17 WHAMPOA - An Interdisciplinary Journal 52(2007) 17-24

COMPUTATIONS OF THE DIVERGENT PORTION OF A TWO-DIMENSIONAL SUPERSONIC NOZZLE

Weihao Chung, JuiYang Wang

Department of Civil Engineering The Academy of Chinese Military

Abstract

The nozzle for supersonic flow is designed in this paper such that it is of minimum length, flow at the exit is uniform and parallel to the centerline of the nozzle, the flow within the nozzle is shock free, and the Mach number at exit equals the desired one. Based on these principles, contour lines representing the wall of the divergent portion of the nozzle are plotted, enabling the evaluation of nozzle lengths which depend on the Mach number. Comparisons between numerical and analytical solutions are made to assure the accuracy of computations. Key words: supersonic flow, nozzle, Mach number, shock wave, Euler equation. 1.Governing Equations The equation applicable in subsonic, supersonic, and transonic range for steady or unsteady, 1-D, 2-D, or 3-D compressible flow is the so-called Euler equation. Of which the continuity equation states that

( ) 0=+ Utvv (1)

the u-momentum equation along the x-axis is

( ) 0=++ ipUutu vvv (2)

and the energy equation reads:

( ) 00 =+ UhtEvv (3)

where E denotes the energy of fluid per unit mass and the total specific enthalpy is given

by 2

222

0wvup

Rpch v ++++= . A full set

of equations which include viscous terms called Navier-Stokes equation are more

applicable in the hypersonic regime. It can be shown that in steady, inviscid and adiabatic flow the total specific enthalpy is constant along a streamline, and for steady, isentropic and potential flow equation (1) is reduced to the following equation with a single variable[1]:

( ) ( ) ( )0222

222222

=++

xzzxyzzyxyyx

zzzyyyxxx aaa

(4) where the gradient of velocity potential

Uvv = with kwjviuU vvvv ++= , the speed of

sound RTa = , the ratio of specific heats

4.1== vp cc for air. It is also verified that potential flow satisfies the u-momentum equation in smooth regions of flow (ie, away from shock waves) by Croccos theorem[2]. Shocks however can be fitted using oblique

18

shock relations.

2. Method of Characteristics (MOC) It can be shown that in 2-D supersonic flow at any given point equation (4) has two characteristics, which are:

( ) = tandxdy (5)

where denotes the flow angle with respect to the x-axis, and the Mach angle defined by M/1sin = with M the Mach number. The compatibility equations of eq.(5) are

( ) .constM = m (6) respectively. The Prandtl-Meyer function of Mach numbers is defined as:

( ) ( )

+

+= 1

11tan

11 21 MM

( )1tan 21 M (7) Mathematically supersonic flow fields can be viewed as a family of left running characteristics and a family of right running ones. A physical view of these characteristic lines is viewing them as waves, across which flow properties change an infinitesimal amount. They behave like shock waves except across them flow is isentropic. There are basically two types of shock waves, namely expansion wave and compression wave. For the former, across it the pressure, density and temperature are all decreased, the Mach number and velocity however increased. 2.1 Flow properties at interior nodes At the nozzle throat where Mach number equals one, fluid flow is still parallel to the

x-axis, i.e., 0= and =0 by eq.(7). This gives a result that the characteristic line at nozzle throat is perpendicular to the flow direction since 090= for M=1. Based on the flow information of this characteristic line, fluid properties for other interior nodes at the divergent portion of the nozzle can be computed by eq.(6) Consider now a two-dimensional supersonic potential flow through a divergent nozzle with given initial flow data along the line at the throat mentioned above. One can draw a right running characteristic from a point on the line, and a left running characteristic from the other point at lower position of the line to determine and by solving simultaneously eq.(6). The respective Mach number at the intersection of the characteristic lines is hence obtained by interpolation from a table listing the relation between and M. Coordinate (x,y) of the intersection point is then ready to be determined. 2.2 Flow properties at nozzle walls In general, the wall shape of a nozzle for supersonic flow is not a straight line but a curve. It is especially true for the portion just at downstream of the nozzle throat. This portion is often prescribed and designed as a part of a circle with radius approaching to zero so as to reduce the entire length/weight of the nozzle. To determine the coordinate of any point (say node 1) where a characteristic line starting from the throat intersects with the prescribed portion, several steps are proceeded: (i)

Weihao Chung, JuiYang Wang: Computations of the divergent portion of a two-dimensional supersonic nozzle 19

assuming 21 = and 21 = , where node 2 is any interior node at the characteristic line across the nozzle throat, (ii) draw a characteristic from 2 to 1 of slope

2tan 2211 +++ , (iii) from the intersection of the characteristic and the prescribed portion and the shape of the wall, compute ( )dxdy /tan 11 = , (iv) find better estimate for 1 using the relation

2211 = , (v) find 1 and 1M , (vi) go to step (ii) until convergence. In case that the Prandtl-Meyer function at nozzle exit is preset and the prescribed part of the nozzle is circular, one can describe flow properties for any node j on the prescribed

part as: = jj , jj = , Nexit 2/ = where j=1~N.

Aside form the wall shape, the entire nozzle length should be designed as short as possible to save materials. To achieve this, a criterion must be satisfied:

2max,M

ML = (8)

which states that to design a nozzle with the minimum length, the maximum expansion angle of the wall downstream of the throat (i.e., the prescribed portion) should be equal to one-half of the Prandtl-Meyer function for the design exit Mach number. Thus, based

on eq.(7), 0max, 189.13=ML for a Mach

number of two, and 0max, 879.24=ML for a Mach number of three. Obviously, eq.(8) provides itself a useful tool to determine the length of the prescribed portion.

