertiesalways begins with the lowest Mach number forwhich a valid solution of the equations exists,that is, M1,lim. Succeeding entries are for Machnumbers greater than M1,lim. In the single tableformat, explanatory text is output under theshock property headings when shock propertydata is not valid.

The table(s) are written to an output file,TPG.out, by default since they can be ratherextensive. The output file can then be viewedon-line or printed via an appropriate systemcommand. However, if the user desires, thetables can be written to standard output (typi-cally the computer screen), as specified inresponse [12] to the question of tabular outputlocation.

Response [13] determines the output of anadditional file for data plotting and postprocess-ing. Because of extensive usage at NASA Lan-gley Research Center, the Tecplot1 (ref. 10) codeformat has been chosen for this file. The basicpostprocessor file TPGpost.dat contains thesame data as that in the TPG.out tabular out-put. The format of this ASCII file, described inreference 10 and in appendix B of reference 2,could be easily modified to satisfy the inputrequirements of many different graphics post-processors. One, two, or three zones of informa-tion for all variables and all table entries areincluded. The first zone includes all subsonicconditions, and always contains, at a minimum,total (stagnation) and sonic entries. The lastzone contains all supersonic conditions for whichthe desired shock wave exists. For oblique shockwaves with M1,lim greater than sonic, an inter-mediate zone is written containing conditionsbetween, and including, sonic and the first shockwave. The second and third zones are omitted ifno supersonic conditions are present. The shockwave properties have no physical meaning atsubsonic conditions (similarly with supersonicconditions below the limiting Mach number) andare defaulted to a large negative number as awarning flag.

1. Tecplot: trademark of Amtec Engineering, Inc.,Bellevue, WA 98009-3633.

Response [14] allows the user to compute asuccession (or train) of shock waves where theupstream conditions of each succeeding shockwave is based upon the downstream conditionsof the preceding one. The default response is toend the current set of calculations. An affirma-tive response [14] returns the user to the menuof options, waiting for a new response [8]. Thus,the mixture ID, composition, and total tempera-ture are retained, and the table entries aredetermined by the downstream conditions of thepreceding calculation.

Following interactive inputs [1]-[13], thedatabase file, if one is requested, is opened, read,and closed. Properties of the gas mixture arecomputed and the coefficients of the cp relation-ship for the mixture are defined (equations (3)and (4)). The TPG.out file is opened and headerinformation is written to it stating the gas mix-ture specifications and the total temperature.The coefficients of the mixture cp relationshipare also listed (equation (4)). A sample header isshown in table 3.

Through calls from the Main program tovarious subroutines (see table 1) the data arraysare initialized, stagnation and sonic conditionsare computed, and the isentropic flow propertiesare computed for each Mach number or statictemperature entry in the table, as requested bythe user in responses [6] and [7]. These proper-ties are then output to TPG.out unless shockproperties are requested in conjunction with thesingle table format. If shock wave properties arerequested (response [8]), the code next deter-mines the limiting Mach number and tempera-ture conditions for existence of the shock wave.Beginning with the first entry at a greater Machnumber (lower static temperature), the shockwave properties are computed to complete thedata set. Tabular output to TPG.out is com-pleted and the file is closed. The same informa-tion is then output in postprocessor format to theTPGpost.dat file.

The default N (no) for response [14] ter-minates the calculation. Code execution is com-pleted by closing all files and recording warningsummaries of the times the valid temperatureranges were exceeded during the calculations.


11

These warnings, if any, are related to the data-base information and are written to an addi-tional file called Extrap.WARN for laterreference by the user.

Include (or Header) Files

Three files contain statements that areincorporated into the Main program and varioussubroutines via FORTRAN include statements atprogram compilation. The Version.h file con-tains a character variable which specifies thecurrent version number of the computer code.This information is output at the beginning ofeach interactive execution and to the first linesof the TPG.out file.

The params.h and dimens.h files containinformation which determines the amount ofcomputer memory required to execute the TPGcode, and, thus, the size of the case which may bespecified. The dimensions of the primary arrayswithin the code as defined below are included ina FORTRAN parameter statement in params.hand have the following values as released in Ver-sion 3.1:

mtrm (=8): number of polynomial termswhich define the relationshipfor cp/R,

mspc (=25): maximum number of gas spe-cies allowed,

mcvf (=3): maximum number of temper-ature ranges per speciesallowed and correspondingsets of polynomial coeffi-cients,

mtncr (=10000): maximum number of com-puted static temperatures.

All variables assigned to the precedingparameters, except mtrm, may be changed tosuit particular user requirements. A change inmtrm may require coding changes in the subrou-tines which perform specific calculations usingthe cp(T)/R relationships. (See the sectionPolynomial Summation and Integration Sub-routines in reference 2 for details on suchchanges.) The dimens.h file specifies common

blocks containing all of the major arrays that aredependent on the parameters described above.

Mixture and Basic Flow PropertiesSubroutines

Subroutine mixture combines the individualgas species data using the user-specified massfractions to compute corresponding mixtureproperties. The database file is opened and read,if needed. The mixture molecular weight andspecific gas constant are computed and the mix-ture polynomial curve fit coefficients are com-puted according to equation (4). The validtemperature limits of this mixture curve fit aredetermined from the individual species tempera-ture limits. The lower temperature limit of themixture is chosen as the greatest minimum tem-perature of all species with nonzero mass frac-tion. Similarly, the upper temperature limit ofthe mixture is chosen as the smallest maximumtemperature of all species with nonzero massfraction.

