Joseph MajdalaniH. H. Arnold Chair of Excellence in Advanced

PropulsionProfessor

e-mail: maji@utsi.edu

Brian A. Maicke

Mechanical, Aerospace and BiomedicalEngineering Department,

University of Tennessee Space Institute,Tullahoma, TN 37388

Explicit Inversion of Stodola’sArea-Mach Number EquationStodola’s area-Mach number relation is one of the most widely used expressions incompressible flow analysis. From academe to aeropropulsion, it has found utility in thedesign and performance characterization of numerous propulsion systems; these includerockets, gas turbines, microcombustors, and microthrusters. In this study, we derive aclosed-form approximation for the inverted and more commonly used solution relatingperformance directly to the nozzle area ratio. The inverted expression provides a com-putationally efficient alternative to solutions based on traditional lookup tables or rootfinding. Here, both subsonic and supersonic Mach numbers are obtained explicitly as afunction of the area ratio and the ratio of specific heats. The corresponding recursiveformulations enable us to specify the desired solution to any level of precision. In closing,a dual verification is achieved using a computational fluid dynamics simulation of atypical nozzle and through Bosley’s formal approach. The latter is intended to confirm thetruncation error entailed in our approximations. In this process, both asymptotic andnumerical solutions are compared for the Mach number and temperature distributionsthroughout the nozzle. �DOI: 10.1115/1.4002596�

Keywords: asymptotic analysis, nozzle theory, compressible motion, analyticalapproximation

1 IntroductionThe design of a supersonic nozzle may be viewed as an opti-

mization study in which the exit Mach number plays the role of atarget variable defined by mission requirements. In this vein, awide variability exists as virtually any smooth curve can be usedin the subsonic region so long as the concavity of the contractingsection does not prompt flow separation. In the throat section,curvature is minimized to promote the establishment of a uniformchoking plane. Downstream of the throat, the flow is acceleratedto supersonic levels across an expanding section. In this region,proper contouring becomes critically important as discontinuitiesin the Prandtl–Meyer angle can cause undesirable shocks and ex-pansions. In the vicinity of the exit plane, the curvature is re-straightened to reduce flow divergence losses. While traditionaltechniques such as the method of characteristics can be employedto arrive at an optimal nozzle shape, computational fluid dynamics�CFD� can also be used as a modern alternative.

In addition to the geometric optimization challenges, thermalprotection issues arise in practical nozzle manufacture. These areoften associated with the temperature distribution and heat fluxesresulting at the walls. On the one hand, enhancing performanceand payload capability entails reducing weight and tapering edges.On the other hand, thermal protection against exhaust gases re-quires strong materials and optimally designed cross sections. Togradually straighten the flow out of the nozzle, a relatively long,slowly sloping skirt is required to the extent that choosing a suit-able length becomes an optimization process in its own right,specifically, one that gauges weight constraints, geometry, and di-vergence losses. It can thus be seen that the initial design phase ofa nozzle relies on an iterative process in which a multitude ofcalculations are required and where isentropic tools can be readilyemployed due to their ease and relative accuracy.

During this iterative process, several isentropic flow approxi-mations in converging-diverging nozzles may be used, connectingthe Mach number to the nozzle area ratio and gas properties. Sinceits inception in 1903 by Stodola �1�, this key expression has led to

a substantial leap in our understanding of supersonic nozzle be-havior �2,3�; today, it can be found in most fundamental textbookson the subject �3–7�.

Stodola’s relation has been verified both experimentally �1� andmore recently via CFD �8,9�. It is used in compressible flow simu-lations of the internal flowfield in solid rocket motors by Cheng etal. �10�, Jackson et al. �11�, and Stewart et al. �12�. It appears inone segment of ROCFLU, a compressible Navier–Stokes solver in-tended for simulating rocket internal ballistics �11–14�. It is alsoused in characterizing turbomachinery �15–17�, scarfed andcontoured-plug nozzles �18,19�, pulse detonation engines �20,21�,and magnetohydrodynamic systems �22�. More recently, it hasbeen employed in applications of constructal theory by Bejan �23�and in modeling microthrusters and microcombustors by Leach�24� and Tosin et al. �25�. Its popularity as a simple design toollies in its ability to predict the area ratio needed to produce adesired exit flow Mach number. This feature is, however, not ana-lytically invertible due to the transcendental nature of the Machnumber dependence. At present, one must resort to tabulation orroot finding in the process of estimating the expected exit Machnumber for fixed area ratio and specific gas properties. In thisstudy, we overcome this difficulty through the use of asymptotics.

