O. N. KATSKOVA, P. I. CHUSHKIN

ON ONE SCHEME OF THE NUMERICAL METHOD OF CHARACTERISTICS

(Presented by Academician A. A. Dorodnitsyn, 27 VII 1963)

For the numerical calculation of a supersonic steady gas flow around three-dimensional bodies in the region between the shock wave and the surface of the body, one may apply the three-dimensional method of characteristics or a finite-difference method. However, the practical realization of these methods on electronic computers requires the creation of very complex programs. In this connection, one numerical method appears highly promising, the idea of which is as follows.

Let us use the variables \(x\), \(\xi=(r-r_{\text{t}})/(r_{\text{w}}-r_{\text{t}})\), \(\varphi\), where \(x,r,\varphi\) are cylindrical coordinates; \(r=r_{\text{t}}(x,\varphi)\) is the prescribed equation of the body; \(r=r_{\text{w}}(x,\varphi)\) is the sought equation of the shock wave. Considering a system of equally spaced meridional planes \(\varphi=\mathrm{const}\), we shall approximate, in the three-dimensional equations of the problem, the functions by trigonometric polynomials in \(\varphi\) with interpolation nodes on these planes. Then the original system of three-dimensional equations is reduced to a system of two-dimensional equations in \(x\) and \(\xi\), relating the values of the functions on the individual meridional planes. To solve this hyperbolic system we shall apply the two-dimensional method of characteristics, in which the calculation is carried out by layers bounded by the planes \(x=\mathrm{const}\), and in which the characteristics on each new layer pass through selected points with values \(\xi=\mathrm{const}\).*

In order to test such a scheme of the numerical method of characteristics by layers, in the present note we shall apply it to a particular case—the calculation of supersonic nonisentropic axisymmetric flow of a perfect gas around a body of revolution with a contour without breaks. For convenience of machine calculations, as in \((^{2,3})\), we take as the principal functions \(\beta=\sqrt{\mathrm{M}^{2}-1}\), \(\zeta=\tan\theta\), the entropy \(S=\ln(p/\rho^{\chi})\), and the radius \(r\), where \(\mathrm{M}\) is the Mach number; \(\theta\) is the angle of inclination of the velocity vector to the flow axis; \(\rho\) is the dimensionless density, referred to the density of the oncoming flow \(\rho_{\infty}\); \(p\) is the dimensionless pressure, referred to \(\rho_{\infty}a_{\mathrm{cr}}^{2}\) (\(a_{\mathrm{cr}}\) is the critical speed of sound); \(\chi\) is the adiabatic exponent. Then, in the axisymmetric case under consideration, in the variables \(x,\xi\), the differential equations and compatibility relations for the characteristics of the first and second families, as well as the differential equation of the streamlines, will have, respectively, the form

\[ \frac{d\xi}{dx} = \frac{1}{\delta} \left( \frac{\beta\zeta+1}{\beta-\zeta} -\lambda \right) \equiv A_{1}, \qquad d\zeta+K\,d\beta+L\,dx-P\,dS=0, \tag{1} \]

\[ \frac{d\xi}{dx} = \frac{1}{\delta} \left( \frac{\beta\zeta-1}{\beta+\zeta} -\lambda \right) \equiv A_{2}, \qquad d\zeta-J\,d\beta-N\,dx+Q\,dS=0, \tag{2} \]

\[ \frac{d\xi}{dx} = \frac{1}{\delta}(\zeta-\lambda) \equiv A_{3}, \tag{3} \]

* In work (1) a scheme of the numerical method of characteristics by layers is considered, but applied to the calculation of one-dimensional unsteady gas flows; moreover, it does not require fulfillment of the last condition. A scheme of the method of characteristics by layers, but only for the case of isentropic one-dimensional gas flows, is also given in (5).

where

\[ \delta=r_{\mathrm{в}}-r_{\mathrm{т}},\qquad \lambda=\left(\frac{dr_{\mathrm{в}}}{dx}-\frac{dr_{\mathrm{т}}}{dx}\right)\xi+\frac{dr_{\mathrm{т}}}{dx}, \]

\[ J=K=-\frac{2\beta^2(\zeta^2+1)} {(\chi+1)(\beta^2+1)(\varepsilon\beta^2+1)},\qquad L=\frac{\zeta(\zeta^2+1)}{r(\beta-\zeta)}, \]

\[ N=\frac{\zeta(\zeta^2+1)}{r(\beta+\zeta)},\qquad P=Q=\frac{\beta(\zeta^2+1)} {\chi(\chi-1)(\beta^2+1)},\qquad \varepsilon=\frac{\chi-1}{\chi+1}, \]

and

\[ r_{\xi}=\delta \xi+r_{\mathrm{т}}. \tag{4} \]

We shall now describe the computational scheme. Suppose that on some layer \(x=x_0\), located between the shock wave \((\xi=1)\) and the body \((\xi=0)\) in the supersonic region of the flow, the values of the basic functions are known at a number of points \(n\), with the corresponding \(\xi=\xi_n=\mathrm{const}\). We shall find the values of these functions on the next layer \(x_0+\Delta x\).