The isntrop subroutine computes the basicproperties of isentropic expansion from total(stagnation) conditions for a given static temper-ature. The shokwav subroutine computes theconditions downstream of a shock wave, relativeto the upstream conditions from known values ofthe shock wave angle , the flow deflection angle, the downstream static temperature T2, andthe downstream velocity V2. These four funda-mental properties are either given as inputs orcomputed in the oblique shock wave subroutines(see the following subsection).

For any known Mach number M the corre-sponding static temperature T can be deter-mined from the nonlinear relationship

(25)

in which is also a function of T. Subroutinet1iter performs this function within the TPGcode. See reference 2 for additional detailsregarding this subroutine and algorithm(unchanged from reference 2).

2 cp TdTT t M2RT=


12

Oblique Shock Wave Subroutines

As stated previously, equations (10) and (15)are nonlinear functions of three variables (, ,and T2) which define the relations across anarbitrary oblique shock wave. These equationsare solved within the TPG code through Newtoniteration algorithms, given one of the two angles or . Subroutine Todelt solves for and T2given , and Tosigm solves for and T2 given .The downstream velocity component u2 (and,thus, V2) is also obtained in the course of theseiterations. Each of these iterations is initializedby the solution of the calorically perfect shockwave equations (see reference 1) using the localratio of specific heats . Subroutines Codelt andCosigm compute such calorically perfect initialestimates.

Subroutines Todelt and Tosigm only con-verge to reasonable answers if the equations arephysically valid for the given upstream condi-tions and desired angles. Thus, the limitingstatic temperature and corresponding Machnumber M1,lim must be determined prior to iter-ating for specific conditions; subroutine sta2lim(short for station 2 limits) determines theselimiting conditions. For a given , the limitingconditions are found at the Mach angle, wherethe shock strength and reduce to zero. For agiven , a two-dimensional iteration proceduredetermines the conditions at which the weakand strong shock wave solutions are identical,that is, the Mach number at which max(M)=.The max for a given Mach number is determinedin subroutine parabt by assuming a parabolicform of as a function of for a given T1. Thequadratic coefficients of the parabola areadjusted until a good fit of the curve apex isfound, and the resulting quadratic equation issolved analytically for the extremum.

Postprocessor Subroutine

The postout subroutine in postout.f simplyopens the TPGpost.dat file, writes the tabulardata in Tecplot ASCII format, and closes the fileupon completion. The FORTRAN open state-ment specifies a status of unknown to allow thefile to be re-opened for additional output.

Other Subroutines

Many of the subroutines in Version 3.1 ofthe TPG code are unchanged since the publica-tion of reference 2. Specifically, these routinesinclude those which evaluate cp, the integral ofcp, and the integral of cp/T, and are found in filesntgrat.f and cpsubs.f. The trangej subroutine,which finds appropriate temperature ranges andthe corresponding polynomial curve fit for agiven T, is also unchanged; it also recordsinstances in which the curve fit temperature lim-its are exceeded and writes warning messages tooutput files. Finally, the initd and Block Dataroutines are basically unchanged from reference2, except that a few isolated variables have beenadded or deleted as the code has evolved. Seereference 2 for additional details regarding anyof these routines.

Database Files

The thermochemical data required by theTPG code for a given mixture of gases is definedin a database file to be read at execution time. Adatabase for the standard composition of air iscontained within the code and may be accessedas the default. However, for mixtures of othergases, or for more complete models of air, a sepa-rate database file may be provided. The formatis simple and modular, grouped by species with atwo-line header, and additional databases areeasily constructed. A sample two-species data-base is shown in Table 4, and the format isdescribed in some detail in references 2 and 3.

Sample Tabular Output

Table 5 shows a sample of the tabular out-put in the single table format for a shock angle of30 in air at Tt=400 K. Following the headerinformation, described previously, are columns ofdata for the isentropic flow properties and theproperties across the oblique shock wave. Thecolumn headings, in order from left to right,refer to the isentropic flow properties of Machnumber M1 (upstream of any shock wave), statictemperature T1, ratio of specific heats , static-to-total pressure ratio p1/pt, static-to-total den-sity ratio 1/t, static-to-total temperature ratioT1/Tt, dynamic-to-total pressure ratio q/pt, arearatio for expansion from sonic flow A/A*, and


13

local velocity divided by the sonic speed-of-soundV/a*. The oblique shock properties continue asshock wave angle , flow deflection angle , Machnumber downstream of the shock wave M2,static pressure ratio across the shock p2/p1, den-sity ratio across the shock 2/1, static tempera-ture ratio across the shock T2/T1, and the ratioof total pressures across the shock pt,2/pt,1. Fora given shock angle, the minimum flow deflec-tion is 0, corresponding to a shock wave of zerostrength, i.e., a Mach wave. Thus, the minimumMach number of 2.0 for a 30 shock is deter-mined by the equation for the Mach angle, equa-tion (24). For all non-subsonic Mach numbersthe code outputs an informative message statingthe minimum (=) for that Mach number. Atsubsonic Mach numbers there is obviously nosolution to the shock relations, as noted in theoutput. These informative messages do notappear in the two-table format; the shock wavetable simply begins with the minimum Machnumber M1,lim for which a solution exists.

Tables 6(a) and (b) illustrate the TPG tabu-lar output for a constant flow deflection angle .Table 6(a) gives the weak shock solutions, and6(b) gives the strong shock solutions (i.e., approaches 90 and M2


14

within the calorically perfect temperatureregime for air. The lower Mach number limitwas defaulted to stagnation conditions and theupper Mach number limit was set to 10, with aMach number increment of 0.1. In the case ofreal air at Tt=400 K, liquefaction would occurwell before Mach 10; the data is presented tothat extreme herein merely for comparison withthe calorically perfect gas tables of NACA Rep.1135 (which, incidentally, extend up to Mach100). (All cases presented herein were run withthe TPG code in the Mach number incrementmode. See reference 2 for examples of the tem-perature increment mode.)