The inversion of a Mach number relation via asymptotics is nota novel concept. In similar context, the Prandtl–Meyer functionhas been treated by a number of researchers such as Probstein �26�who introduced an analytical inversion of the Prandtl–Meyerfunction for particular values of the specific heat ratio. Day �27� isalso known for developing a hybrid inversion using a mix ofanalytical and computational tools. While these methods varyfrom those employed here, they lend support to the usefulness ofanalytical approximations for problems that arise in similar physi-cal settings. The present relation will be derived with a sufficientdegree of accuracy to serve as a direct and practical design alter-native. Not only will this solution increase our repertoire ofknown approximations for compressible flow models but it willalso provide a simple alternative to tabulation and root finding.

2 FormulationFor isentropic flow through a converging-diverging nozzle with

throat area At, a transcendental equation relates the area ratio, �

Contributed by the Heat Transfer Division of ASME for publication in the JOUR-

NAL OF HEAT TRANSFER. Manuscript received May 26, 2009; final manuscript receivedSeptember 11, 2010; published online March 30, 2011. Assoc. Editor: Wei Tong.

Journal of Heat Transfer JULY 2011, Vol. 133 / 071702-1Copyright © 2011 by ASME

=At /A, and the Mach number M at any cross section of surfacearea A. We are particularly interested in the so-called nozzle ex-pansion ratio for which A corresponds to the maximum area Ae inthe nozzle exit plane. From an asymptotic perspective, this condi-tion results in the smallest possible �=�e. In order to calculate theexit Mach number for a given nozzle design ratio, one can solvefor M using a numerical root finding technique appropriate for

��1 + 12 �� − 1�M2���+1�/�2��−1�� − M�1 + 1

2 �� − 1����+1�/�2��−1�� = 0

�1�

In general, � varies between 1.1 and 1.67. In rocket motors, �varies between 1.1 and 1.4 with an average value of 1.25. Forexample, the reusable solid rocket motor �RSRM� has an averagemolecular weight Mw=28.46 kg /kmol, constant specific heatsCp=1966.54 J /kg K and Cv=1674.4 J /kg K, a specific gas con-stant R=292.14 J /kg K, a mean chamber pressure p0=6.28 MPa, a ratio of specific heats �=1.17, and an exit arearatio �e=0.13. The exit Mach number is typically larger than 2and can, of course, be calculated from Eq. �1�. According to Sut-ton �7�, the exit value can range from as low as 2.5 to as high as10, with most being in a 4–6 range. What is important is that inmany applications, the exit area ratio �e varies between 1/3 and1/25 �7�. For high altitude applications �100 km or higher�, �eranges between 1/400 and 1/40; here, the exit areas are increasedto accommodate expansions to lower external pressures. To beconservative, one may then assume

0.002 � �e � 0.4 �2�

The natural emergence of a small quantity � is the motivationfor using perturbation methods. These permit the construction ofasymptotic solutions for the area-Mach number M=M�� ,�� thatencapsulate both roots of Eq. �1�.

Once the Mach number is determined, it can be used to predictthe spatial evolution of the temperature, pressure, and densitythroughout the nozzle. Because Stodola’s equation remains an ide-alized case, it may be used in conjunction with isentropic relationssuch as

T = TC/�1 + 12 �� − 1�M2�, p = pC/�1 + 1

2 �� − 1�M2��/��−1�

� = �C/�1 + 12 �� − 1�M2�1/��−1� �3�

where pC, TC, and �C stand for stagnation chamber properties.Clearly, substitution of M=M�� ,�� in the above will give rise, bydirect extension, to T=T�� ,� ,TC�, p=P�� ,� ,pC�, and �=��� ,� ,�C�.

