User Guide For Compressible Flow Toolbox
NASA/TM—2006-214086User Guide for Compressible Flow ToolboxVersion 2.1 for Use With MATLAB Version 7Kevin J. MelcherGlenn Research Center, Cleveland, OhioJanuary 2006
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NASA/TM—2006-214086User Guide for Compressible Flow ToolboxVersion 2.1 for Use With MATLAB Version 7Kevin J. MelcherGlenn Research Center, Cleveland, OhioNational Aeronautics andSpace AdministrationGlenn Research CenterJanuary 2006
AcknowledgmentsThe author acknowledges the significant contribution of Jonathan DeCastro, QSS Group, Inc.Mr. DeCastro conducted comprehensive testing of the algorithms comprising the Compressible Flow Toolbox,and completed the tedious task of reviewing this document in detail prior to publication.Trade names or manufacturers’ names are used in this report foridentification only. This usage does not constitute an officialendorsement, either expressed or implied, by the NationalAeronautics and Space Administration.Available fromNASA Center for Aerospace Information7121 Standard DriveHanover, MD 21076National Technical Information Service5285 Port Royal RoadSpringfield, VA 22100Available electronically at http://gltrs.grc.nasa.gov
ContentsAbstract . 1Introduction. 1Nomenclature . 3Quick Reference Tables . 5Function Reference Guide . 7ames . 7ameserr . 11amesplt . 17deltamax . 19deltason . 23fanno . 27fannoerr . 31fannoplt . 35fannotbl . 37isentbl. 39nshktbl. 41oblqshck . 43oblqw12. 45oblqw21. 49rayleigh . 53raylerr. 57raylplt . 61rayltbl . 635. References. 651.2.3.4.NASA/TM—2006-214086iii
User Guide for Compressible Flow ToolboxVersion 2.1 for Use With MATLAB Version 7Kevin J. MelcherNational Aeronautics and Space AdministrationGlenn Research CenterCleveland, Ohio 44135AbstractThis report provides a user guide for the Compressible Flow Toolbox, a collection of algorithmsthat solve almost 300 linear and nonlinear classical compressible flow relations. The algorithms,implemented in the popular MATLAB programming language, are useful for analysis of onedimensional steady flow with constant entropy, friction, heat transfer, or shock discontinuities.The solutions do not include any gas dissociative effects. The toolbox also contains functions forcomparing and validating the equation-solving algorithms against solutions previously publishedin the open literature. The classical equations solved by the Compressible Flow Toolbox are:isentropic-flow equations, Fanno flow equations (pertaining to flow of an ideal gas in a pipe withfriction), Rayleigh flow equations (pertaining to frictionless flow of an ideal gas, with heattransfer, in a pipe of constant cross section.), normal-shock equations, oblique-shock equations,and Prandtl-Meyer expansion equations. At the time this report was published, the CompressibleFlow Toolbox was available without cost from the NASA Software Repository.1.IntroductionDescriptionThis paper provides a User Guide for the Compressible Flow Toolbox, a collection of algorithmsthat solve almost 300 linear and nonlinear classical compressible flow relations. The algorithms,implemented in the popular MATLAB programming language, are useful for analysis of onedimensional steady flow with constant entropy, friction, heat transfer, or shock discontinuities.The solutions do not include any gas dissociative effects. The toolbox also contains functions forcomparing and validating the equation-solving algorithms against solutions previously publishedin the open literature. The classical equations solved by the Compressible Flow Toolbox are: The isentropic-flow equations, The Fanno flow equations (pertaining to flow of an ideal gas in a pipe with friction), The Rayleigh flow equations (pertaining to frictionless flow of an ideal gas, with heattransfer, in a pipe of constant cross section.) The normal-shock equations, The oblique-shock equations, and The Prandtl-Meyer expansion equations.The user should note that the scope of this guide is limited to documenting the individualfunctions and providing instruction in using them to solve simple compressible flow examples.Functions in the toolbox can be used together to solve more complex compressible flowproblems—that is why they were created. However, instructing the user in the broader context ofcompressible flow is not the intended purpose of this guide.NASA/TM—2006-2140861