Figures 3(a-c) and 4(a-c) compare TPG cal-culations of M2, , 2/1, T2/T1, p2/p1, and pt,2/pt,1for air with calorically perfect (=1.40) solutionsat shock wave angles of 30 and 50, respectively.Each variable is plotted versus upstream Machnumber M1. The differences are negligible asshown. Figures 5 and 6 show the same data pre-sented as ratios of the thermally perfect valuesto the calorically perfect values. At =30, thedifferences are less than 0.23% for all variablesexcept total pressure ratio pt,2/pt,1. Similarly, for=50 the differences are less than 0.25%, exceptfor total pressure ratio. Even for this most sensi-tive variable, the differences are less than 1%,and that value is approached only at the largestshock angle and for Mach numbers approaching10. As noted in reference 2 for normal shockwaves, these small differences actually representthe small amount of error associated with thecalorically perfect gas assumption. A slight vari-ation with temperature actually exists in theheat capacities even at such a low Tt, and is thecause of the variations seen in figures 5 and 6.

Thermally Perfect TemperatureRegime

The properties for arbitrary oblique shockwaves were verified for test cases involving stan-dard four-species air at total temperatures of1500 K and 3000 K, and at shock wave angles of30 and 50. Comparisons were made withresults obtained via the imperfect gas relationsof NACA Rep. 1135 (i.e, the -equations for adiatomic gas) and with selected results from thecalorically perfect gas assumption. The proper-

ties of , M2, p2/p1, T2/T1, 2/1, and pt,2/pt,1 areplotted versus the upstream Mach number M1 infigures 7 and 8. Figures 7(a)-(f) show results forTt=1500 K, a temperature well above the limitsof the calorically perfect assumption for air, andfigures 8(a)-(f) are for Tt=3000 K, an even moreextreme case. The two thermally perfect gasmethods agree extremely well. Cursory exami-nation of the plots shows that imperfect gaseffects increase with both total temperature andshock angle. For some flow conditions the differ-ences appear to be negligible, but, for other con-ditions, the differences can be large. Also, not allproperties appear to be equally sensitive tocaloric imperfections.

The calculation for a total temperature of3000 K is presented as an extreme case to illus-trate application of the TPG code over anextended temperature range under the assump-tion of chemically frozen flow, i.e., fixed composi-tion. The presentation is in the same spirit asthat of references 5 and 6, where data is pre-sented to 5000 K and 6000 K, respectively. Asalways, users must keep in mind the applicabil-ity of the assumption of chemically frozen flow toa particular problem.

A more descriptive mode for examining thedifferences between the thermally perfect gascalculations and calorically perfect results isshown in figures 9(a)-(f). In these plots, thethermally perfect properties (, M2, p2/p1, T2/T1,2/1, and pt,2/pt,1) are divided by the caloricallyperfect properties at identical values of M1 toyield a measure of the imperfect gas effects.Results are shown for both total temperaturesand both shock angles, for both the TPG codeand the -equation method of NACA Rep. 1135(ref. 1). In this format differences due to caloricimperfections are observed of 10% or greater fortemperature ratio, density ratio, and total pres-sure ratio at the higher total temperature andgreatest shock angle. Caloric imperfection dif-ferences on the order of 4-5% are observed for allvariables except static pressure ratio, even atthe lower total temperature. Also, the magni-tudes of the caloric imperfections do not all varyuniformly with Mach number M1. The effects on are largest at lower M1, but are larger for other


15

variables at higher M1. For static pressure ratiothe sign of the caloric imperfection effectchanges as M1 increases. Thus, the use of thecalorically perfect relationships could result insubstantial errors, particularly in the design oranalysis of flows with a sequence of such shockwaves. Finally, these results are shown assum-ing that M1 is the control variable, i.e., the vari-able held fixed for both calorically and thermallyperfect calculations. For other applications, thecontrol variable might be the flow deflectionangle , for example, and the magnitude of thecaloric imperfection could be even greater.(Comparisons relative to the total, state 0, condi-tions, rather than state 1, would also yield largereffects of caloric imperfection.)

The differences between the two thermallyperfect gas methods can be seen to be small (typ-ically less than 0.25%). This excellent agree-ment verifies the accuracy of the TPG code withthe derived thermally perfect gas relations foroblique shock waves based upon polynomialexpressions for cp. Although these test caseswere all for air, which is a primarily diatomicgas, the TPG code is also valid for polyatomicgases. The utilization of a polynomial curve fitfor cp makes the TPG code applicable to anymolecular structure of the gas, not just thediatomic structure upon which the -equationsare based.

