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Applied Gas Dynamics

Method of Characteristics

Ethirajan Rathakrishnan

Applied Gas Dynamics, John Wiley & Sons (Asia) Pte Ltd c© 2010 Ethirajan Rathakrishnan 1 / 98

Introduction

Method of characteristics is a numerical method for solving the full non-linear equations of motion for inviscid, irrotational flow. As we havealready discussed, except the Prandtl-Meyer expansion, all other prob-lems have been solved with the linear theory. If we are looking for betteraccuracy of results than that obtained by using the approximate lin-earized equations, it is necessary to work out improved solutions, byincluding higher-order terms in the approximate equations or by con-sidering the exact equations. However, in the latter case, it is rarelypossible to get solutions in analytical form because of the nonlinear na-ture of the equations. We must then resort to numerical techniques; themethod of characteristics being one such technique.


The Concepts of Characteristics

In the previous chapters, Mach lines were identified as characteristiclines, and they have been labeled as ′left-running′ and ′right-running′,depending upon whether they run to the left or right with respect toan observer looking in the flow direction. Now, let us see some of theimportant features of the characteristics. From the earlier discussionson the properties of Mach lines and expansion flows, we may infer thefollowing general features of characteristics.


1 They exist only in supersonic flow field.2 Characteristics are coincident with Mach lines. (Mach lines are

lines along which very weak disturbances propagate and across aMach line the flow properties change are small but finite.)

3 While the derivatives of the flow properties are discontinuous, theflow properties themselves are continuous on the characteristics.

4 Given the characteristics or Mach lines, the dependent variablessatisfy a relation known as the compatibility relation. This providesthe key to the method of computation.

5 Because the characteristics are lines across which there is a jump(even though small) in flow properties, the downstream flow doesnot affect the upstream flow. Therefore, it is sufficient to calculatethe flow for different regions of the flow field and then they can bepatched up. But in subsonic flow, any downstream flow affects theupstream flow. So the entire flow has to be solved simultaneously.


The Compatibility Relation

Consider a steady, adiabatic, two-dimensional, irrotational supersonicflow. The governing equations for this flow are

(V 2x − a2)

∂Vx

∂x+ Vx Vz

∂Vx

∂z+

∂Vz

∂x

+ (V 2z − a2)

∂Vz

∂z= 0 (9.1)

∂Vz

∂x− ∂Vx

∂z= 0 (9.2)

If (V 2x +V 2

z )/a2 < 1, the equations are of elliptic type, and the relaxationmethod of solution is appropriate. If (V 2

x +V 2z )/a2 > 1, the equations are

of hyperbolic type. The numerical solution may be obtained by methodof characteristics.


Using the natural coordinate system, in which the velocity is expressedin terms of its magnitude and direction (V , θ), and the independentvariables are the streamline coordinates (l , n), with l varying along thestreamline and n varying normal to streamline, Eqs. (9.1) and (9.2) canbe written as

V 2

a2 − 1

1V

∂V∂l

− ∂θ

∂n= 0 (9.3)

∂V∂n

− V∂θ

∂l= 0 (9.4)

where Eqs. (9.3) and (9.4) are respectively the momentum and irrota-tionality equations.


With the introduction of Mach angle µ, we can write Eqs. (9.3) and (9.4)as

cot2 µ

V∂V∂l

− ∂θ

∂n= 0 (9.5)

1V

∂V∂n

− ∂θ

∂l= 0 (9.6)

wherecot2 µ = M2 − 1 (9.7)

In Section 4.6, it was shown that for a finite deflection angle θ, the direc-tion of a weak oblique shock wave differs from the Mach wave directionµ by an amount ε, which is of the same order as θ. The change of flowspeed across such a wave may be found as follows. From Figure 4.2,we have

V 22

V 21

=V 2

x2 + V 2y

V 2x1 + V 2

y=

(Vx2/Vy )2 + 1(Vx1/Vy )2 + 1

=tan2(β − θ) + 1

tan2 β + 1=

cos2 β

cos2(β − θ)


From Eq. (4.27) we have

cos2 β = 1 − sin2 β =M2

1 − 1

M21

1 − 2ε√

M21 − 1

A similar expression for cos2(β − θ) can be obtained by replacing ε by(ε − θ) in the above expression. Substituting the expressions for cos2 βand cos2 (β − θ) in terms of ε and θ in the above expression for velocityratio and dropping all terms of order θ2 and higher, we obtain

V2

V1≈ 1 − θ

√

M21 − 1

∆VV1

≈ − θ√

M21 − 1

(9.8)


In Section 4.9, it was defined that the Prandtl-Meyer function ν = ±θ,where the plus sign holds across a right-running characteristic and theminus sign holds across a left-running characteristic.

Now, the Prandtl-Meyer function ν, which is a dimensionless measureof the speed may be defined, using Eq. (9.8), as

ν =

∫

cot µV

dV (9.9)

where cot µ =√

M2 − 1. In differential form, Eq. (9.9) becomes

dν = cot µdVV

(9.10)

Now it will be seen that for the method of characteristics the Prandtl-Meyer function ν is the most appropriate one of the many functions thatare related to the velocity V (or the Mach number M).