BackgroundAlgorithms included in the Compressible Flow Toolbox were originally developed to supportcontrols and dynamics research under the NASA’s High Speed Research Program. They wereinspired by NACA Report 1135 “Equations Tables, and Charts for Compressible Flow” (ref. 1)which the author studied extensively as part of that research. Early implementations were firstpublished as part of the author’s Masters Thesis in 1996. They were subsequently made publiclyavailable via a MATLAB third party software web site hosted by the Mathworks, Inc. Afterseveral years, the toolbox was removed from the web site for a variety of reasons, including theneed to upgrade the algorithms for compatibility with newer versions of MATLAB . Finally, toappease a number of recent requests for the software, the toolbox has been updated, expanded,and made available to the general public via the NASA Software Repository.All of the numerical and graphical results shown in this report were generated using functionsincluded in the Compressible Flow Toolbox version 2.1and running MATLAB version 7.04 onan MS Windows XP, 2.2 GHz Intel Pentium 4 processor-based personal computer. Results mayvary slightly based on the precision of the floating point processor used to perform thecalculations.OrganizationThis User’s Guide is organized in five sections. Introduction, Nomenclature, Quick ReferenceGuide, Function Reference Guide, and References. Section 1. Introduction provides a generaldescription of the User Guide along with historical information on the origin of the toolbox andavailability of the software. Section 2. Nomenclature describes the symbols and specialformatting conventions used throughout the text. Section 3. Quick Reference Guide provides acomprehensive list of the functions contained in the toolbox and provides a brief description ofeach function listed. Section 4. Function Reference Guide provides a detailed description ofeach function in the toolbox including its purpose, syntax, a discussion of how the algorithmworks, and examples demonstrating its use. Finally, Section 5. References contains a list ofreferences used in developing and documenting the toolbox.AvailabilityAt the time this report was published, the Compressible Flow Toolbox was available to thegeneral public without cost through the NASA Software e/NASA/TM—2006-2140862
2. NomenclatureFormats and ConvensionsMonospaceMATLAB commands, functions names and screen output aredisplayed in this font. For example: rayleigh.ItalicsBook titles and names of book sections, mathematical symbolsand notation, and the introduction of new terms. For example:Introduction.Bold Initial CapsKey names, menu names, and items that are selected frommenus. For example: the File menu.SymbolsThis document uses the following symbols and notations:Roman SymbolsADHIMPPtTTtVfqGreek -214086DescriptionCross-sectional area of stream tube or channelHydraulic diameter of the flow cross-sectional areaImpulse functionMach number, V/aStatic PressureTotal PressureStatic TemperatureTotal TemperatureVelocityAverage friction factorDynamic pressure, ρV2/2DescriptionM 2 1Ratio of specific heats of the working fluid (default 1.4)Turning angle (degrees)Oblique shock angle (degrees)Mach Angle (degrees)Prandtl-Meyer angle (degrees)Static mass densityMass densityDescriptionCritical flow condition (i.e., conditions where the local fluidvelocity is equal to the local speed of sound)Upstream flow propertyDownstream flow property3
