Elements of the Method of Characteristics

In this section, we only introduce the basic elements of the method of characteristics. A full discussion is beyond the scope of this book; see References 21, 25, and 34 for more details.

Consider a two-dimensional, steady, inviscid, supersonic flow in xy space, as given in Figure 13.2a. The flow variables (p, и, T, etc.) are continuous throughout this space. However, there are certain lines in xy space along which the derivatives of the flow-field variables (др/дх, du/dy, etc.) are indeterminate and across which may even be discontinuous. Such lines are called characteristic lines. This may sound strange at first; however, let us prove that such lines exist, and let us find their precise directions in the xy plane.

In addition to the flow being supersonic, steady, inviscid, and two-dimensional, assume that it is also irrotational. The exact governing equation for such a flow is given by Equation (11.12):

ІІ 1.12]

[Keep in mind that we are dealing with the full velocity potential ф in Equation

(11.12) , not the perturbation potential.] Since дф/dx = и and дф/ду — v, Equation

(11.12) can be written as

The velocity potential and its derivatives are functions of x and y, for example,

3 ф

— = f{x, y) dx

Hence, from the relation for an exact differential,

Examine Equations (13.1) to (13.3) closely. Note that they contain the second deriva­tives д2ф/дх2, д2ф/ду2, and д2ф/дхду. If we imagine these derivatives as “un­knowns,” then Equations (13.1), (13.2), and (13.3) represent three equations with three unknowns. For example, to solve for д2ф/дх dy, use Cramer’s rule as follows:

where N and D represent the numerator and denominator determinants, respectively. The physical meaning of Equation (13.4) can be seen by considering point A and its surrounding neighborhood in the flow, as sketched in Figure 13.3. The derivative Ь2ф/Ьх dy has a specific value at point A. Equation (13.4) gives the solution for д2ф/дх dy for an arbitrary choice of dx and dy. The combination of dx and dy defines an arbitrary direction ds away from point A as shown in Figure 13.3. In general, this direction is different from the streamline direction going through point A. In Equation (13.4), the differentials du and dv represent the changes in velocity that take place over the increments dx and dy. Hence, although the choice of dx and dy is arbitrary, the values of du and dv in Equation (13.4) must correspond to this choice. No matter what values of dx and dy are arbitrarily chosen, the corresponding values of du and dv will always ensure obtaining the same value of d2ф/dx dy at point A from Equation (13.4).

The single exception to the above comments occurs when dx and dy are chosen so that D = 0 in Equation (13.4). In this case, d2ф/дxdy is not defined. This situation will occur for a specific direction ds away from point A in Figure 13.3, defined for that specific combination of dx and dy for which I) = 0. However, we know that d2ф/dx 3у has a specific defined value at point A. Therefore, the only

consistent result associated with D = 0 is that N = 0, also; that is,

д2ф _ N _ 0 Эх ду ~ Ъ ~ 0

Here, д2ф/дх ду is an indeterminate form, which is allowed to be a finite value, that is, that value of д2ф/дх ду which we know exists at point A. The important conclusion here is that there is some direction (or directions) through point A along which д2ф/Эх ду is indeterminate. Since д2ф/дх ду — ди/ду = dv/dx, this implies that the derivatives of the flow variables are indeterminate along these lines. Hence, we have proven that lines do exist in the flow field along which derivatives of the flow variables are indeterminate; earlier, we defined such lines as characteristic lines.

Consider again point A in Figure 13.3. From our previous discussion, there are one or more characteristic lines through point A. Question: How can we calculate the precise direction of these characteristic lines? The answer can be obtained by setting D = 0 in Equation (13.4). Expanding the denominator determinant in Equation

(13.4) , and setting it equal to zero, we have

In Equation (13.6), dy/dx is the slope of the characteristic lines; hence, the subscript “char” has been added to emphasize this fact. Solving Equation (13.6) for (dy /dx )ctm by means of the quadratic formula, we obtain

/dy —2uv/a2 ± y/(2uv/a2)2 — 4(1 — u2/a2)( 1 — v2/a2)

dx)c har 2(1 — u2/a2)

/dy —uv/a2 ± л/(и2 + v2)/a2 — 1

dx ) char 1-м2 /a2

From Figure 13.3, we see that и = V cos в and v = V sin в. Hence, Equation (13.7) becomes

/dy (—V2cos0 sin0)/a2 ± _

W/char 1 – [(V2/a2)cos20] l3’8

Recall that the local Mach angle p. is given by p. = sin^'(l/M), or sin p, = 1 /М. Thus, V2/a2 = M2 = 1/ sin2 p,, and Equation (13.8) becomes

/ dy (—cost? sin в)/sin2 p. ± vTcos^’+^in^X/sin^Ti^^

dx )char 1 – (COS2 0)/Sin2/Г

After considerable algebraic and trigonometric manipulation, Equation (13.9) reduces to

[13.10]