Substitution of Eq. (9.10) into Eqs. (9.5) and (9.6) results in

∂ν

∂l− tan µ

∂θ

∂n= 0 (9.11)

tan µ∂ν

∂n− ∂θ

∂l= 0 (9.12)

The objective here is to find the compatibility relation between the Prandtl-Meyer function ν and the flow turning angle θ which, according to thetheory of hyperbolic equations, must exist on the characteristics, orMach lines. Though the theory gives rules for finding the compatibilitycondition, here we shall obtain it only by inspection, since our interestis only from the application point of view.


We are familiar with the fact that the Mach lines are inclined to thestreamlines at an angle ±µ. Therefore, we may expect to get the com-patibility relations by rewriting Eqs. (9.11) and (9.12) in a coordinatesystem (ξ, η) consisting of the network of Mach lines, as shown in Fig-ure 9.1. The change in any function f , in going from point P to point P ′

along η coordinate may be written as


∆n

(b) Natural coordinates (l, n)(a) Characteristic coordinates (ξ, η)

ξ

∆ξ

∆η

P ′

l

η

n n η

∆n

l

ξ

µ∆lµ

P

P ′′

Figure 9.1Characteristic and natural coordinate systems.


∆f =∂f∂η

∆η

∆f may also be calculated by going along the streamline coordinatesystem as

∆f =∂f∂l

∆l +∂f∂n

∆n =

∂f∂l

+∂f∂n

∆n∆l

∆l

From the above two equations,

∂f∂η

∆η

∆l=

∂f∂l

+∂f∂η

∆n∆l


From the geometry of Figure 9.1(b), this may be expressed as

sec µ∂f∂η

=∂f∂l

+ tan µ∂f∂n

(9.13)

Similarly, we can write

sec µ∂f∂ξ

=∂f∂l

− tan µ∂f∂n

(9.14)

Equations (9.13) and (9.14) give the rules that relate the derivatives ofany function f , in the two coordinate systems. Adding and subtractingEqs. (9.11) and (9.12), we get

∂

∂l(ν − θ) + tan µ

∂

∂n(ν − θ) = 0

∂

∂l(ν + θ) − tan µ

∂

∂n(ν + θ) = 0


Comparing the above equations with Eqs. (9.13) and (9.14) respec-tively, we obtain

∂

∂η(ν − θ) = 0,

∂

∂ξ(ν + θ) = 0

That is,ν − θ = R (constant) along η−characteristic

(9.15)

ν + θ = Q (constant) along ξ−characteristic

These are the compatibility relations between ν and θ. They simplymean that the functions Q = ν + θ and R = ν − θ are invariants on theξ and η characteristics, respectively.


Note: At this juncture we should note that the compatibility relations arenot always obtained in such convenient form as Eq. (9.15). Generally,they are obtained in differential form, and cannot always be integratedin this way, independently of the specific flow field to be solved.


The Numerical Computational Method

Consider the element from a characteristic network, illustrated in Fig-ure 9.2. Flow properties along AB are given. To find the flow proper-ties at point P, consider the right-running characteristic AP, with Q =constant, and left-running characteristic BP, with R = constant.


B

P

A Q = constant

R = consta

ntData

curv

e

Figure 9.2Characteristic network element.


For the above element we can write

QP = QA, RP = RB

But by Eq. (9.15),Q = ν + θ, R = ν − θ

Hence,

ν =12(Q + R), θ =

12(Q − R)

Therefore,

νP =12

(QP + RP), θP =12

(Qp − Rp)


Now, the problem is solved since, once ν is known, M, µ, p/p0 are allknown from the isentropic relations (or table). Also, once θ is known,the flow inclination with respect to the data line is known. The locationof the point P, which is an unknown, is found by a numerical technique.In this technique, the space is divided into parts to result in a charac-teristic network, as illustrated in Figure 9.3. As a first approximation,the characteristics are replaced by straight line segments. Because ofthis linear approximation, we arrive at point 3′ instead of 3, as shown inFigure 9.3. The error adds up and finally we get a point P ′ instead of P.


To minimize the error, the dimensions of the meshes should be of smallersize. Applying a step-by-step procedure, starting from the data curve,giving the data, or boundary conditions, we can identify the flow fieldat point P. For instance, point 3 is located by using the known Machangles and flow directions at points 1 and A to draw the characteristicsegments. Flow conditions at point 3 are determined from the data atpoints A and 1. Similarly, point 4 is found, and then point 6 is foundfrom points 3 and 4. Thus, starting from the data curve the computationproceeds outwards.


1

B

µ

2

A

1

A

P

3

4

5

7

6

3

3′

Figure 9.3Characteristic network.


The ′working outward′ computation from the data curve indicates thatthe nature of the boundary condition is such that it influences the flowonly in the downstream direction. This is in contrast to the Laplacianor elliptic type of field, in which the region of computation must be com-pletely bounded, and in which each point is influenced by all other pointsin the region.