3. Quick Reference TablesPROPERTIES OF ISENTROPIC FLOW, PRANDTL-MEYER FLOW,AND NORMAL SHOCKSamesamespltameserrisentblnshktblSolves the equations for isentropic flow, Prandtl-Meyer flow, andnormal shocks to obtain flow properties.Plots the properties for isentropic flow, Prandtl-Meyer flow, and thenormal shocks as a function of Mach number.Consistency check for function ames. Computes and plots, as a functionof Mach number, errors in ames calculations.Generates text file containing a table of the isentropic flow properties.Generates text file containing a table of Prandtl-Meyer flow and normalshock properties.PROPERTIES OF OBLIQUE SHOCKSoblqshckoblqw12oblqw21deltasondeltamaxSolves the oblique shock equations for both weak and strong shockangles.Solves the oblique shock equations to obtain downstream flowproperties as a function of upstream flow properties.Solves the oblique shock equations to obtain upstream flow propertiesas a function of downstream flow properties.Computes the theoretical deflection angle that reduces supersonic flowto sonic conditions.Computes the theoretical maximum angle through which supersonicflow may be deflected or turned without separation.PROPERTIES OF FANNO-LINE 4086Solves the Fanno line equations to obtain properties of flow withfriction.Plots the Fanno line flow properties as a function of Mach number.Consistency check for function fanno. Computes and plots, as afunction of Mach number, errors in fanno calculations.Generates text file containing a table of the Fanno-line flow properties.5
PROPERTIES OF RAYLEIGH-LINE 4086Solves the Rayleigh-line equations to obtain properties of flow heatingor cooling.Plots the Rayleigh-line flow properties as a function of Mach number.Consistency check for function rayleigh. Computes and plots, as afunction of Mach number, errors in rayleigh calculations.Generates text file containing a table of the Rayleigh-line flowproperties.6
4. Function Reference GuideamesPurposeSolve the equations for isentropic flow, both subsonic and supersonic, Prandtl-Meyerexpansion, and normal shocks.SynopsisamesProperties ames(VarIn,ValuesIn,VarsOut)Properties bls] ames(VarIn,ValuesIn,VarsOut,Gamma)Descriptionames by itself calls amesplt which plots normalized versions of the isentropic flow,Prandtl-Meyer, and normal shock functions versus Mach number.Properties ames(VarIn,ValuesIn,VarsOut), given a number designating one of theflow properties listed in Table 4.1 and a value or vector of values for that flow property,ames computes corresponding values for isentropic flow, Prandtl-Meyer flow, andnormal shock functions. VarIn is a scalar that specifies the property used as the input(independent variable). ValuesIn may be a scalar or vector and contains values of theindependent variable for which the other properties will be computed. VarsOut contains alist of Indices corresponding to the flow properties listed in Table 4.1. Indices specifiedin VarsOut may be in any order and may be repeated as desired by the user. Results arereturned in the Properties matrix. Columns in this matrix correspond to indicesspecified in VarsOut. Rows of the Properties matrix contain results corresponding tothe elements of ValuesIn.Note that, when properties 5, 6, or 7 are used as the independent variable, the solution isdouble-valued. The double-valued solution is provided by making Properties a cellarray. Properties{1} contains values of the solution associated with the smaller Machnumber, while Properties{2} contains the solution associated with the larger Machnumber.Properties ames(VarIn,ValuesIn,VarsOut,Gamma) provides a mechanism forspecifying values for the ratio of specific heats of the working fluid via Gamma. Gamma isoptional. If unspecified, a value of 1.4, the value of the ratio of specific heats of air atstandard temperature and pressure, is used. If specified, Gamma may be defined as either ascalar or a vector. If it is a vector, it must have the same length as ValuesIn.[Properties,PltLbls] ames(VarIn,ValuesIn,VarsOut,Gamma), in addition toreturning the properties of the fluid at user specified conditions, also returns a cell array,PltLbls, containing text strings that may be used when plotting the results.NASA/TM—2006-2140867