The calculation begins with the determination of point 3 on the shock wave (Fig. 1a), and is carried out here by selecting the value \(\tau=dr_{\mathrm{в}}/dx\), the tangent of the angle of inclination of the shock wave to the \(x\)-axis. Assigning a value \(\tau_3\) close to the value of \(\tau\) at point \(n+1\) on the shock wave on the layer \(x_0\), we compute, from the relations on the shock wave, the quantities

Fig. 1

\[ \rho_3=\frac{w_\infty^2\tau_3^2}{1-\varepsilon w_\infty^2+\tau_3^2},\qquad \zeta_3=\frac{\tau_3(\rho_3-1)}{\tau_3^2+\rho_3} \left( w_\infty^2=\frac{(\chi+1)M_\infty^2}{2+(\chi-1)M_\infty^2} \right), \]

\[ \beta_3=\sqrt{\frac{w_3^2-1}{1-\varepsilon w_3^2}},\qquad w_3^2=\frac{1}{1+\tau_3^2} \left[ w_\infty^2+\frac{1}{\rho_3}(1-\varepsilon w_\infty^2+\tau_3^2) \right], \]

\[ S_3=\ln\left[ \frac{2}{\chi+1}\, \frac{w_\infty^2\tau_3^2}{1+\tau_3^2} -\frac{\chi-1}{2\chi}(1-\varepsilon w_\infty^2) \right] -\chi\ln\rho_3, \]

\[ r_3=r_{\mathrm{в}}=r_{n-1}+\frac{\Delta x}{2}(\tau_{n-1}+\tau_3). \]

Next, drawing from point 3 a characteristic of the first family \(I\), we write equations (1) in finite-difference form:

\[ \xi_1=\xi_3-A_1\Delta x\qquad(\xi_3=1), \tag{5} \]

\[ \zeta_3-\zeta_1+K(\beta_3-\beta_1)+L\Delta x-P(S_3-S_1)=0. \tag{6} \]

Formula (5), in which \(A_1\) is computed from the quantities at point 3, gives \(\xi_1\). By quadratic interpolation to this value \(\xi_1\) from the points \(n-1\), \(n\), and \(n+1\) on the known layer \(x_0\), we determine \(\beta_1\), \(\zeta_1\), and \(S_1\), and from formula (4) we find \(r_1\). Then, by satisfying equality (6), the correctness of the choice of \(\tau_3\) is checked. The subsequent approximations, in which the values of \(A_1\) in (5) averaged over points 1 and 3 are now taken, make it possible to determine \(\tau_3\) with the required accuracy. After the selection is completed, one can compute the generalized stream function \(\Psi\), which is related to the ordinary stream function \(\psi\) by the relation \(d\Psi=(\chi e^S)^{1/(\chi-1)}d\psi\). On the shock wave at point 3 we have

\[ \Psi_{\mathrm{в}}=\Psi_{n-1} +\chi^{\frac{1}{\chi-1}}\frac{w_\infty}{4} \left( \exp\frac{S_1}{\chi-1} +\exp\frac{S_3}{\chi-1} \right) (r_3^2-r_{n-1}^2). \tag{7} \]

After computing the point on the shock wave, we proceed to the determination of the functions at an interior point 3 of the flow field, corresponding to some prescribed \(\xi_3=\xi_n\) (Fig. 1b). From point 3 we draw the characteristics of the first family \(I\) and of the second family \(II\), and the streamline \(III\). Writing equations (1)—(3) in finite-difference form, we obtain, for determining the unknown quantities \(\beta_3\), \(\xi_3\), and \(S_3\), a system that must be solved by iteration; in the first iteration the values of the functions at point 3 are taken to be the same as at point \(n\) on the layer \(x_0\). First \(\xi_1\), \(\xi_2\), and \(\xi_4\) are computed from the formulas