The pressure and temperature are of chief interest because thepressure determines, in part, how the nozzle performs for a givengeometry, and the construction of the nozzle is dependent, in part,on the wall temperature distribution and its induced thermal gra-dients. To illustrate these effects, Fig. 1 is used to describe thepressure and temperature distributions in an isentropic nozzle. Inthis graph, the subsonic pressures and temperatures are seen toexhibit qualitatively similar shapes, starting at a maximum valueand decreasing as the flow enters the converging section of thenozzle. The temperature decreases rather gradually as the accel-eration in the subsonic region builds progressively. After the flowcrosses the throat at �=1, the pressures and temperatures continueto decrease. In the case of the pressure, the supersonic branchappears to mirror its subsonic counterpart, with the slight asym-metry being caused by the location of the critical pressure at thethroat, here shown for �=1.4. The temperature, however, experi-ences visibly dissimilar trends. As the flow enters the supersonicregion, the temperature steadily decreases, for ��1, and thendrops precipitously as the area ratio approaches zero. This behav-ior can be attributed to the thermal-to-kinetic energy conversionand its sensitivity to the degree of area expansion. For small val-

ues of �, the substantial gain in Mach number is seen to translateinto appreciable temperature differentials, specifically in thenozzle exit section.

3 AnalysisFor a fixed area ratio, two Mach numbers are possible depend-

ing on whether the axial position is located upstream or down-stream of the throat section. In what follows, the asymptoticanalysis leading to each of these roots is described.

3.1 Subsonic Solution. Using � as our perturbation param-eter, an explicit series approximation for the subsonic Mach num-

ber M=M�� ,�� can be pursued. The subsonic series approxima-tion can be obtained using regular perturbations. To begin, Eq. �1�can be written in subsonic notation. In the interest of clarity, it ishelpful to define two constants and a variable,

� � 12 �� + 1�/�� − 1�, � � � 1

2 �� + 1���, ��M� � 12 �� − 1�M2

�4�

These transform Eq. �1� into

M =1

���1 + ��M��� �5�

Equation �5� represents the reduced relation that needs to be

solved for M. First, one may express M in a series of diminishingterms, specifically

M = x0 + �x1 + �2x2 + �3x3 + �4x4 + �5x5 + �6x6 + O��7� �6�

Second, by inserting Eq. �6� into Eq. �5�, one may expand andsegregate terms of the same order. At the outset, one finds thatterms of even order must strictly vanish:

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1C

pp

supersonic

(a) At /A

subsonic

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1C

TT

(b) At /A

subsonic

supersonic

Fig. 1 Sensitivity of the pressures and temperatures to thenozzle expansion ratio at �=1.4

071702-2 / Vol. 133, JULY 2011 Transactions of the ASME

x0 = x2 = x4 = x6 = 0 �7�

In view of Eq. �7�, the expansion of M will be accurate to O��7�when the first three nonzero corrections are retained. The effectiveexpansion reduces to

M = �x1 + �3x3 + �5x5 + O��7� �8�

Subsequently, one may substitute Eq. �8� into Eq. �4� to retrieve aseries expression for �,

� = 12�2x1

2�� − 1� + �4x1x3�� − 1� + O��6� �9�

One may also use the binomial expansion of Eq. �5� to evaluate

�1 + ��� = 1 + �� + 12!��� − 1��2 + 1

3!��� − 1��� − 2��3 + ¯

�10�

and so

�−1��1 + ��� = �−1��1 + �� + 12!��� − 1��2 + 1

3!��� − 1��� − 2��3

+ ¯� �11�

The binomial expansion in Eq. �11� will be valid for ���= 12 ��

−1�M21, an expression that remains true in the subsonic case.This is especially true when considering that the ratio of specificheats varies between 1.1 and 1.4 in most propulsive applications.This ensures that ���1, which, in turn, prompts the swift conver-gence of the series. The left-hand side of Eq. �5� can now be

replaced by the perturbed form of M such that

�−1�1 + ��� = �x1 + �3x3 + �5x5 + O��7� �12�

The final step is to insert Eq. �11� into Eq. �12�. Collecting termsof the same order yields