Comparisons with CFD

As an additional validation of the TPGcodes oblique shock wave capability, resultswere compared to computational fluid dynamicsEuler solutions (i.e., inviscid flows). The CFDsolutions were obtained using the General Aero-dynamic Simulation Program (GASP), ref. 11, ofAeroSoft, Inc. GASP solves the integral form ofthe governing equations, including the full time-dependent Reynolds-averaged Navier-Stokesequations and various subsets: the Thin-LayerNavier-Stokes equations, the ParabolizedNavier-Stokes equations, and the Euler equa-tions. A generalized chemistry model and bothequilibrium and non-equilibrium thermodynam-ics models are included. The discretized equa-tions may be solved by space marching or byglobal iteration. The current computations uti-

lized space marching (i.e., totally supersonicflow) for the inviscid Euler equations. Third-order upwind inviscid fluxes were calculatedusing Roes split flux normal to the flow directionwith Min-Mod flux limiting. The NASA LewisResearch Center equilibrium curve fits for spe-cific heat (references 7, 8 and 9), of the sameform employed within TPG, defined the thermo-dynamic characteristics of the gases. A 2-D gridof 20 uniformly-spaced computational cells perunit length was defined on a nondimensionaldomain (10 units long) above a simple wedge.The domain height was chosen so that theresulting oblique shock wave would passthrough the downstream boundary (5 and 8.4units for the two test cases, respectively).

The first CFD test case involved conditionsrepresentative of a generic high-speed inlet:thermally perfect air at M1=4.8, T1=1670 K andp1=101325 N/m

2 (1 atmosphere) flowing over a10 wedge. Figure 10 shows contours of pressureratio p/p1 for the GASP Euler solution with theshock wave computed by the TPG code(=19.415) overlaid as a thick dashed line. Themagnified view (see the inset) shows that theagreement in shock position is excellent, consid-ering that GASP captures the shock over a num-ber of cells and the TPG solution is a precisepoint-solution, that is, the shock wave is a truediscontinuity. More detailed comparisons of M,p/p1, T/T1, and /1 are shown in the line plots offigure 11(a)-(d). The data points shown wereinterpolated from the GASP solution field (e.g.,figure 10) along a horizontal line at z=2.6 unitsabove the wedge leading edge. The TPG solutionis shown as a discontinuous solid line with theshock position calculated geometrically at thatsame lateral location of 2.6 units. Again, theagreement is excellent outside the region ofshock capturing.

The -equations of NACA Rep. 1135 arevalid only for a single-species diatomic gas.Therefore, a second comparison with GASP wascomputed at arbitrary conditions designed tohighlight the flexibility of the TPG code. The gaschosen was a two-species mixture of 20% steamand 80% CO2 by mass, neither species beingdiatomic. Upstream conditions were set at


16

M1=3.5, T1=1200 K and p1=101325 N/m2. A

rather severe wedge angle of 25 was selectedand the 2-D GASP grid was constructed asbefore: 20 uniformly-spaced computational cellsper unit length defined on a nondimensionaldomain (10 by 8.4 units to contain the expectedshock wave). GASP required 282.4 cpu secondsto compute the solution on a Cray2 Y-MP com-puter, not including grid generation and GASPinput setup time (about 2 days). The TPG solu-tion was practically instantaneous, computed ona Sun3 Sparcstation 20 workstation. The result-ing solutions are shown in figure 12, again withthe TPG solution (=37.3037) overlaid as athick dashed line on the GASP pressure ratiocontours. The magnified view in the inset againconfirms the excellent agreement of the shockpositions. The line plots of M, p/p1, T/T1, and/1 shown in figure 13(a)-(d) confirm the accu-racy of the TPG code for arbitrary mixtures ofpolyatomic gases. Since the TPG result is a pre-cise point-solution, the overshoots seen infigure 13, and common to shock-capturing CFDcodes, are eliminated. Similar to figure 11(a)-(d),the data points were interpolated from theGASP solution (figure 12) along a horizontal lineat z=5.0 units above the wedge leading edge, andthe TPG solution is shown as a discontinuoussolid line with the shock position calculated atthe same lateral location of 5 units.

Temperature Limits of the TPGCode

The TPG code provides valid results as longas its application is within the thermally perfecttemperature regime for the gas of interest. Ifthe code is used outside of the region for which cpis a function of temperature only, then the ther-mally perfect results produced by this code willno longer accurately reflect what actually hap-pens in nature. Dissociation will occur abovesome upper temperature limit, and will cause adeviation from thermally perfect theory. At theopposite extreme, below some lower temperaturereal gas effects will become important (i.e., inter-molecular forces will not be negligible). The

2. Cray Research, Inc., Minneapolis, MN 55402.3. Sun Microsystems, Inc., Mountain View, CA

94043.

TPG code user must remain aware of these highand low temperature boundaries associated withthe thermally perfect assumption for the partic-ular gas(es) under consideration. Note thatthese are boundaries associated with the physi-cal properties of the gas(es), and are distinctfrom the upper and lower temperature limitsassociated with the polynomial curve fits utilizedwithin the TPG code. These latter curve fit lim-its are tracked within the code, and users arewarned by the TPG code when these limits areexceeded. The limits of the thermally perfectassumption are the users responsibility, and aredependent upon the specific gas properties andflow conditions. See reference 2 for a moredetailed discussion of these temperature limits.

Conclusions

A set of compressible flow relations describ-ing flow properties across oblique shock waveshas been derived for a thermally perfect, calori-cally imperfect gas, and applied within the exist-ing thermally perfect gas (TPG) computer code.The relations are based upon a value of cpexpressed as a polynomial function of tempera-ture. The code now produces tables of compress-ible flow properties of oblique shock waves, aswell as the properties of normal shock waves andbasic isentropic flow from the original code, in aformat similar to the tables for normal shockwaves found in NACA Rep. 1135. The coderesults were validated in both the caloricallyperfect and the calorically imperfect, thermallyperfect temperature regimes through compari-sons with the theoretical methods of NACA Rep.1135, and with a state-of-the-art computationalfluid dynamics code. The TPG code is applicableto any type of gas or mixture of gases and is notrestricted to only diatomic gases as are the ther-mally perfect methods of NACA Rep. 1135.