Fig. 2

\[ \xi_i=\xi_3-A_i\Delta x \qquad (i=1,\,2,\,4), \tag{8} \]

where in the first iteration \(A_i\) are determined from point \(n\). With the aid of quadratic interpolation to these values \(\xi_i\) at points \(n-1\), \(n\), and \(n+1\), one finds \(\beta_1,\beta_2,\xi_i\), and \(S_i\) \((i=1,\,2,\,4)\), and from formula (4), \(r_1\) and \(r_2\). The refined values of the sought functions at point 3 are obtained from the expressions

\[ \beta_3=\frac{1}{K+J}\{\xi_1+K\beta_1-(L+N)\Delta x-P(S_3-S_1)-\xi_2+J\beta_2+Q(S_3-S_2)\}, \]

\[ \xi_3=J(\beta_3-\beta_2)+N\Delta x-Q(S_3-S_2), \qquad S_3=S_4, \]

where the coefficients are taken averaged over points 1 and 3 or 2 and 3. Subsequent iterations are computed analogously, but in them the coefficients \(A_i\) are taken averaged. Correct signs are usually obtained after three iterations.

When computing point 3 on the body contour \((\xi_3=0)\), it is necessary to determine only the quantity \(\beta_3\), since the quantities \(r_3=r_t\), \(\zeta_3=dr_t/dx\), and \(S_3\) are known here. Draw the characteristic of the second family \(II\) from point 3 (Fig. 1b). Taking in the first iteration \(\beta_3=\beta_{n+1}\), we find from formula (8), for \(i=2\), the quantity \(\xi_2\), and by quadratic interpolation over the points \(n-1\), \(n\), and \(n+1\) we obtain \(\beta_2,\zeta_2\), and \(S_2\), and from expression (4), \(r_2\). The refined value \(\beta_3\) is determined as follows:

\[ \beta_3=\beta_2[\xi_3-\xi_2-N\Delta x+Q(S_3-S_2)]. \]

Subsequent iterations are carried out analogously. From the final value of \(\beta_3\), the pressure on the body is found:

\[ p=e^{-S/(\varkappa-1)}[\varkappa(\varepsilon\beta^2+1)]^{-\varkappa/(\varkappa-1)}, \]

and by integrating \(p\) along the body contour the drag coefficient \(c_x\) is obtained.

After the functions have been computed on the entire layer, the value of the stream function \(\Psi_t\) on the body is computed from them:

\[ \Psi_t=\Psi_{\mathrm{B}}-\delta\int_0^1 r\sqrt{\frac{\beta^2+1}{(\varepsilon\beta^2+1)^{1/\varepsilon}}}\,d\xi, \]

where \(\Psi_{\mathrm{B}}\) is given by equality (7). The obtained \(\Psi_t\) is compared with its known value on the body, which serves as a check on the accuracy of the entire computation. The step size \(\Delta x\) must be related to the step \(\Delta\xi\) and is chosen from considerations of accuracy and stability of the computations.

The developed scheme of the numerical method of characteristics by layers was tested on the example of computing a supersonic flow region past blunt bodies placed in an air stream ($\chi = 1.4$) with Mach number $M_\infty=\infty$. The cases considered were a cone with semi-vertex angle $\omega = 5^\circ$ and a cylinder $\omega = 0^\circ$ with spherical blunt noses. The flow was computed from the known initial data (4) in the section $x/R = 1$, where $R$ is the radius of the blunt nose (the origin of coordinates is placed at the front point of the body). The results of the computations are given in Fig. 2, where the pressure distribution over the body (solid line) and the shape of the shock wave (dashed line) are plotted. On the same graphs, the corresponding computational data (4), obtained by the ordinary method of characteristics, are plotted as points. Comparison of the results shows that the considered scheme of the method of characteristics by layers gives good accuracy. Let us note that the same accuracy is also obtained when comparing the flow fields. We also point out that the scheme presented is readily generalized to the case of a real gas in a state of thermodynamic equilibrium.

Computing Center
Academy of Sciences of the USSR

Received
23 VII 1963

CITED LITERATURE

1. S. K. Godunov, Difference Methods for Solving Equations of Gas Dynamics, Novosibirsk, 1962.
2. P. I. Chushkin, Prikl. matem. i mekh., 24, no. 5 (1960).
3. O. N. Kachkova, I. N. Naumova, Yu. D. Shmyglevskii, N. P. Shulishnina, Experience in Computing Plane and Axisymmetric Supersonic Gas Flows by the Method of Characteristics, Moscow, 1961.
4. P. I. Chushkin, N. P. Shulishnina, Tables of Supersonic Flow around Blunted Cones, Moscow, 1961.
5. M. Lister, “Mathematical Methods for Digital Computers,” Sec. 15, N. Y.—London, 1960, p. 165.