x1 = � 12 �� + 1��−�, x3 = 1

4 �� + 1�x13, x5 = 1

32�� + 1��3� + 7�x15

�13�

The corresponding Mach number may be obtained to arbitraryorder from

Mn = �i=0

n

�2i+1x2i+1 + O��2n+3�

= �i=0

n

�2i+1 �2i�!i!2i

�� − 1�i

�2i + 1�!�2i+1j=0

i−1

��2i + 1�� − j� + O��2n+3�

�14�

By way of example, a three-term subsonic series approximationbecomes

M2 = �x1 + �3 14 �� + 1�x1

3 + �5 132�� + 1��3� + 7�x1

5 + O��7�

�15�

It may be later shown that this expression is virtually indiscerniblefrom the numerical solution of the problem. In fact, for opera-tional area ratios up to �=0.47 and �=1.7, only one term needs tobe calculated and still secure a practical approximation that ac-crues a less than 5% error. Table 1 catalogs the maximum value of� for which the error will remain bracketed under 5% at differentvalues of � and the first three asymptotic orders. Note that therange of validity broadens when more terms are retained or when� is lowered. The range extends to �=0.50 for the lowest valuereported, �=1.1. Using a two-term correction, the maximum op-erational range increases to 0.64���0.79 for 1.1���1.7, withthe largest � corresponding to the smallest �. From an engineeringperspective, Eq. �15� mirrors the exact solution because its errorremains smaller than that associated with the governing equationitself; the latter is attendant on the isentropic flow idealization.Hence, by virtue of the physical range defined in Eq. �2�, a one-

term approximation is sufficient to provide an accurate approxi-mation up to �=0.40, ∀�. This key result is

M �� 12 �� + 1����+1�/�2��−1�� + O��3� �16�

Note that � is not limited to the nozzle expansion ratio. It maycorrespond to any cross section so long as the inverted ratio withthe throat area remains small.

3.2 Supersonic Solution. Using the method of successive ap-proximations, an explicit series expansion for M=M�� ,�� may beobtained. Successive approximations may be applied to polynomi-als and transcendental relations in which roots are not deduciblefrom a regular expansion, namely, from a predetermined sequenceof gauge functions �28�. The ensuing linearization may beachieved by striking a balance in Stodola’s equation betweenterms that dominate for M�1. To this end, we rewrite Eq. �1� insupersonic notation by introducing the three constants �, , and�.

� � �� + 1�/�� − 1�, � �12 �� − 1�, � � �1/�/�1 + 2

�17�

These may be substituted into Eq. �1� to render the simple form

�2�1 + 2M2� = M2/� �18�

Next, M must be expanded carefully. In identifying the leadingorder part of Eq. �18�, we substitute M0=X0 into Eq. �18� andwrite

X02/� − �2 2X0

2 − �2 = 0 �19�

A scaling analysis reveals that the first two members of Eq. �19�dominate with the third member representing a secondary contri-bution. This behavior can be confirmed through the use of a nu-merically calculated root. For the supersonic case at �0.1, theMach number is O�1�. When substituting the ordered quantitiesback into Eq. �19�, the first member will be seen to carry thelargest contribution. Along similar lines, the second member willexhibit a non-negligible, albeit smaller contribution, owing to itscoefficient �2=�2/�, multiplying its X0

2�1 part. The third memberin Eq. �19� contains an �2/� term only; as such, it can be viewed asa higher order quantity. It is important to note that by balancingthe first two members, a different leading order equation isachieved. At leading order, the two largest terms will balancewhen

X0 = �� ��/�1−�� + O����−1�/2� �20�

Successive expansions of M may be similarly undertaken by set-ting M1=X0+X1. Backward substitution into Eq. �18� leads to asolution for X1 bearing a truncation error of O��4��−1�/��+1��. Forall approximations past X0, a recurrence relation may be writtenfor Xi in terms of Xi−1:

Table 1 Maximum nozzle area ratio with less than 5% relativeerror at increasing asymptotic orders

�

Subsonic Supersonic

n

0 1 2 0 1 2

1.1 0.495 0.794 0.889 n/a 0.948 0.9871.2 0.491 0.670 0.885 n/a 0.767 0.9991.3 0.486 0.664 0.883 n/a 0.729 0.9621.4 0.482 0.658 0.881 0.00631 0.746 0.9401.5 0.478 0.653 0.879 0.01890 0.773 0.9311.7 0.470 0.644 0.875 0.06470 0.820 0.930