Table 4.1—Description of Flow Properties Computed by Function amesREF.INDEXPROPERTYREF. 1DESCRIPTIONISENTROPIC FLOW PROPERTIES (VALID FOR ALL M):1.2.M or M1P PtEq. 44Mach numberRatio of static to total pressure3.ρ / ρtT TtEq. 45Eq. 43Ratio of static to total densityRatio of static to total temperatureβpg. 16.7.q PtEq. 48A A*Eq. 80Ratio of dynamic to total pressureRatio of flow area to critical flow area8.*Eq. 50Ratio of flow velocity to critical flow velocity4.5.V VM21PRANDTL-MEYER FLOW (VALID FOR M 1):9.10.νμEq. 171pg. 1Prandtl-Meyer angle (degrees)Mach Angle (degrees), sin 1 (1 / M )NORMAL SHOCK PROPERTIES (VALID FOR M 1):11.12.13.14.15.16.M2P2 P1ρ 2 / ρ1T2 T1Pt , 2 Pt ,1P1 Pt , 2Eq. 96Eq. 93Eq. 94Eq. 95Eq. 99Eq. 100Mach number downstream of a normal shockStatic pressure ratio across a normal shockStatic density ratio across a normal shockStatic temperature ratio across a normal shockTotal pressure ratio across a normal shockRatio of static pressure upstream of a normal shockto total pressure downstream of the same shockAlgorithmames determines the desired flow properties by first obtaining a Mach number solutionfor each value, ValuesIn, of the user specified flow property, VarIn. These Machnumbers are then used to compute the other properties, VarsOut, specified by the user.Most of the flow equations may be manipulated analytically to obtain Mach number as afunction of the other properties. However, some nonlinear relationships exist which haveno simple analytical solution. In these cases, MATLAB’s fminbnd function is useddetermine an approximate solution for Mach number from the nonlinear equations. Thesearch is arbitrarily constrained to Mach numbers less than 100. Solutions associated withMach numbers larger than 100 are returned as NaN (i.e., not a number).See Alsoameserr, amesplt, isentbl, and nshcktblExample 4.1:Determine the properties of air at Mach 2.NASA/TM—2006-2140868
ames(1,2,1:16)ans Columns 1 through 52.00000.12780.2300Columns 6 through 100.35791.68751.6330Columns 11 through 150.57744.50002.6667Column ple 4.2:Given a normal shock with downstream Mach number of 0.85, determine the Machnumber upstream of the shock. ames(11,0.85,1)ans 1.1876Example 4.3:Determine the properties of air when A A* 3.007. properties ames(7,3.007,1:16)properties [1x16 double][1x16 double] properties{1}ans Columns 1 through 50.19700.97330.9809Columns 6 through 100.02643.00700.2149Columns 11 through 15NaNNaNNaNColumn 16NaN properties{2}ans Columns 1 through 62.63990.04710.1128Columns 6 through 100.22993.00701.8691Columns 11 through 150.50057.96383.4935Column NNaN0.41772.443242.304922.25972.27960.4453
Example 4.4:Plot the Mach number downstream of a normal shock as a function of the Mach numberupstream of the shock.M1 1:0.1:10;[M2,Lbls] ames(1,M1,11);plot(M1,M2);xlabel('M 1'); gure 4.1.—Result of using function ames to computeMach number variations across a normal shock.NASA/TM—2006-2140861010
ameserrPurposeShow the computational errors that result when using function ames to solve theequations for isentropic flow, Prandtl-Meyer expansion, and normal shocks.Synopsisameserr[error,M1] ameserrDescriptionameserr computes the error between Mach numbers used as inputs to function ames andMach numbers calculated from the output of function ames. The results are plotted asabsolute and percent errors versus Mach number for each of the flow functions shown inTable 4.1.[error,M1] ameserr returns the computed error in error. If specified, M1 contains theinitial vector of Mach numbers.Algorithmameserr first generates a logarithmically spaced vector of 250 Mach numbers from 0.01to 10. This vector also includes critical Mach number values where numerical stability isimportant, such as saddle points. ameserr then uses function ames to calculate each ofthe isentropic flow properties and the normal shock properties corresponding to thoseMach numbers. The functions of Mach number, obtained from ames, are then used asinput to the ames function in order to obtain a Mach number which corresponds to thefunction value. Theoretically, the initial and computed Mach numbers should be thesame. In general, they are not due to round off, truncation, convergence, and/oroptimization errors. The difference in the two Mach numbers is returned as the error inthe calculations.See Alsoames, amesplt, isentbl, and nshcktblExample 4.5:Compute and plot the errors the errors that result from running ameserr. Plots are shownin Figure 4.2(a to g). ameserrNASA/TM—2006-21408611