Solid and Free Boundary Points

From the characteristic network shown in Figure 9.3, it is seen that forcomputing ν and θ at point 3, the invariants Q and R at points A and1 must be known. These points A and 1 may lie on a solid wall or freeboundary, like the edge of a jet, i.e. the boundary conditions fit into thecomputation quite readily. Consider the characteristic network elementshown in Figure 9.4.


µ1

Data

curv

eQ

1 = ν1 + θ

1

(ν1, θ1)

(ν3, θ3)

R2= ν2

−θ2

(ν2, θ2)

3

2

µ1

1

µ1

µ1

µ1

µ1

Figure 9.4Characteristic network element.


In Figure 9.4, the properties on the data curve arc 1-2 is known, i.e. ν1,θ1, ν2, θ2 are known. In other words, the invariants Q on the arc 1-3 andR on arc 2-3 are known. Therefore, the flow properties (ν3, θ3) at point3 can easily be obtained from the relations of Eq. (9.15) as follows.

Q1 = ν1 + θ1, R2 = ν2 − θ2 (9.16a)

Hence,

ν3 =12

(Q1 + R2), θ3 =12

(Q1 − R2) (9.16b)

With Eqs. (9.16), we can form the data given in Figure 9.5, which shouldbe known for computing data for point 3, depending on whether 3 is aninterior point or a point on a solid wall, or a point on a free boundary.


Table

1

3

Q1Q1

Interior point Solid boundary Free boundary

R2Q1RQ

ν θ θ3 ν3

1

3

1

3

2

Figure 9.5Known data for computing flow at point 3.

From Figure 9.5, it is seen that if any two of the quantities in the tableare known, the other two may be calculated with the relations in Eqs.(9.16).


Sometimes it may be necessary to compute flows in which shocks ap-pear. On such occasions, the method illustrated in Figure 9.6 may beemployed. As seen from the figure, point 3 is just behind the shock.One invariant R is obtained from point 1, since arc 1-3 is a left-runningcharacteristic. The other is determined by the shock equations (seeSection 4.3); it is not given explicitly, but as a relation between ν3 andθ3. Thus the flow at point 3 may be solved. These then determine theshock angle β, which is used to draw the next shock segment.


3

Shock

(β − θ)

1

Figure 9.6Shock in a flow.

If the shock is strongly curved, the flow downstream of it will have vor-ticity and the isentropic equations are not valid in this region, and theymust be replaced by appropriate equations accounting for vorticity ef-fects.


Example 9.1

Computation of flow in a diverging channel shown in Figure E9.1 withwalls diverging by 15◦ and Mdata = 1.348. Divide the data curve intothree equal segments, i.e. ∆θ = 5◦.

1

2

4

3

M = 1.348Data curve

13

5

14

12

10

9

µ1

6

7

11

8

Figure E9.1


The values of ν and θ are known at points 1 to 4 (data curve). There-fore, the invariants Q and R on all left- and right-running characteristicsoriginating from the above points may be calculated with the relations

Q = ν + θ, R = ν − θ

Hence, with the Prandtl-Meyer function ν and turning angle θ at points5to 14 may be obtained from the corresponding values of Q and R foreach point using the relations

ν =12

(Q + R), θ =12

(Q − R)

Table E9.1 gives the computed values of ν, θ, µ and M at points 5 to14 following the above procedure.


Table E9.1

Boundary conditions Point Q R ν◦ θ◦ µ◦ MGiven Derived 5 15 5 10 5 44.2 1.434

Point µ◦ ν◦ θ◦ Q R 6 10 10 10 0 44.2 1.4347 5 15 10 −5 44.2 1.434

1 47.9 7.5 7.5 15 0 8 20 5 12.5 7.5 41.1 1.5202 47.9 7.5 2.5 10 5 9 15 10 12.5 2.5 41.1 1.5203 47.9 7.5 −2.5 5 10 10 10 15 12.5 −2.5 41.1 1.5204 47.9 7.5 −7.5 0 15 11 5 20 12.5 −7.5 41.1 1.520

12 20 10 15 5 38.5 1.60613 15 15 15 0 38.5 1.60614 10 20 15 −5 38.5 1.606


Remarks

1 The compatibility conditions Q = ν + θ and R = ν − θ make thewhole procedure simple. These conditions are simple only for two-dimensional irrotational flows.

2 A drawing should always be made along with the computation, tolocate the points.


Sources of error

1 In actual flow, there will be boundary layer. This introduces error tothe results obtained, since it is not correct to assume the charac-teristics to be straight near the wall. The error may be corrected bycalculating the displacement thickness at different stations and byadding it to the contour already calculated.

2 Values obtained with ∆θ = 1◦ and ∆θ = 0.5◦ are almost the same.Theodore, there is no necessity to take ∆θ less than 1◦.


Axisymmetric Flow

The important features of the method of characteristics have been al-ready described in our discussion of plane flow. The computations fortwo-dimensional flow were seen to be very easy because of the simplenature of the compatibility relations (9.15). But the theory of character-istics for general three-dimensional flow is quite involved and the com-putations are cumbersome. However, for axisymmetric flow, the methodis easily extended from two-dimensional flow case.


Consider the fluid element shown in Figure 9.7.