The TPG code computes properties ofoblique shock waves for given shock wave angles,or for given flow deflection (wedge) angles,including both strong and weak shock waves.Normal shock waves are a special case of anoblique shock wave. The code also allows calcu-lation of a sequence of shocks where each succes-sive shock wave is dependent upon the flow


17

conditions from the preceding one. Both tabularoutput and output for plotting and post-process-ing are available. Typical computation time ison the order of 1-5 seconds for most computerworkstations or personal computers. The usereffort required is minimal, and consists of onlysimple answers to 14 interactive questions (orless) per table generated. Thermally perfectproperties of flows through oblique shock wavesmay be computed with less effort than has tradi-tionally been expended to compute caloricallyperfect properties. Errors incurred from usingcalorically perfect gas relations, instead of thethermally perfect gas equations, in the ther-mally perfect temperature regime have beenshown to approach 10% for some applications.

The TPG code is available from the NASALangley Software Server (LSS) on the World-Wide-Web at the following address:

http://www.larc.nasa.gov/LSS

References

1. Ames Research Staff: Equations, Tables, andCharts for Compressible Flow. NACA Rep.1135, 1953. (Supersedes NACA TN 1428.)

2. Witte, David W.; and Tatum, Kenneth E.:Computer Code for Determination ofThermally Perfect Gas Properties. NASA TP-3447, September 1994.

3. Witte, David W.; Tatum, Kenneth E.; andWilliams, S. Blake: Computation ofThermally Perfect Compressible FlowProperties. AIAA-96-0681, January 1996.

4. Donaldson, Coleman duP.: Note on theImportance of Imperfect-Gas Effects andVariation of Heat Capacities on the IsentropicFlow of Gases. NACA RM L8J14, 1948.

5. Hilsenrath, Joseph; Beckett, Charles W.;Benedict, William S.; Fano, Lilla; Hoge,Harold J.; Masi, Joseph F.; Nuttall, Ralph L.;Touloukian, Yeram S.; and Woolley, HaroldW.: Tables of Thermal Properties of Gases,National Bureau of Standards Circular 564,U.S. Dep. Commerce, November 1, 1955.

6. JANAF Thermochemical Tables, Second ed.,U.S. Standard Reference Data SystemNSRDS-NBS 37, U.S. Dep. Commerce, June1971.

7. Rate Constant Committee, NASP High-SpeedPropulsion Technology Team: HypersonicCombustion Kinetics. NASP TM-1107, NASPJPO, Wright-Patterson AFB, May 1990.

8. McBride, Bonnie J.; Heimel, Sheldon; Ehlers,Janet G.; and Gordon, Sanford:Thermodynamic Properties to 6000K for 210Substances Involving the First 18 Elements.NASA SP-3001, 1963.

9. McBride, Bonnie J.; Gordon, Sanford; andReno, Martin A.: Thermodynamic Data forFifty Reference Elements. NASA TP-3287,January 1993.

10. TecplotVersion 6 Users Manual. AmtecEng. Inc., 1993.

11. McGrory, William F.; Huebner, Lawrence D.;Slack, David C.; and Walters, Robert W.:Development and Application of GASP 2.0.AIAA-92-5067, December 1992.


18

Appendix A

Equations for NumericalImplementation

Equations (7) and (14) may be written as

(A1)

and

(A2)

where state 1 conditions T1 and u1 are assumedto be known, and u2 is a function of , , and T2,as seen from equations (9) and (16), reproducedhere for completeness.

(A3)

(A4)

Thus, there are two nonlinear equations forthree unknowns. Typically, either or isknown, and the equations may be solved numer-ically for the remaining angle and T2 by Newtoniteration.

In the case of the shock wave angle beinggiven, the equations may be rewritten as

, for i=1,2. (A5)

That is,

(A6)

and

(A7)

Now u2 can also be written as

(A8)

Therefore, u2 is a function of only T2 if is given,

and equation (A7) is actually

(A9)

Once equation (A9) is solved for T2, V2 is knownfrom equation (A4), u2 from equation (A3), and may be found from equation (A6) or simply from

(A10)

In the case of the flow deflection angle being given, equations (A1) and (A2) are a sys-tem of two nonlinear equations for the twounknowns and T2, and may be written as

(A11)

and

(A12)

The solution can be found by application of New-tons method to a system of equations, similar tothe application to equation (A9).

For equation (A9), a single equation of asingle unknown, Newtons iterative method canbe written

(A13)

where the superscripts denote the iteration step.That is, the function G2 and the derivative of G2are evaluated using values at the current, orkth, iterative approximation step in order tosolve for the k+1th correction to the approxi-mate solution. The differential equation is thenreplaced by the following algebraic equation

(A14)

For a system of equations, the functionalevaluation G2

k becomes a vector, and the deriva-tive becomes a matrix of partial derivativeswhich must be inverted. The correction term

F1 T 2, ,( ) ( )tan

tan-------------------------

u2u1----- 0= =

F2 T 2, ,( )RT 1u1

---------- u1RT 2u2

---------- u2+ 0= =

u2 V 2 ( )sin=

V 22 2 cp TdT 2

T t=

Fi T 2, ,( ) Gi T 2,( ) =

G1 T 2,( ) ( )tan

tan-------------------------

u2u1----- 0= =

G2 T 2,( )RT 1u1

---------- u1RT 2u2

---------- u2+ 0= =

u22 V 2

2w

2 2 cp T w

2d

T 2

T t= =

G2 T 2,( ) G2 T 2( ) 0= =

( )cos wV 2------=

F1 T 2,( ) ( )tan

tan-------------------------

V 2 ( )sinV 1 sin

-------------------------------- 0= =

F2 T 2,( )RT 1

V 1 sin------------------ V 1 sin

RT 2V 2 ( )sin--------------------------------+=

V 2 ( )sin

T 2ddG2

k

T 2( )k 1+ G2

k+ 0=

T 2( )k 1+ T 2

k 1+ T 2k

G2k

T 2ddG2

k 1

= =


19

T2 becomes a vector of the unknowns and T2,in the case of equations (A11) and (A12). Equa-tion (A14) is thus replaced by a system of twoalgebraic equations for the two unknowns.