Journal of Heat Transfer JULY 2011, Vol. 133 / 071702-3

Xi = ��Mi−1��Mi−12/� − Mi−1

2 � 2�2 + �Mi−12+2/� 2�2��2 − 4� + 4�

− Mi−12 �2 2�4 + Mi−1

4/��2� − 3� + 2Mi−12/��2�2 − ���1/2� , i � 1

�21�where

� =1

2Mi−12/��� − 2� + Mi−1

2 �2 2�2 �22�

The higher order solutions may be sequentially obtained from thestraightforward sum

Mn = X0 + X1 + ¯ + Xn �23�Table 1 also lists the area ratios for which the relative error in

Eq. �23� will reach 5% at different perturbation orders. To main-tain a less than 5% error, a two-term approximation is clearlynecessary as the validity of one-term expressions is limited. UsingM1=X0+X1, the relevant range of area ratios extends to 0.73–0.95, depending on the value of �. When three terms are held, thevalid range is extended to 0.99 at �=1.2. Considering that practi-cal nozzles do not exceed �=0.4, a two-term approximation of theform M�M1 may be relied on.

Before leaving this section, it may be interesting to note that the�-dependence differs in behavior between the subsonic and super-sonic solutions. In the former, the range of validity in � improvesat lower values of �. In the latter, more than one local minimumcan appear due to the nonlinearity of the supersonic formulation.For M=M1, a local minimum appears in the middle of the feasiblerange of �, occurring approximately near �=1.3. The correspond-ing range of validity in � subsequently expands as ��−1.3� isincreased.

4 Comparison and Order Verification

4.1 Numerical Verification. The approximation for Mn iscompared in Fig. 2 to MN, the numerical solution of Eq. �1�, atincreasing asymptotic orders n=0,1 ,2. Both MN and Mn areshown in subsonic and supersonic regimes at a representativevalue of �=1.4. These trends remain unchanged at other �. Theinset in Fig. 2 enhances the differences between numerics andasymptotics in the transonic region. In the meaningful range of��0.4, the visible discrepancies become indiscernible when us-ing either the one-term subsonic or the two-term supersonic ap-proximations. This behavior concurs with the relative error pre-dictions �reflecting a less than 5% deviation� furnished in Table 1

for M0 and M1. In the vicinity of �=1, the discrepancies do notpose a concern as they only denote impractically small nozzledivergences.

4.2 CFD Verification. An inviscid, axisymmetric, density-based, double precision, finite volume solver is employed to simu-late the motion of an ideal gas in a fully flowing supersonicnozzle. The nozzle has inlet and outlet area ratios of 2.42 and5.44, respectively. Given an overall nozzle length of L=0.3 m,the computational domain is resolved using a mesh comprised of26,801 quadrilateral cells, as shown in Fig. 3. The nozzle contourconsists of a simple three-point spline. Using air as the workingfluid, two separate validation runs are undertaken. The first trialuses a low pressure, low temperature flow with TC=300 K, whilethe second simulates a higher chamber pressure and a temperatureof TC=2200 K. The thermodynamic properties specific to eachtrial are posted in Table 2.

Figure 4 describes the evolution of the velocity vectors andcorresponding Mach numbers for the low temperature trial. Thevelocity scatter and Mach number contours characterize the flowmagnification through the converging-diverging nozzle �Figs. 4�a�and 4�b��. Downstream of the throat section, in the supersonicacceleration stage, the particles at the wall outrun the centerlineparticles. This can be attributed to the absence of friction at the

wall and the enhanced radial and axial expansions near the walldue to nozzle divergence; these exceed the centerline expansionthat is limited to axial acceleration alone. Being expanded bothaxially and radially, the flow near the wall experiences increasedvalues of the local Mach number. Conversely, the axially domi-nated centerline motion undergoes no radial acceleration and, assuch, remains limited to a relatively smaller increase in its localMach number.

The wall, centerline, and average Mach numbers obtained atdifferent stations in the nozzle are compared in Fig. 4�c� to thethree-term analytical approximations. This comparison is carriedout point-by-point and shows that the 1D analytical model liesbetween the wall and centerline curves, thus providing excellentagreement with the average computed Mach number at any givenstation. This agreement holds everywhere except in the close vi-cinity of the throat, where the solution begins to diverge as thearea ratio approaches unity. This region can be further shrunk byincluding more corrective terms, despite the fact the three-termapproximation remains adequate over the vast majority of thenozzle.