V

r1 r2 r3r

(a)

r + ∆r

θ

(b)

∆ξ13

∆η 23

1

3∆n

2

∆l

Figure 9.7Axisymmetric flow coordinates. (a) Natural coordinates, (b) Characteristic

network.


The governing equation for this motion may be shown as

cot2 µ

V∂V∂l

− ∂θ

∂n=

sin θ

r(9.17)

1V

∂V∂n

− ∂θ

∂l= 0 (9.18)

where (V , θ) define the velocity in the natural coordinates plane. Equa-tion (9.17) differs from Eq. (9.5) only in the last term. The irrotationalityequation [(Eq. 9.18)] is the same as Eq. (9.6). Multiplication of Eq.(9.17) by tan µ and Eq. (9.18) by tan µ cot µ yields

cot µV

∂V∂l

− tan µ∂θ

∂n= tan µ

sin θ

r(9.17a)


tan µcot µ

V∂V∂n

− ∂θ

∂l= 0 (9.18a)

With the help of relations (9.9) and (9.10), Eqs. (9.17a) and (9.18a)become

∂ν

∂l− tan µ

∂θ

∂n= tan µ

sin θ

r

tan µ∂ν

∂n− ∂θ

∂l= 0

The above two relations correspond to Eqs. (9.11) and (9.12) for thetwo-dimensional case. Following the same procedure as that adoptedfor two-dimensional flow, we can obtain the following equations.


∂

∂η(ν − θ) = sin µ

sin θ

r(9.19)

∂

∂ξ(ν + θ) = sin µ

sin θ

r(9.20)

Now the integration has to be done numerically, step by step, simulta-neously with the construction of the characteristic network.

Consider the characteristic mesh element shown in Figure 9.7(b). Point3 is to be solved from the known data at 2 and 1. From Eqs. (9.19) and(9.20) we may write


∫ 3

2d(ν − θ) =

∫ 3

2

sin µsin θ

r

dη

∫ 3

1d(ν + θ) =

∫ 3

1

sin µsin θ

r

dξ

Now assume that for small size mesh, the quantities in parentheses onthe RHS to be approximately constant, over the interval of integration,and to have the known values at 1 and 2, respectively. The integrationsyield

(ν3 − θ3) − (ν2 − θ2) = sin µ2sin θ2

r2∆ η23

(ν3 + θ3) − (ν1 + θ1) = sin µ1sin θ1

r1∆ ξ13


From the above two equations, we get

ν3 =12

(ν1+ν2)+12(θ1−θ2)+

12

sin µ1sin θ1

r1∆ξ13 + sin µ2

sin θ2

r2∆η23

(9.21)

θ3 =12(ν1−ν2)+

12

(θ1+θ2)+12

sin µ1sin θ1

r1∆ξ13 − sin µ2

sin θ2

r2∆η23

(9.22)Equations (9.21) and (9.22) differ from the two-dimensional equations(9.15) only in the additional terms which depend on the geometry ofthe particular problem. In these terms, the radial distances r1 and r2

of the points in consideration, and the length of the mesh sides, ∆η23

and ∆ξ13, must be obtained from the flow field by measurement on adrawing or by computation.


Nonisentropic flow

For a nonisentropic flow, the governing equation of motion becomes

cot2 µ

V∂V∂l

− ∂θ

∂n=

sin θ

r(9.23)

and the vorticity equation follows from relation (5.6) as

1V

∂V∂n

− ∂θ

∂l= − T

V 2

dsdn

+1

V 2

dh0

dn(9.24)


Transforming Eqs. (9.23) and (9.24) into characteristic coordinates, weget

∂

∂η(ν − θ) = sin µ

sin θ

r− cos µ

V 2

Tdsdn

− dh0

dn

∂

∂ξ(ν + θ) = sin µ

sin θ

r+

cos µ

V 2

Tdsdn

− dh0

dn

The last terms in each equation may be written as derivatives along thecharacteristics with geometry shown in Figure 9.1(b);

∂η

∂n= cosec µ,

∂ξ

∂n= − cosec µ


Integration of the above governing equations over a small-mesh ele-ment yields

ν3−θ3 = ν2−θ2+sin µ2sin θ2

r2∆η23−

cot µ2

V 22

[T2(s3 − s2) − (h03 − h02)]

(9.25)

ν3 +θ3 = ν1 +θ1 +sin µ1sin θ1

r1∆ξ13−

cot µ1

V 21

[T1(s3 − s1) − (h03 − h01)]

(9.26)ν3 and θ3 may be obtained from Eqs. (9.25) and (9.26). From the aboverelations, it is seen that values of s3 and h03 at point 3 are neededfor computation. These may be determined as follows: In Figure 9.8,once point 3 is located, the streamline through it may be approximatelylocated by drawing a line with slope θ3′ = (θ1 + θ2)/2, intersecting thedata curve at 3′. Since s and h0 are invariant along streamlines, theirvalues at 3 are the same as at 3′ on the data curve, where they areknown.


curve

θ1

1

2

33′

θ3′

θ2

Data

Figure 9.8Characteristic mesh and a streamline.