Test Consistency of AMES.MError (M1)10 1%Error (M1)10 1 14Error (P/Pt)2x 100 2 4%Error (P/Pt) 102x 100 2 4 210 110010010101Mach No.(a)Test Consistency of AMES.M 14Error (ρ/ρt)2x 100 2 4%Error (ρ/ρt) 102x 100 2 4 14Error (T/Tt)2x 100 2 4%Error (T/Tt) 102x 100 2 4 210 11010Mach No.(b)Figure 4.2.—Output of function ameserr as computed onan Intel Pentium4 processor-based computer runningMATLAB 7.NASA/TM—2006-214086121
Test Consistency of AMES.M 14Error (β)x 100.50 0.5 11%Error (β)5x 100 5 8Error (q/Pt)5x 100 5%Error (q/Pt) 52x 100 2 210 110010010101Mach No.(c)Test Consistency of AMES.M 7Error (A/A*)1x 100 1%Error (A/A*) 52x 100 2 4 14Error (V/a*)2x 100 2%Error (V/a*) 135x 100 5 210 110101Mach No.(d)Figure 4.2.—Output of function ameserr as computed on anIntel Pentium4 processor-based computer runningMATLAB 7 (continued).NASA/TM—2006-21408613
8Error (ν)5x 10Test Consistency of AMES.M0 5 10 6%Error (ν)4x 1020 2 15Error (μ)2x 100 2 4 14%Error (μ)5x 100 5010110Mach No.(e) 14Error (M2)2x 10Test Consistency of AMES.M0 2 4 13%Error (M2)5x 100 5Error (P2/P1) 151.5x 1010.50%Error (P2/P1) 142x 1010010110Mach No.(f)Figure 4.2.—Output of function ameserr as computed onan Intel Pentium4 processor-based computer runningMATLAB 7 (continued).NASA/TM—2006-21408614
14Error (ρ2/ρ1)2x 10Test Consistency of AMES.M0 2%Error (ρ2/ρ1) 132x 100 2 7Error (T2/T1)1x 100 1%Error (T2/T1) 64x 1020 2010110Mach No.(g)Error (Pt,2/Pt,1) 64x 10Test Consistency of AMES.M20 2%Error (Pt,2/Pt,1) 44x 1020 2Error (P1/Pt,2) 71x 100 1%Error (P1/Pt,2) 64x 1020 2010110Mach No.(h)Figure 4.2.—Output of function ameserr as computed on anIntel Pentium4 processor-based computer runningMATLAB 7 (continued).NASA/TM—2006-21408615
amespltPurposePlots normalized properties for isentropic flow, Prandtl-Meyer expansion, and normalshocks as a function of Mach scriptionamesplt uses function ames to compute and plot the isentropic and normal shock flowproperties at 250 points between Mach 0.01 and Mach 10 when the ratio of specific heatsof the fluid is 1.4.amesplt(MNmin,MNmax) plots results for a range of user specified Mach numbers where:MNmin is the minimum Mach number; and MNmax is the maximum Mach number.amesplt(MNmin,MNmax,Npts) in addition to allowing the user to specify the range ofMach numbers used, this form allows the user to specify the number of data points, Npts,used to plot each curve.amesplt(MNmin,MNmax,Npts,Gamma) in addition to allowing the user to specify Machnumber. and number of points per curve, this form also allows the user to specify a scalarvalue for the ratio of specific heats, Gamma, of the fluid.Algorithmamesplt first generates a logarithmically spaced vector of 250 Mach numbers from 0.01to 10. This vector also includes critical Mach number values where numerical stability isimportant, such as solution saddle points. amesplt then uses this vector as inputs tofunction ames which is used to calculate each of the isentropic flow properties and thenormal shock properties corresponding to those Mach numbers. The resulting values arenormalized and plotted versus Mach number to provide the user a graphicalunderstanding of the relationship between flow properties and Mach number.See Alsoames, amesplt, isentbl, and nshcktblExample 4.6:Plot normalized isentropic flow and normal shock properties as a function of Machnumber. The resulting plots are shown in Figure 4.3 (a and b). amespltNASA/TM—2006-21408617
AMES.M: Isentropic Functions, Table Columns 2 810.9P/Ptρ/ρNormalized 0.10012345Mach No.678910(a)AMES.M: Normal Shock Functions, Table Columns 9 161νμM20.90.8P /P21ρ /ρ2Normalized Parameters0.71T /T21P /Pt,20.6t,1P1/Pt,20.50.40.30.20.10123456789Mach No.(b)Figure 4.3.—Normalized isentropic and normal shockfunctions as generated by function amesplt.NASA/TM—2006-2140861810