Newtons method requires an initial esti-mate of the solution. Within the TPG code, theinitial approximation is given by the solution ofthe oblique shock wave relations for a caloricallyperfect gas at the local conditions (i.e., M1, and1). NACA Rep. 1135 summarizes these equa-tions, as do many compressible flow textbooks.

Newtons method applied to equations (A9)and (A11-12) does not converge for M1


20

Table 3: Sample TPG.out header information

| Thermally Perfect Gas Properties Code || TPG, Version 3.1.0 || || NASA Langley Research Center |

Table of Thermally Perfect Compressible Flow PropertiesGaseous Mixture: Air: 4-Species Mixture of N2, O2, Argon, and CO2Database file name: [ Default database of Air Mixture ]Species Names are:

N2 , O2 , Argon, CO2 ,Species Mass Fractions are:

0.75530 0.23140 0.01290 0.00040Species Mole Fractions are:

0.78092 0.20946 0.00935 0.00026Mixture Properties:

Molecular Weight = 28.9663Gas Constant = 2.87035E+02 J/(kg*K)

Polynomial Coefficients : cp/R = Sum{A(i)*T^(i2)}80.0 < T < 1000.0 degrees K

3.7319821D+00 5.4196701D01 3.4693586D+00 3.8353798D042.7653130D06 8.1886142D09 7.7368690D12 2.4348723D15

1000.0 < T < 6000.0 degrees K 2.3769851D+05 1.2440527D+03 5.1266690D+00 2.0400644D04 6.8321801D08 1.0553542D11 6.6450071D16 0.0000000D+00

Total Temperature = 400.000 K


21

Table 4: Sample thermochemical database file

# of species #of Temperature ranges : NASP TM 1107 + Mods. 2 2Name: Molecular Wt.N2 28.0160tmin tmax 20. 1000.Cp/R Coefficients: c1(2) > c1(5)1.33984200E+01 1.34280300E+00 3.45742000e+00 5.74727600E043.21711900E06 7.50775400E09 5.90150500E12 1.50979900E15

tmin tmax1000. 6000.Cp/R Coefficients: c2(2) > c2(5)

5.87702841E+05 2.23921563E+03 6.06686971E+00 6.13957913E041.49178026E07 1.92307130E11 1.06193594E15 0.00000000E+00

Name: Molecular Wt.O2 32.0000tmin tmax 30. 1000.Cp/R Coefficients: c1(2) > c1(5)

3.88517500E+01 2.70630800E+00 3.56119600E+00 3.32782400E041.18148000E06 1.10853500E08 1.49299400E11 5.99553800E15

tmin tmax1000. 6000.Cp/R Coefficients: c2(2) > c2(5)1.05642070E+06 2.41123849E+03 1.73474238E+00 1.31512292E032.29995151E07 2.13144378E11 7.87498771E16 0.00000000E+00


22

Table 5: Sample TPG tabular output for constant shock wave angle


23

Table 6: Sample TPG tabular outputs for constant flow deflection angle

(a) Weak shock wave solutions


24

Table 6: Sample TPG tabular outputs for constant flow deflection angle

(b) Strong shock wave solutions


25

Table 7: Sample TPG output for a sequence of shock waves


26

Figure 1. Geometric relationships of oblique shock waves.

Figure 2. Contours of upstream Mach number M1 for given shock wave angle and flow deflectionangle : thermally perfect air at an arbitrary Tt=3000 K.

V1, M1

w =

w 1 u1

w =

w 2u2

V2,

M2

Shock Wave

0 10 20 30 40 500

10

20

30

40

50

60

70

80

90

2.0

3.0

5.0

50.0

2.0 3.0 5.0

1.5

1.5 50.0

1.21.1

Flow deflection angle (deg)

Sho

ck w

ave

a

ngl

e

(d

eg)

Weak shocks

Strong shocksM1=

max


27

(a) Downstream Mach number M2 and flow deflection angle

(b) Density ratio 2/1 and temperature ratio T2/T1Figure 3. Thermally and calorically perfect calculations of oblique shock wave properties within the

calorically perfect temperature regime: air, Tt=400 K, =30.

1 2 3 4 5 6 7 8 9 101.5

2.0

2.5

3.0

3.5

4.0

0

5

10

15

20

25

Flow

defle

ction

angle

(deg)

Do

wn

stre

am

M

ach

n

um

ber

M

2

M2

Upstream Mach number M1

1 2 3 4 5 6 7 8 9 101.5

2.0

2.5

3.0

3.5

4.0

0

5

10

15

20

25

TPG CPG

1 2 3 4 5 6 7 8 9 101.5

2.0

2.5

3.0

3.5

4.0

0

5

10

15

20

25

Flow

defle

ction

angle

(deg)

Do

wn

stre

am

M

ach

n

um

ber

M

2

M2


1 2 3 4 5 6 7 8 9 101.5

2.0

2.5

3.0

3.5

4.0

0

5

10

15

20

25

TPG CPG

1 2 3 4 5 6 7 8 9 101

2

3

4

5

1

2

3

4

5

6


De

nsi

ty ra

tio 2

/

1T

em

pe

rature

ratio T

2 / T

1

2/1T2/T1

1 2 3 4 5 6 7 8 9 101

2

3

4

5

1

2

3

4

5

6

TPG CPG


28

(c) Total and static pressure ratios pt,2/pt,1 and p2/p1

Figure 3. Concluded.