Theorems for Two-Dimensional Flow

For two-dimensional (plane) supersonic flow, from the compatibility re-lations (9.15), we have

ν − θ = R (along η−characteristics)

ν + θ = Q (along ξ−characteristics)

These two relations, which are independent of the specific flow geome-try, lead to three useful theorems which we now give under three typesof flow.


1 General or nonsimple region – Characteristics of both the familiesare curved and are physically significant.

2 Simple region (simple wave) – One family of characteristics is straight.The other family (curved) is physically insignificant and not shown(by convention).

3 Uniform flow – Both families are straight and physically insignificantand not shown (by convention).


Consider the general region shown in Figure 9.9. The value of ν and θat the intersection of any two characteristics is found from the solutionof the above equations, as

ν =12

(Q + R), θ =12

(Q − R)

R1

Nonsimple region

Q1

Q2

Q3

Q4

Q5

R5

R4

R3

R2

Figure 9.9Characteristics of a general region.


Along any η−characteristic, R is constant and so the changes in ν andθ depend only on the changes in Q. Thus,

∆ν =12

∆Q = ∆θ (9.27a)

Similarly, along ξ−characteristics,

∆ν =12

∆R = −∆θ (9.27b)

Thus, the entire flow field is known if the values of R and Q on thecharacteristics are known.


Q1

Q3

R0

R0

R0

R0

Q2

Q4

Figure 9.10Characteristics of a simple region.

Consider next the simple region shown in Figure 9.10. In a simple re-gion, by definition, either Q or R is constant throughout the region. Inthe figure, all the η−characteristics have the same value of R (= R0).


Then, by Eq. (9.27b), ν and θ are individually constant along a ξ−characterwhich must be straight. Thus, in a simple region, one set of character-istics are straight lines, with uniform conditions on each one. The flowchanges encountered in crossing the straight characteristics are givenby

∆ν = ±∆θ (9.28)

In Eq. (9.28), the plus sign is for ξ−characteristics and the minus signfor η−characteristics. The relation given by Eq. (9.28) is different fromthat given by Eq. (9.27) in the sense that Eq. (9.28) is valid on any linethat crosses the straight characteristics, and in particular on a stream-line.


Consider now the uniform flow shown in Figure 9.11. By definition,uniform flow is that for which R = R0 and Q = Q0 throughout. Thatis, ν and θ are uniform, and both ξ−type and η−type characteristicsare straight lines constituting a parallel network, as illustrated in Figure9.11.

Figure 9.11Characteristics in a uniform flow region.


A flow field in which all three regions coexist is given in Figure 4.19. Asshown in the figure, the usual convention is to omit the Mach lines in theuniform region, to show only the straight lines in a simple region, andboth sets in the nonsimple region.

It is seen that the uniform region does not adjoin the nonsimple region(except at one point). This is a general theorem, which may be easilyproved by trying to construct contrary case, if we remember the defini-tions given above.


Numerical Computation with Weak Finite Waves

The method of constructing two-dimensional, supersonic flows by usingwaves was outlined in Chapter 4. If the waves are weak, we can set upa computing procedure which is equivalent to the characteristic method(see Section 4.6). In computation with weak waves, it is assumed thatthe entire gradual change in flow is assumed to occur discontinuouslyalong a single line given by Mach angle µ, as shown in Figure 9.12.


µ2

µ1

µ̄ =µ1 + µ2

2

M2

M1

θ

Figure 9.12Centered expansion.

It is further assumed that the strength of weak finite waves (∆θ) doesnot change in intersections. This assumption is valid only for two-dimensional flow.


Reflection of Waves

(a) On rigid walls a wave is reflected as a wave of the same sense (ofopposite family), as illustrated in Figure 9.13 (see also Section 4.11).

M1M1

βi

∆θ = 2◦

Compression waveExpansion waveµ2 > µ1

M3 < M2

M2 < M1

∆θ = −2◦

µ2 < µ1

βrµ2

M3 > M2

M2 > M1

µ1

Figure 9.13Reflections of waves from solid wall.


(b) On an open or free boundary (say the boundary of a free jet), a waveis reflected as a wave of opposite sense, as illustrated in Figure 9.14.

p = constant (free boundary)

M1

Expansion M2 > M1

M3 < M2

Compression

M1

Compression ExpansionM2 < M1

M3 > M2

p = constant (free boundary)

Figure 9.14Reflection of waves from free boundary.


It is seen from Figure 9.14 that the free boundary itself is deflected atthe wave incident point. The deflection of the free boundary is down-wards if an expansion wave hits it and the deflection is upwards whena compression wave hits it. It is important to note that in Figure 9.14,an expansion ray is shown to reflect as a compression wave from thefree boundary and a compression wave is shown to reflect as a singleexpansion wave. This should be taken as a representative expansionwave. In reality, the compression wave will reflect as an expansion fanfrom a free boundary and not as a single expansion ray.


From the above reflection, it is seen that on reflection from a wall, awave of ξ−type is changed to wave of η−type. The turning strengthof the reflected wave is the same as that of the incident wave, sincethe flow must return to the original direction, parallel to the wall. Fromthis process we can visualize that the wave reflection may be ′canceled′

by suitable accommodation of the portion of the wall after the incidentwave, i.e. there will not be reflection of wave as shown in Figure 9.15.The wall deflection is equal to the strength of the wave.