2.3 The rest portion Up to now, only one principle of

designing the nozzle is remained unsolved. It is the flow should be uniform and parallel at the exit. This can be done by following two steps to design the rest portion of the nozzle: (i) compute the interior node properties by repeatedly calling an interior point analysis, i.e., eq.(6). (ii) design the rest

of the nozzle using elations ij =

and ij = (with i the subscript for interior nodes) to avoid the generation of expansion and compression waves when characteristics strikes nozzle walls. Step (ii) guarantees that flow angles at the exit are zero, since 0=i along a nozzle centerline, meaning a supersonic flow parallel to the centerline line of the nozzle occurs at exit.

3. Solution Procedures (1) Read the preset Mach number at exit in

the expansion region. In order to smoothen the nozzle contour, more than 5 nodes in the prescribed portion are suggested. In this paper, a total node number N=11 is adopted.

(2) Radius for the prescribed portion is set to as a small value as possible (0.001 for this paper) to reduce the nozzle length. Interior nodes in the simple interior region are endowed with double-subscript notations so that the validity of simple relations described at the procedure (iv) (which would become very complicated if only use of single subscript notation) is ensured.

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(3) single subscript notation is adopted at the wall of the rest portion of the nozzle, and only half/top of the nozzle is necessary to be considered in this project because of its symmetry about the x-axis.

(4) Mach number at each node can be obtained by trial-and-error or the Lagrange interpolation formula. For the sake that possible solution range of Mach numbers during trial-and-error procedure is already known if M at exit is given, the half-interval method is a not bad and efficient way to solve this problem. Usually ten iteration steps are required to make the assumed Mach number converge to some tolerant criteria.

(5) Numerical results are correct by checking that local area ratio A/A* (A*=the area of the throat) matches the value from the 1-D nozzle table based on analytical solutions. A list of comparison is made and shown by table1: Errors of A/A* among the last three columns are small enough to be accepted for N=11. Eventually, accuracy can be improved by introducing finer step size on the prescribed portion of the nozzle. Note that the required CPU time is proportional to N2 due to the N*N matrix which consists of the interior points plus nodes at the wall.

4. ResultsNozzle Contours Figure 1 shows the variation of Mach numbers along the centerline of the nozzles with different designed flow velocities at exits. For all cases, the Mach numbers increase with the increasing x values,

meaning flow velocities keep increasing as shock waves move towards the exits. Figure 2 shows the contours of the upper part of nozzles with different exit Mach numbers. Note that all figures are conceptually plotted since plots are not to scale. Figure 2 however, reveals that each nozzle seems to expand itself abruptly right after the throat, and then becomes more smoothen at the downstream end. The computed maximum lengths of the nozzle for M=2, 3, and 4 at exit are 4.817m, 16.803m ,and 53.017m, respectively. The maximum deflection angle also occurs at the throat of the nozzle which coincides with the result from eq.(8). Results are summarized in table 2. From table 2, it is realized that the length or weight of a nozzle increases dramatically with the prescribed Mach number at exit. It is therefore necessary to keep developing new techniques for design purposes and seeking some strong but low-density materials for constructing supersonic flow nozzles if one needs to fly an air-vehicle with great speeds , say more than Mach ten. References [1] Anderson, J. D. Jr, Modern

compressible flow with historical perspective, McGraw-Hill, New York, 1982.

[2] Crocco L., Eine neue Stromfunktion fur die Erforschung der Bewegung der Gase mit Rotation, Z. Angew. Math. Mech. Vol. 17, 1-7, 1937.

Weihao Chung, JuiYang Wang: Computations of the divergent portion of a two-dimensional supersonic nozzle 21

Table 1: Comparisons of local area ratios for different Mach numbers.

Table 2: The maximum expansion angle at throat and nozzle sizes at exit. Mach no.

MLmax, (degree)

Maximum length(m)

Maximum width (m)

M=2 13.189 4.817 3.280 M=3 24.879 16.803 8.408 M=4 32.892 53.017 21.440

M at throat M at exit A/A* (M.O.C.) (N=3)

A/A* (M.O.C.) (N=11)

A/A* (theory)

1.0 2.0 1.64 1.68 1.68 1.0 3.0 4.29 4.20 4.23 1.0 4.0 14.46 10.72 10.72

22

Fig 1: Figure 1: Computed Mach numbers along the x-axis for

different flow velocities of design at nozzle exits. (top: Mexit=2.0; middle: Mexit=3.0; bottom: Mexit=4.0)

01234

0.00 1.00 2.00 3.00 4.00 5.00

x(m)

Mac

h no

.

0

1

2

3

0 0.5 1 1.5 2X(m)

Mach

no.

012345

0.00 2.00 4.00 6.00 8.00 10.00 12.00

x(m)

Mac

h no

.

Weihao Chung, JuiYang Wang: Computations of the divergent portion of a two-dimensional supersonic nozzle 23

Figure 2: Upper nozzle contours for different Mach numbers of design at exits.(top: Mexit=2.0; middle: Mexit=3.0; bottom: Mexit=4.0)

02468

1012

0.00 15.00 30.00 45.00 60.00x(m)

y(m)

00.51

1.52

0.00 1.00 2.00 3.00 4.00 5.00x(m)

y(m)

0246

0.00 4.50 9.00 13.50 18.00x(m)

y(m)

24

: , , , ,