(a) Downstream Mach number M2 and flow deflection angle

Figure 4. Thermally and calorically perfect calculations of oblique shock wave properties within thecalorically perfect temperature regime: air, Tt=400 K, =50.

1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0

5

10

15

20

25

30


Static

pre

ssure

ratio p

2 / p

1

p2/p1

Tota

l pre

ssu

re ra

tio p t

,2 / p

t,1

pt,2/pt,1

1 2 3 4 5 6 7 8 9 100.0

0.5

1.0

0

5

10

15

20

25

30

TPG CPG

1 2 3 4 5 6 7 8 9 101.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

0

5

10

15

20

25

30

35

40

Flow

defle

ction

angle

(deg)

Do

wn

stre

am

M

ach

n

um

ber

M

2

M2


1 2 3 4 5 6 7 8 9 10

1.5

2.0

0

5

10

15

20

25

30

35

40

TPG CPG


29

(b) Density ratio 2/1 and temperature ratio T2/T1

(c) Total and static pressure ratios pt,2/pt,1 and p2/p1


1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

1

2

3

4

5

6

7

8

9

10

11

12

13


De

nsi

ty ra

tio 2

/

1T

em

pe

rature

ratio T

2 / T

12/1T2/T1

1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

1

2

3

4

5

6

7

8

9

10

11

12

13

TPG CPG

1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0

10

20

30

40

50

60

70


Static

pre

ssure

ratio p

2 / p

1

p2/p1pt,2/pt,1

Tota

l pre

ssu

re ra

tio p t

,2 / p

t,1

1 2 3 4 5 6 7 8 9 100.0

0.5

1.0

0

10

20

30

40

50

60

70

TPG CPG


30

Figure 5. Effect of caloric imperfections on oblique shock wave properties within the caloricallyperfect temperature regime: air, Tt=400 K, =30.

Figure 6. Effect of caloric imperfections on oblique shock wave properties within the caloricallyperfect temperature regime: air, Tt=400 K, =50.

1 2 3 4 5 6 7 8 9 100.997

0.998

0.999

1.000

1.001

1.002

1.003

1.004

1.005

1.006

The

rma

lly Pe

rfect

Calo

rica

lly Pe

rfect

M2

pt,2 / pt,1T2 / T12 / 1p2 / p1


1 2 3 4 5 6 7 8 9 10

0.998

1.000

1.002

1.004

1.006

1.008

1.010

The

rma

lly Pe

rfect

Calo

rica

lly Pe

rfect

M2


pt,2 / pt,1T2 / T12 / 1p2 / p1


31

(a) Flow deflection angle

(b) Downstream Mach number M2

Figure 7. Comparison of thermally and calorically perfect calculations for oblique shock waves in airat Tt=1500 K for =30 and 50.

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

= 30

= 50

TPG equation


Flo

w de

flect

ion

a

ngl

e

(d

eg)

CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 101.0

1.5

2.0

2.5

3.0

3.5

4.0TPG equation

Do

wn

stre

am

M

ach

n

um

ber

M

2

= 30

= 50


CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 101.0

1.5

2.0

2.5

3.0

3.5

4.0


32

(c) Static pressure ratio p2/p1

(d) Static temperature ratio T2/T1

Figure 7. Continued.

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

TPG equation

= 30

= 50

Pre

ssu

re ra

tio p 2

/ p

1


CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

TPG equation

Tem

pera

ture

ra

tio T 2

/ T

1

= 30

= 50


CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12


33

(e) Density ratio 2/1

(f) Total pressure ratio pt,2/pt,1


1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

TPG equation

= 30


= 50

Tota

l pre

ssu

re ra

tio p t

,2 / p

t,1

CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

TPG equation

= 30

= 50

De

nsi

ty ra

tio 2

/

1


CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 10

2

4

6


34



Figure 8. Comparison of thermally and calorically perfect calculations for oblique shock waves in airat Tt=3000 K for =30 and 50.

1 2 3 4 5 6 7 8 9 101.0

1.5

2.0

2.5

3.0

3.5

4.0TPG equation

Do

wn

stre

am

M

ach

n

um

ber

M

2


= 50

= 30

CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 101.0

1.5

2.0

2.5

3.0

3.5

4.0

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

= 50

TPG equation


Flo

w de

flect

ion

a

ngl

e

(d

eg)

= 30

CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40


35




1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

TPG equation

Tem

pera

ture

ra

tio T 2

/ T

1

= 30

= 50


CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

TPG equation

= 30

= 50

Pre

ssu

re ra

tio p 2

/ p

1


CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70


36




1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

TPG equation

= 50

De

nsi

ty ra

tio 2

/

1


= 30

CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 10

2

4

6

1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

TPG equation

= 30

= 50


Tota

l pre

ssu

re ra

tio p t

,2 / p

t,1

CPG (=30)CPG (=50)

1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


37



Figure 9. Effect of caloric imperfections on oblique shock wave properties: comparisons between TPGand NACA Rep. 1135 for two total temperatures and two shock wave angles

1 2 3 4 5 6 7 8 9 101.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

TPG eqn.