Expansion wave

∆θ

∆θ

∆θ

Figure 9.15Cancellation of wave reflection.


We can summarize the above reflection patterns as follows (Figure9.16).

∆θ1

Reflected wave

(∆θ1 − ∆θ2)

∆θ2

Incident waveM1

Figure 9.16Wave reflection.


In Figure 9.16, if ∆θ2 = ∆θ1, then there will not be any reflection. If∆θ2 < ∆θ1, then the reflected wave will be an expansion wave andwhen ∆θ2 > ∆θ1, the reflected wave will be a compression wave.

In a process involving a large number of reflections of expansion waves,the flow condition at any point in the flow field is given by

θ − θ1 = m − n, ν − ν1 = m + n (9.29)

where m is the number of expansion waves of equal strength, say 1◦,i.e. one family (ξ−type) crossed by the flow, and n is the number ofexpansion waves of equal strength, say 1◦, the other family (η−type)crossed by the flow. The initial flow field is given by (ν1, θ1).


More generally, the flow field condition (M, p, ρ, T ) at any point in aflow involving multiple reflections of expansion and compression wavesof both ξ−type and η − type is given by

θ − θ1 = m − n − k + l

(9.30)

ν − ν1 = m + n − k − l

where k is the number of compression waves on one family (ξ−type)crossed by the flow, l is the number of compression waves of otherfamily (η−type) crossed by the flow.


Example 9.2

Solve the flow field at the exit of an underexpanded two-dimensionalnozzle with air flow, shown in Figure E9.2. At the nozzle exit, MA =1.435 and θA = 0◦.

Exp

ansion

A

Com

pression

B C D E

Constant pressure free boundary

MA = 1.435 Exp

ansion

Com

pression

5

3

1

p/pA = 0.605

θA = 0◦

2 4

Figure E9.2


Solution

Because of symmetry, the streamline along the axis of the nozzle mustbe straight, and may be replaced by a solid wall, as shown in the figure.Given MA = 1.435 and θA = 0. For MA = 1.435, from isentropic table,we get νA = 10◦ and pA/p0 = 0.299. In this example, it is assumed thatthe entire expansion is taking place through a single expansion wave.

From our discussion on the Prandtl-Meyer function (Section 4.9), weknow that ν and θ are connected by the relation ν = ±θ, where theplus sign holds across a right-running characteristic and the minus signholds across a left-running characteristic.


Now, the entire expansion from region A to region B is taking placeacross a left-running characteristic and, therefore, θB = −νA = −10◦.

Also, by Eq. (4.49a), for the expansion

νB = νA + |θB − θA| = 10 + 10 = 20◦

With the value of the Prandtl-Meyer function, we can get the flow prop-erties. For νB = 20◦, from isentropic table,

MB = 1.775, pB/p0 = 0.181


After the expansion, the pressure ratio in region B is 0.181. At thefree boundary, the pressure outside the boundary must be equal to pB.Therefore, the pressure ratio at the free boundary for the given exitpressure pA is

ppA

=pB/p0

pA/p0=

0.1810.299

= 0.605

Following the above procedure, we can get the flow field as given inTable E9.2.


Table E9.2

Field ν M µ p/p0 θA 10◦ 1.435 44.2◦ 0.299 0◦

B 20◦ 1.775 34.3◦ 0.181 −10◦

C 30◦ 2.134 27.9◦ 0.104 0◦

D 20◦ 1.775 34.3◦ 0.181 10◦

E 10◦ 1.435 44.2◦ 0.299 0◦


The above solution can yield the wave and deflection angles also. Themean Mach angle µ for an expansion fan with µ1 and µ2 as the Machangles at the beginning and end of the fan is given by

µ =µ1 + µ2

2

Similarly, the mean deflection angle θ is given by

θ =θ1 + θ2

2


With the above relations, for the present flow field, we have

Wave µ θ

1 − 2 39.25◦ −5◦

2 − 3 31.1◦ 5◦

3 − 4 31.1◦ 5◦

4 − 5 39.25◦ −5◦

Note: In the above problem, all the regions are assumed to be simpleregions. There is a 10◦ deflection of the jet boundary as it leaves the nozzle.


Design of Supersonic Nozzle

In the design of Laval nozzle, we are looking for a proper geometry ofthe nozzle to accelerate the flow to result in uniform, parallel and wave-free supersonic flow. In Section 2.4, it has been highlighted that only ashape like the one shown in Figure 9.17 can produce such an uniformand unidirectional flow.


yM > 1

x

P

Q

N

Uniform flow

Sonic lineM = 1

M < 1 Me

θwmax

θw

Figure 9.17Supersonic (Laval) nozzle.


In order to accelerate a flow from subsonic to supersonic speed, theduct has to be convergent-divergent in shape, as shown in Figure 9.17.Further, for a supersonic convergent-divergent nozzle, it is essentialto have wave-free and parallel flow in the test-section (at the nozzleexit) at the desired Mach number. An improper contour will result inthe presence of weak waves, which may coalesce to form a finite shockand prevent a uniform flow in the test-section. Therefore, it is imperativeto have a proper design of nozzle contours for generation of uniformsupersonic flows.