Equation (13.10) is an important result; it states that two characteristic lines run through point A in Figure 13.3, namely, one line with a slope equal to tan(6 — ;u) and the other with a slope equal to tan (б + /і). The physical significance of this result is illustrated in Figure 13.4. Here, a streamline through point A is inclined at the angle в with respect to the horizontal. The velocity at point A is V, which also makes the angle в with respect to the horizontal. Equation (13.10) states that one characteristic line at point A is inclined below the streamline direction by the angle /x this characteristic line is labeled as C in Figure 13.4. Equation (13.10) also states that the other characteristic line at point A is inclined above the streamline direction by the angle /x this characteristic line is labeled as C+ in Figure 13.4. Examining Figure 13.4, we see that the characteristic lines through point A are simply the left – and right-running Mach waves through point A. Hence, the characteristic lines are Mach lines. In Figure 13.4, the left-running Mach wave is denoted by C+, and the right-running Mach wave is denoted by C_. Hence, returning to Figure 13.2a, the characteristics mesh consists of left- and right-running Mach waves which crisscross the flow field. There are an infinite number of these waves; however, for practical calculations we deal with a finite number of waves, the intersections of which define the grid points shown in Figure 13.2a. Note that the characteristic lines are curved in space because (1) the local Mach angle depends on the local Mach number, which is

a function of x and y, and (2) the local streamline direction 9 varies throughout the flow.

The characteristic lines in Figure 13.2a are of no use to us by themselves. The practical consequence of these lines is that the governing partial differential equations which describe the flow reduce to ordinary differential equations along the charac­teristic lines. These equations are called the compatibility equations, which can be found by setting N = 0 in Equation (13.4), as follows. When N = 0, the numerator determinant yields

/ u2 ( v2

I 1—— — J du dy + I 1——- 1 dx dv = 0

dv —(1 — u2/a2) dy

or — = ———- —Ц—— [13.11]

du 1 — Vі/a2 dx

Keep in mind that N is set to zero only when D = 0 in order to keep the flow – field derivatives finite, albeit of the indeterminate form 0/0. When D = 0, we are restricted to considering directions only along the characteristic lines, as explained earlier. Hence, when N = 0, we are held to the same restriction. Therefore, Equation

(13.11) holds only along the characteristic lines. Therefore, in Equation (13.11),

<У = / dy

dx ~ у dx ) ch^

Substituting Equations (13.12) and (13.7) into (13.11), we obtain

dv 1 — и2 /a2 —uv/a2 ± (u2 + v2)/a2 — 1

du 1 — v2/a2 1 — u2/a2

dv uv/a2 =F у/(и2 + v2)/a2 — 1

du 1 — v2/a2

Recall from Figure 13.3 that и = V cos 9 and v = V sind. Also, (и2 + v2)/a2 = V2/а2 = M2. Hence, Equation (13.13) becomes

£?(Vsin0) M2 cos 9 sin в VM2 — 1 d(V cos9) 1-М2 sin2 9

which, after some algebraic manipulations, reduces to

Examine Equation (13.14). It is an ordinary differential equation obtained from the original governing partial differential equation, Equation (13.1). However, Equation (13.14) contains the restriction given by Equation (13.12); that is, Equation (13.14) holds only along the characteristic lines. Hence, Equation (13.14) gives the com­patibility relations along the characteristic lines. In particular, comparing Equation

(13.14) with Equation (13.10), we see that

(applies along the C characteristic) [13.15] (applies along the C+ characteristic) [13.16]

Examine Equation (13.14) further. It should look familiar; indeed, Equation (13.14) is identical to the expression obtained for Prandtl-Meyer flow in Section 9.6, namely, Equation (9.32). Hence, Equation (13.14) can be integrated to obtain a result in terms of the Prandtl-Meyer function, given by Equation (9.42). In particular, the integration of Equations (13.15) and (13.16) yields

в + v(M) = const = K_ (along the C_ characteristic) [13.17]

в — v(M) = const = K+ (along the C+ characteristic) [13.18]

In Equation (13.17), K – is a constant along a given C_ characteristic; it has different values for different C_ characteristics. In Equation (13.18), К t is a constant along a given C+ characteristic; it has different values for different C+ characteristics. Note that our compatibility relations are now given by Equations (13.17) and (13.18), which are algebraic equations which hold only along the characteristic lines. In a general inviscid, supersonic, steady flow, the compatibility equations are ordinary differential equations; only in the case of two-dimensional irrotational flow do they further reduce to algebraic equations.

What is the advantage of the characteristic lines and their associated compatibility equations discussed above? Simply this—to solve the nonlinear supersonic flow, we need deal only with ordinary differential equations (or in the present case, algebraic equations) instead of the original partial differential equations. Finding the solution of such ordinary differential equations is usually much simpler than dealing with partial differential equations.

How do we use the above results to solve a practical problem? The purpose of the next section is to give such an example, namely, the calculation of the supersonic flow inside a nozzle and the determination of a proper wall contour so that shock waves do not appear inside the nozzle. To carry out this calculation, we deal with two types of grid points: (1) internal points, away from the wall, and (2) wall points. Characteristics calculations at these two sets of points are carried out as follows.