Tt = 1500 K

th

erm

pe

rf

Tt = 3000 K

1 2 3 4 5 6 7 8 9 101.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

= 50

= 30

1 2 3 4 5 6 7 8 9 101.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

(Ma

ch2) th

erm

(Ma

ch2) p

erf

TPG eqn.


Tt = 1500 K

Tt = 3000 K

1 2 3 4 5 6 7 8 9 101.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

= 30

= 50


38




1 2 3 4 5 6 7 8 9 100.98

0.99

1.00

1.01

1.02

1.03

TPG eqn.

(p2

/ p1) th

erm

(p2

/ p1) p

erf


Tt = 3000 K

Tt = 1500 K

1 2 3 4 5 6 7 8 9 100.98

0.99

1.00

1.01

1.02

1.03

= 30

= 50

1 2 3 4 5 6 7 8 9 100.88

0.90

0.92

0.94

0.96

0.98

1.00

TPG eqn.

(T2

/ T1) th

erm

(T2

/ T1) p

erf


Tt = 1500 K

Tt = 3000 K

1 2 3 4 5 6 7 8 9 100.88

0.90

0.92

0.94

0.96

0.98

1.00

= 30

= 50


39




1 2 3 4 5 6 7 8 9 101.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14 TPG eqn.

( 2 /

1) the

rm

( 2 /

1) pe

rf


Tt = 1500 K

Tt = 3000 K

1 2 3 4 5 6 7 8 9 101.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

= 30

= 50

1 2 3 4 5 6 7 8 9 10

0.80

0.85

0.90

0.95

1.00

TPG eqn.


Tt = 1500 K

Tt = 3000 K(pt,2

/ p

t,1) the

rm

(pt,2

/ p

t,1) pe

rf

1 2 3 4 5 6 7 8 9 10

0.80

0.85

0.90

0.95

1.00 = 30

= 50


40

Figure 10. GASP 2-D Euler solution pressure ratio contours and TPG oblique shock wave forthermally perfect air: M1=4.8, T1=1670 K over a 10 wedge.

(a) Mach number M

Figure 11. GASP 2-D Euler and TPG oblique shock wave calculations for thermally perfect air:M1=4.8, T1=1670 K over a 10 wedge: at z=2.6 units above the wedge leading edge.

0.0 2.5 5.0 7.5 10.00

1

2

3

4

5

Late

ral d

ista

nce

fro

m W

edg

e Le

adi

ng

Edge

TPG shock wave

p / p1

z=2.6

Wedge angle, = 10

Longitudinal distance from Wedge Leading Edge

2.72.62.52.42.32.22.121.91.81.71.61.51.41.31.21.1

7.0 7.1 7.2 7.3 7.4 7.52.4

2.5

2.6

2.7

6 7 8 9 104.0

4.2

4.4

4.6

4.8

Ma

ch N

um

ber


GASP 2-D Euler at z=2.6TPG Solution

6 7 8 9 104.0

4.2

4.4

4.6

4.8


41

(b) Static pressure ratio p/p1

(c) Static temperature ratio T/T1


6 7 8 9 10

1.0

1.5

2.0

2.5Pr

ess

ure

ra

tio p

/ p1



6 7 8 9 10

1.0

1.5

2.0

2.5

6 7 8 9 100.90

1.00

1.10

1.20

1.30

1.40

Tem

pera

ture

ra

tio T

/ T1

GASP 2-D Euler at z=2.6


TPG Solution

6 7 8 9 100.90

1.00

1.10

1.20

1.30

1.40


42

(d) Density ratio 2/1Figure 11. Concluded.

Figure 12. GASP 2-D Euler solution pressure ratio contours and TPG oblique shock wave for 20%Steam/80% CO2, by mass: M1=3.5, T1=1200 K over a 25 wedge.

6 7 8 9 10

1.0

1.2

1.4

1.6

1.8

2.0

2.2

De

nsi

ty ra

tio

/ 1



6 7 8 9 10

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 2.5 5.0 7.5 10.00

1

2

3

4

5

6

7

8

z=5.0

p / p1

Late

ral d

ista

nce

fro

m W

edg

e Le

adi

ng

Edge

TPG shock wave


Wedge angle, = 25

6.756.255.755.254.754.253.753.252.752.251.751.25

6.2 6.3 6.4 6.5 6.6 6.74.7

4.8

4.9

5.0


43

(a) Mach number M

(b) Static pressure ratio p/p1

Figure 13. GASP 2-D Euler and TPG oblique shock wave calculations for 20% Steam/80% CO2, bymass: M1=3.5, T1=1200 K over a 25 wedge: at z=5.0 units above the wedge leading edge.

5 6 7 8 92.0

2.5

3.0

3.5

4.0

Ma

ch n

um

ber



5 6 7 8 92.0

2.5

3.0

3.5

4.0

5 6 7 8 9

1

2

3

4

5

Pre

ssu

re ra

tio p

/ p1



5 6 7 8 9

1

2

3

4

5


44

(c) Static temperature ratio T/T1

(d) Density ratio /1Figure 13. Concluded.

5 6 7 8 90.9

1.0

1.1

1.2

1.3

1.4

1.5

Tem

pera

ture

ra

tio T

/ T1



5 6 7 8 90.9

1.0

1.1

1.2

1.3

1.4

1.5

5 6 7 8 9

1.0

1.5

2.0

2.5

3.0

3.5



De

nsi

ty ra

tio

/ 1

5 6 7 8 9

1.0

1.5

2.0

2.5

3.0

3.5

NASA-96-cr4749

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Transcript of NASA-96-cr4749