Figures 5 and 6 display the thermal data for the low and highchamber temperature runs. As expected, the spatial distribution ofthe temperature isolines are qualitatively comparable to the Machnumber contours, both for the cold �Fig. 5� and hot �Fig. 6� simu-lations. The temperatures undergo a constant decline as the fluid isaccelerated throughout the nozzle, owing to the continual thermal-to-kinetic energy conversion. Figures 5�b� and 6�b� compare bothtrials to the present study using the isentropic relation in Eq. �3�.As with the Mach number, the asymptotic approximations areseen to provide an excellent average throughout the nozzle, withsmall discrepancies being detected in the neighborhood of thethroat. The resulting expressions can thus be used as analyticalalternatives to CFD computations in simplified thermal analysesof the nozzle. For example, both Stodola’s and CFD solutionspredict the occurrence of the maximum heat flux just upstream of

0.001 0.01 0.1 10

2

4

6

8

10

12

(a)

MNM0M1M2

ε

M

0.1 0.4 10

1

2

3M

subsonic

ε

supersonic

(b)

Fig. 2 Comparison between numerics and asymptotics inboth subsonic and supersonic regimes for �=1.4; the selectedarea in „a… is magnified in „b… to showcase the degree of agree-ment at increasing asymptotic orders as ε\1

071702-4 / Vol. 133, JULY 2011 Transactions of the ASME

the sonic point �where the centerline velocity first reaches thespeed of sound�. This behavior is consistent with available experi-mental data �7� and may be seen in Fig. 7, where the normalizedheat flux indicator, defined as q=T / �2�r�, is plotted over thelength of the nozzle using Stodola’s numerical solution, thepresent approximation, and CFD predictions. Also shown on thesame graph is the spatial variation of the Mach number obtainedfrom Stodola’s 1D model previously described in Fig. 4.

4.3 Asymptotic Error Verification. To verify the order of theerror associated with the subsonic and supersonic perturbation ex-pansions of M, one may apply Bosley’s graphical technique �29�.Accordingly, one may confirm that the expansion is asymptoti-cally valid by showing that its absolute error exhibits, in somerange of �, a constant logarithmic rate. For that purpose, the ab-solute error may be defined as

En��,�� = �Mn − MN�, En = �Mn − MN� �24�

In either of the subsonic or supersonic cases, if the truncationerror appears at order r, one can put

En��,�� = C�r �25�Graphically, the order of a given approximation can then be in-ferred from the log-log plot of En versus � at constant �. Accord-ing to Bosley �29�, it is important for the error to approach zero atthe correct rate �i.e., r→const� as �→0.

Forthwith, graphs of log En versus log � are given in Fig. 8over a practical range of nozzle area ratios and a representativevalue of �=1.4. Corresponding asymptotic slopes are obtainedusing linear least-squares and are posted in Table 3, includingthose for �=1.2. It is gratifying to note that all slopes approachconstant rates, thus reflecting an “error-free” analysis.

In the subsonic case, the best fit slopes obtained from least-squares match quite closely the order of the theoretical truncationerror given by Eq. �14�.

In the supersonic case, the slope begins at a modest rate ofdescent that can be approximated by 1

2 ��−1�. It then increasesrapidly as more terms are added. By retaining two terms, theasymptotic order in M1 jumps to 4��−1� / ��+1�. Being almostquadrupled in comparison to the leading order, M1 is accompa-nied by a less than 5% error for practical values of � and up to0.73���0.95 �Table 1�. In the supersonic case, the order in-

(a)

x (m)

y(m)

0.1 0.12 0.14 0.16 0.18 0.20

0.02

0.04

0.06

0.08

(b)

Fig. 3 CFD nozzle geometry and mesh selection „a… showingenlarged inset in „b…

Table 2 Parameters used in the low and high temperaturesimulations with air as the working fluid

Parameter Cold Hot

Chamber pressure �Pa� 101,325 506,625Outlet pressure �Pa� 3,740 50,660Chamber temperature �K� 300 2,200Ratio of specific heats 1.4 1.4Exit area ratio �At /Ae� 0.184 0.184

(a)

(b)

0 0.5 1

0.5

1

1.5

2.0

2.5

3.0 wall CFDcenterlineCFDaverageCFD3-termsolution

M

x/L

γ=1.4

(c)

Fig. 4 Cold flow simulation results depicting „a… the velocityvector distribution, „b… the Mach number contours, and „c… acomparison of the wall, centerline, and average CFD predic-tions to the present three-term analytical approximation overthe length of the nozzle. Here, air is used with TC=300 K and�=1.4.