The method of characteristics provides a technique for properly design-ing the contour of supersonic nozzle for shock-free, isentropic flow, tak-ing into account the multidimensional flow inside the duct. The purposehere is to illustrate the design of a supersonic nozzle by the method ofcomputation with weak waves (characteristics).


Consider the supersonic nozzle shown in Figure 9.17. The subsonicflow in the convergent portion of the nozzle is accelerated to sonicspeed at the throat. Generally, because of the multi-dimensionality ofthe converging subsonic flow, the sonic line is gently curved. However,in most applications, we assume the sonic line to be straight, as shownin Figure 9.17. In the divergent portion downstream of the throat, let θw

be the angle at any point P on the duct wall.


The portion of the nozzle with increasing θw is called the expansionsection, where expansion waves are generated and propagate in thedownstream direction, reflecting from the opposite wall. In Figure 9.17,because of symmetry, waves above the centre-line only are shown. Atpoint Q, there is an inflection of the duct wall contour and θw is maxi-mum. Downstream of Q, θw decreases until the wall becomes parallelto the x−direction at point N.


Supersonic nozzles with gradual expansions as illustrated in Figure9.17 are characteristics of wind tunnel nozzle where high-quality, uni-form and unidirectional flow is required in the test-section. Hence, windtunnel nozzles are long, with very smooth gradual expansion. But in ap-plications like rocket motors, nozzles are comparatively short in order tominimize weight. Also, in applications where rapid expansions are therequirements, such as the nonequilibrium flow in gas dynamic lasers,the nozzle length should be as short as possible.


In such cases, the expansion portion of the nozzle is shrunk to a pointand the expansion takes place through a centered Prandtl-Meyer waveemanating from a sharp-corner throat with an angle θwmax, as shown inFigure 9.18. The length L shown in Figure 9.18 is the minimum lengthpossible for shock-free, isentropic flow. If the contour is made within alength shorter than L, shocks will develop inside the nozzle.


Me

x

θwmax

y

M = 1Sonic line

L

Figure 9.18Minimum-length nozzle.


Contour design details

To illustrate the application of the method of characteristics for super-sonic nozzle design, let us consider the specific problem of designing aminimum length nozzle to expand the flow from M = 1 at the throat toM = 2.0 in the test-section (nozzle exit) where the flow is to be uniformand parallel to the flow direction at the throat.


Let us employ the region-to-region Method of characteristics for design-ing the contour. Since minimum length nozzle is to be designed, sharpcornered nozzle assumption will be made.

For characteristics it can be proved that,

1 along left-running characteristics or across right-running character-istic, ν − θ = constant;

2 along right-running characteristic or across left-running character-istic, ν + θ = constant;where ν is the Prandtl-Meyer function and θ is the flow turning an-gle.


In the region-to-region method, flow is divided into various regions bythe incident and reflected characteristics (from the centerline). Now,with the help of ν and θ, Mach numbers in the regions can be calculatedusing the above mentioned relations between ν and θ.

The sonic line at the throat is assumed to be straight and the design isdone for one-half of the nozzle, as the other half is only a mirror imageof the first, because of symmetry.


The characteristic lines and contour points for the proposed nozzle areshown in Figure 9.19. For a sharp-cornered nozzle,

θfan =νTS

2

where the subscript ′TS′ refers to the test-section.


θfan

0

B

A

AB = Sonic line

(13,

1)

(12,

2)

1 2 34 5 6

78

910 11

12 13 14

(14,0)

y

x

(13, 0)

O

M = 2

Figure 9.19Characteristic lines and contour points.


From Table 1 in the Appendix, for MTS = 2.0, νTS = 26.38◦. Hence,

θfan = 13.19◦

A total of 14 characteristics are considered in the fan, with the first beingat an angle of 0.19◦ with the sonic line and the rest being at a differenceof 1◦ to each other, as illustrated in Figure 9.19, i.e. all the waves (ex-cept the first) are of strength 1◦. The reflections from the centerline formthe regions. For example, characteristic 1 gives rise to 15 regions from(0,0) to (0,14), as shown in the figure.


The values of ν and θ at every region can be calculated as follows. Forthe regions formed by the first wave, we have

θ = 0◦, ν = 0 [region (0,0)]

θ = 0.19◦, ν = 0.19◦ [region (0,1)]

θ = 1.19◦, ν = 1.19◦ [region (0, 2)]

For the regions formed by the second wave,

θ = 0◦, ν = 0.38◦ [region (1,0)]

θ = 1◦, ν = 1.38◦ [region (1,1)]

etc.


Similarly, we can go up to region (14,0).

A computer program can be written for the calculation of θ and ν in everyregion. The program can take the input of a number n and divides theθfan into (13/n + 1) characteristics. In the present calculation n is takenas unity. (The program listing for calculating the value of ν and θ in theregions considered are given in the Appendix of GAS DYNAMICS by E.Rathakrishnan). Once ν is known, the Mach number can be obtainedfrom Table 1 in the Appendix. The values of ν and θ at different regionsshown in Figure 9.19 and the corresponding Mach numbers can beobtained in this way.