deltamaxPurposeFor steady state supersonic flow with compressive turning, deltamax computes themaximum flow deflection angle (δ) that can occur without producing separation of theflow from the turning surface. Also, optionally calculates the angle of the oblique shock(θ) that results from turning the flow. Both angles have units of degrees. See Figure 4.4for a graphical representation of the flow situation.SynopsisdeltamaxDelta deltamax(M1)[Delta,Theta] deltamax(M1,Gamma)Descriptiondeltamax by itself, computes and plots the maximum flow deflection and resultingoblique shock angle for a range of Mach numbers from 1.0 to 15.Delta deltamax(M1) computes and returns the maximum flow deflection angle, delta,in degrees for user specified Mach numbers, M1. M1 may be a scalar, vector, or matrix.[Delta,Theta] deltamax(M1,Gamma) uses optional input Gamma, the ratio of specificheats for the working fluid, to calculate the turning angle Delta and additionally theangle, Theta, of the oblique shock that results from turning the flow. Gamma has adefault value of 1.4 and must be a scalar or have dimensions equivalent to M1. Thedimensions of Delta and Theta, and the values therein, correspond to the dimensions ofM1.AlgorithmIf no input parameters are specified by the user, deltamax first generates a vector ofupstream Mach numbers. The function then uses the Mach number(s) to calculate themaximum angle, θmax, of an oblique shock that can occur without separation. The shockangle is then used with the Mach number(s) to calculate the associated flow deflectionangle, δmax.Shock waveRegion 2M2 M 1Region 1?θM1 1dδStreamlineFigure 4.4.—Oblique shock diagram.NASA/TM—2006-21408619
The equation used to calculate θmax is:1γM 12θ max sin 1 (γ 1)M 12 4 (γ 1) 1 (γ 1) M 12 (γ 1) M 14 1 216 (4.1)The equation used to calculate δmax is:δ max tan 1(M 12 sin 2 θmax 1)cot θ max12(4.2)(γ 1)M 12 M 12 sin 2 θ max 1Similar equations may be found in reference 1, pp. 9 and 12; (ref. 2), p. 586; and (ref. 4),pp. 315 to 316.See Alsodeltason, oblqshck, oblqw12, and oblqw21Example 4.7:Calculate and plot the maximum compressive turning angle and oblique shock angle forairflow over a range of Mach numbers from 1 to 15. deltamaxMaximum Deflection Angle vs. Mach NumberDeflection Angle (deg)50Asymptote @ 45.5842 Degrees4030201000510151015M190Shock Angle (deg)85807570656005M1Figure 4.5.—Results of function deltamaxshowing maximum turning angle and the angleof the resulting oblique shock as a function ofupstream Mach number.NASA/TM—2006-21408620
Example 4.8:Calculate the maximum compressive turning angle and oblique shock angle for steamflowing at Mach numbers from 1.5 to 3.0. The ratio of specific heats for steam is 1.327 atstandard temperature. [Delta,Theta] deltamax(1.5:0.1:3.0,1.327)Delta Columns 1 through 512.672615.359817.866020.178022.2960Columns 6 through 1024.228225.986927.586229.040430.3634Columns 11 through 1531.568432.667333.671234.589635.4314Column 1636.2042Theta Columns 1 through 566.782066.077465.6264Columns 6 through 1065.150965.158765.2116Columns 11 through 1565.520665.648365.7803Column 95965.401365.913766.0466
3. Quick Reference Tables PROPERTIES OF ISENTROPIC FLOW, PRANDTL-MEYER FLOW, AND NORMAL SHOCKS ames Solves the equations for isentropic flow, Prandtl-Meyer flow, and normal shocks to obtain flow properties. amesplt Plots the properties for isentropic flow, Prandtl-Meyer flow, and the normal shocks as a function of Mach number.
1. Introduction to Compressible Flow. Thermodynamics review. Review of governing equations. 2. Quasi-one-dimensional Compressible Flow. Steady one-dimensional flow of compressible fluids. Isentropic flow. Normal shocks. Nozzles, diffusers and supersonic wind tunnels. 3. Two-dimensional Compressible Flow. Oblique shocks. Shock reflections .
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An isentropic process is the one for which the entropy is constant and the process is reversible and adiabatic. The isentropic relation is given by the following relation; (1) 22 2 11 1. p T pT. γ γγ. ρ ρ (4.1.7) Important Properties of Compressible Flows . The simple definition of compressible flow is the variable density flows .
Solution We are to define incompressible and compressible flow, and discuss fluid compressibility. Analysis A fluid flow during which the density of the fluid remains nearly constant is called incompressible flow. A flow in which density varies significantly is called compressible flow. A fluid whose density is practically independent