Journal of Heat Transfer JULY 2011, Vol. 133 / 071702-5

creases with �. In every case, the slope approaches a constantvalue with a higher correlation coefficient at higher values of nand in ranges of smaller �. The slopes shown in Table 3 areevaluated in the �� �0.001–0.01� domain. Their agreement withthe theoretical orders only improves in ranges of smaller �.

5 ConclusionsIn this work, two asymptotic approximations are presented as

practical equivalents to the numerically-inverted area-Mach num-ber relation. The present analysis provides the explicit dependenceof the Mach number on the nozzle area ratio and ratio of specificheats. This dependence illuminates the influence of each of theseparameters on the maximum achievable Mach number for isentro-pic motion. They also facilitate the efficient evaluation of thermo-dynamic properties while carrying out a performance analysis of aDe Laval nozzle. The present results are hoped to be further usedin deriving closed-form expressions of other related parameters incompressible flow studies, which often assume inviscid motions.Were viscous effects included in the analysis, they would haveresulted in a reduced average velocity, especially near the wallwhere the flow would be decelerated. If such were the conditions,then Stodola’s relation could still provide a good approximation

(a)

0 0.5 1

100

150

200

250

300

wall CFDcenterlineCFDaverageCFD3-termsolution

T

x/L(b)

Fig. 5 Cold flow simulation results depicting „a… the thermalmap in K and „b… a comparison of the wall, centerline, and av-erage CFD predictions to the present three-term analytical ap-proximation of the temperature over the length of the nozzle.Here, air is used with TC=300 K and �=1.4.

(a)

0 0.5 1500

1000

1500

2000

2500

wall CFDcenterlineCFDaverageCFD3-termsolution

x/L(b)

T

Fig. 6 Hot flow simulation results depicting „a… the nozzlethermal map in K and „b… a comparison of the wall, centerline,and average CFD predictions to the present three-term analyti-cal approximation of the temperature over the length of thenozzle. Here, air is used with TC=2200 K and �=1.4.

0 0.5 1

0.2

0.4

0.6

0.8

1

1

2

3

sonicpoint

M

Wall CFDStodola's relation3-termsolution

q

x/L

qmax

Fig. 7 Normalized heat flux obtained from Stodola’s equation,the present approximation, and CFD predictions using the hotflow simulation; the spatial evolution of Stodola’s Mach num-ber is also shown on the right

10-3 10-2 10-1 10010-6

10-4

10-2

γ =1.4

E0

E1

E 2

E

ε

E 2

E 0E 1

Fig. 8 Asymptotic error entailed in supersonic En and sub-sonic En shown for �=1.4

Table 3 Asymptotic slopes in EnÈεr using the method ofleast-squares

� �subsonic� E0 E1 E2

1.2 3 5 71.4 3 5 7

� �supersonic� E0 E1 E2

1.2 0.120 0.361 0.9291.4 0.210 0.635 1.518

071702-6 / Vol. 133, JULY 2011 Transactions of the ASME

for the Mach number and corresponding temperature distributionsnear the centerline, where viscous effects are least significant.

AcknowledgmentThis project was funded in part by the National Science Foun-

dation under Grant No. CMMI-0928762, Program Director, Dr.Eduardo A. Misawa. The authors thank Mr. Timothy A. Barber forhis assistance in producing some of the illustrations needed tofinalize the manuscript.

NomenclatureA � cross-sectional areaAt � throat area� � perturbation parameter, At /A� � ratio of specific heats

Mn � Mach number with n+1 termsMN � numeric value of the Mach numberEn � absolute error between MN and Mn

Subscripts and Symbolse � condition at the exitn � level of the asymptotic sequencet � condition at the nozzle throat− � an overbar denotes a subsonic term

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