Note that, in the present nozzle design for Mach 2, the computed arearatio (considering the width in the z-direction as unity) is

Ae

A∗=

ye × 1y∗ × 1

=14.41

8.8

= 1.6375

The corresponding area ratio (for Mach 2 nozzle) from isentropic area-Mach number relation is Ae/A∗ = 1.688. Thus, the error in the result ofAe/A∗ computed by the method of characteristics is

error =(Ae/A∗)isen − (Ae/A∗)moc

(Ae/A∗)isen

× 100

=1.688 − 1.6375

1.688× 100

= 2.99%


For calculating the x−location of a contour point i , the following formulamay be used:

xi =(A/A∗)i − (A/A∗)i−1

2 tan (θi−1)y∗ + xi−1

yi = (A/A∗)i y∗

where i = 1, 2, . . . , 14. In these equations,

θi−1 = turning angle in region i − 1

(A/A∗)i = area ratio at point i

(A/A∗)i−1 = area ratio at point i − 1

A∗ = area at throat

y∗ = y−coordinate at the throat


Also,

(A/A∗)0 = area ratio at throat = 1

θ0 = 13.19◦ (in the present case)

The area ratio at a particular point may be calculated from Eq. (2.32)or Table 1 in the Appendix as the Mach number at that point is known.The area ratio also gives the y−location of the point. Table 9.1 showsthe x and y coordinates of contour points. These contour points alongwith characteristics are shown in Figure 9.19.


Table 9.1 Coordinates of contour points

Contour points Throat 1 2 3 4 5 6 7x mm 0.0 7.77 7.95 9.01 10.25 11.69 13.38 15.40y mm 8.8 10.62 10.66 10.89 11.13 11.38 11.65 11.93

8 9 10 11 12 13 1417.84 20.85 24.66 29.69 36.78 48.01 71.7212.23 12.55 12.88 13.24 13.61 14.00 14.41


The resulting nozzle contour given by the calculated points is shownin Figure 9.20. A nozzle fabricated as per the contour in this figuregenerated a uniform parallel supersonic stream with Mach number 1.97at the gas dynamics laboratory at the Indian Institute of TechnologyKanpur.

M = 28.8 M∗ = 1

Nozzle contour

14

20 77.17 50Test-section

15

Figure 9.20Laval nozzle contour generated by method of characteristics.


In the above calculations, viscosity has been neglected. But in actualflow, the boundary layer on the nozzle and side walls will have a displac-ing effect which will reduce the effective height and width of the nozzle.Allowance for this should be made by adding a correction for boundarylayer growth to the designed contour. Finally, we should note that inthe present example, calculation procedure has been very much sim-plified by assuming the flow in the nozzle to be two-dimensional. Foraxisymmetric and three-dimensional flow, the strength of a wave, gen-erally, varies continuously in space and, therefore, the simple relationsbetween ν and θ used in the present case is no longer valid.


Summary

Method of characteristics is basically a numerical technique. Charac-teristics are weak waves across which there is a jump in the gradientsof flow properties.

The general features of the characteristics are the following.

They exist only in supersonic flows.

They are coincident with Mach lines.

On the characteristics the derivatives of flow properties are discon-tinuous, while the flow properties themselves are continuous.

On the characteristics the dependent variables satisfy a certain re-lation known as the compatibility relation.


The compatibility relations are

ν − θ = R (constant) along η−characteristics

(9.15)ν + θ = Q (constant) along ξ−characteristics

These relations provide the key to the method at computation. Thecompatibility relations, which are independent of the specific flow ge-ometry, lead to the result that the flow changes encountered in crossingcharacteristics which are straight are given by

∆ν = ±∆θ (9.28)

where the plus sign is for ξ−characteristics and the minus sign is forη−characteristics.


On a rigid boundary a wave is reflected as a wave of the same sense(of opposite family) and on an open or free boundary a wave is reflectedas a wave of opposite sense.

The deflection of a free boundary is downwards if an expansion wavehits it and the deflection is upwards when a compression wave hits it.

The wave reflection may be canceled by suitable accommodation of theportion of the wall after the incident wave.

The method of characteristics provides a technique for the proper de-sign of supersonic nozzle for shock-free, isentropic flow. Centered ex-pansion of the flow at the throat results in a short length nozzle andcontinuous expansion at the throat results in a long nozzle.


For calculating the x and y coordinates of a contour point i , the followingformula may be used.

xi =(A/A∗)i − (A/A∗)i−1

2 tan (θi−1)y∗ + xi−1

yi = (A/A∗)i y∗

where

θi−1 = flow turning angle in region i − 1

(A/A∗)i = area ratio at point i

(A/A∗)i−1 = area ratio at point i − 1

A∗ = area at throat

y∗ = y−coordinate at the throat

(A/A∗)0 = area ratio at throat = 1

θ0 = θfan


The close agreement between the design and measured Mach num-bers experienced by experimental researchers highlights the validity ofthis method for the design of supersonic nozzles for practical applica-tions such as supersonic wind tunnels.


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