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HYDROMECHANICS
O. S. VOROB'EV
APPROXIMATE ANALYTICAL REPRESENTATION OF PLANE SUPERSONIC GAS FLOWS
(Presented by Academician L. I. Sedov on 3 VI 1958)
Integration of the equations
1. The equations of motion describing supersonic flows of a compressible gas have the form \((^1)\):
\[ \frac{\partial^2\varphi}{\partial\lambda\,\partial\mu} = F(t)\left(\frac{\partial\varphi}{\partial\lambda} + \frac{\partial\varphi}{\partial\mu}\right), \qquad \frac{\partial^2\psi}{\partial\lambda\,\partial\mu} = -F(t)\left(\frac{\partial\psi}{\partial\lambda} + \frac{\partial\psi}{\partial\mu}\right), \tag{1,1} \]
where \(\psi\) and \(\varphi\) are, respectively, the stream function and the velocity potential; \(\lambda\) and \(\mu\) are characteristic variables connected with the inclination angle of the velocity \(\theta\) and the variable \(t\) as follows: \(t-\theta=2\lambda,\ t+\theta=2\mu\).
The relation between the variable \(t\) and the velocity is determined from the system
\[ \frac{d\rho}{dt} = -\frac{\rho^2K+1}{\sqrt{K}}, \qquad \frac{1}{\rho v} = -\sqrt{K}\,\frac{d}{dt}\left(\frac{1}{v}\right), \qquad F(t)=\frac{1}{2}\frac{d}{dt}(\ln\sqrt{K}), \tag{1,2} \]
where \(\rho\) is the ratio of the gas density to the stagnation density; \(v\) is the velocity magnitude.
In the case of adiabatic gas flow \(t=\nu(M)\), where
\[ \nu = \sqrt{\frac{\chi+1}{\chi-1}}\, \operatorname{arctg} \sqrt{\frac{\chi-1}{\chi+1}\,(M^2-1)} - \operatorname{arctg}\sqrt{M^2-1}, \tag{1,3} \]
\[ F= \frac{\chi+1}{4}\, \frac{M^4}{(M^2-1)^{3/2}} . \tag{1,4} \]
Let us represent approximately
\[ F=\frac{k}{c-\nu}, \tag{1,5} \]
where \(k\) and \(c\) are arbitrary constants.
The solution of system \((1,1)\) for integral \(k\), given by Darboux, is represented in the form
\[ \varphi = \frac{\partial^{2k-2}}{\partial\lambda^{k-1}\partial\mu^{k-1}} \left[ \frac{f_1(\lambda)+f_2(\mu)}{\lambda+\mu-c} \right]. \tag{1,6} \]
Equating the derivatives with respect to \(\nu\) of the inverse functions \(F\), defined by relations \((1,4)\) and \((1,5)\), for \(M\to\infty\), we obtain \(k=(\chi+1)/2(\chi-1)\). For \(\chi=1.4\) and \(\chi=1.67\), \(k\) is equal to the integers 3 and 2, respectively.
S. A. Khristianovich \((^2)\) proposed approximating the solution of system \((1,1)\) by a solution of the type \((1,6)\) with \(k=\pm1\). Figure 1 shows the graph of the exact function \(F\) for \(\chi=1.4\) according to \((1,4)\), and points are plotted corresponding to the approximate values of \(F\) computed from \((1,5)\) for \(k=3\). The approximating curves chosen by S. A. Khristianovich \((^2)\) and G. A. Dombrovskii \((^3)\) are also indicated there. The best agreement with the exact values occurs in the case of approximation by \((1,5)\).
By choosing the constant \(c\), the approximation curve can be drawn in such a way that it will have either one point of intersection with the exact curve, or a point of intersection and tangency as \(M \to \infty\), or two points of intersection, or one point of tangency at \(M=4.57\), or no common point at all.
Fig. 1. \(1\)—according to Khristianovich (2); \(2, 3, 4\)—according to Dombrovskii (3);
\(5\)—exact dependence; points—the proposed approximation
- We integrate the system (1,3), using the fact that \(F(t)=3/(c-t)\). We obtain
\[ \frac{1}{\rho \bar v}=-(c_1y_1+c_2y_2); \tag{2,1} \]
\[ \frac{1}{\bar v}=-\frac{1}{A}(c_1z_1+c_2z_2); \tag{2,2} \]
\[ \sqrt{K}=\frac{A}{(c-t)^6}, \tag{2,3} \]
where \(\bar v\) is the ratio of the velocity to the critical velocity; \(A, c_1\), and \(c_2\) are arbitrary constants; \(y_1, y_2, z_1\), and \(z_2\) are functions of the argument \(\sigma=c-t\):
\[ y_1=\frac{1}{\sigma^3} \left[ \left(1-\frac{3}{\sigma^2}\right)\sin\sigma+\frac{3}{\sigma}\cos\sigma \right], \qquad y_2=\frac{1}{\sigma^3} \left[ \left(1-\frac{3}{\sigma^2}\right)\cos\sigma-\frac{3}{\sigma}\sin\sigma \right], \]
\[ z_1=\sigma(15-\sigma^2)\cos\sigma-(15-6\sigma^2)\sin\sigma,\qquad z_2=-\sigma(15-\sigma^2)\sin\sigma-(15-6\sigma^2)\cos\sigma . \]
Fig. 2. \(1\)—\(c=2.28,\ A=294,\ c_1=8.0,\ c_2=8.0\); \(2\)—\(c=2.28,\ A=294,\)
\(c_1=8.5,\ c_2=8.0\); \(3\)—\(c=2.28\); \(4\)—\(c=2.278\)
Using Bernoulli’s equation and relations (1,2), it is easy to obtain an expression for the pressure:
\[ p=p_*-\rho_0\int_{t_*}^{t}\frac{\bar v^2}{\sqrt{K}}\,dt . \tag{2,4} \]
Choosing the constants \(c, A, c_1, c_2\) and \(p_*\), one can obtain fourth-order tangency of the approximation curve \(p(\rho)\) with the curve of adiabatic pressure variation.
One can draw a more general conclusion: a broad class of functions \(F(t)\) chosen by us, with an arbitrary constant, will make it possible to construct a function \(p(\rho)\) having fourth-order contact with the adiabatic dependence of \(p\) on \(\rho\).
The general solution of system (1.1) will depend essentially on the form of the function \(F(t)\), and, consequently, if the function \(F(t)\) is chosen insufficiently close to the adiabatic one, the solution obtained will give the correct result only in a narrow range of variation of the density.
Figure 2 gives the relative deviations of the approximating function \(F\) from the adiabatic one for different \(c\). There are also given the relative deviations of the functions \(1/\vartheta\), computed from (2.1) and (2.2), from the corresponding functions of the adiabatic motion. The indicated approximation gives a very accurate result for all Mach numbers, beginning with \(M=2.5\).
Boundary-value problems
We shall carry out the solution of boundary-value problems for the case \(x=1.4\). In this case the solution has the form
\[ \varphi(\lambda,\mu)=\frac{2}{(\lambda+\mu-c)^3} \left\{ f_1''(\lambda)+f_2''(\mu) -\frac{6[f_1'(\lambda)+f_2'(\mu)]}{\lambda+\mu-c} +\frac{12[f_1(\lambda)+f_2(\mu)]}{(\lambda+\mu-c)^2} \right\}, \tag{3.0} \]
where \(f_1(\lambda)\) and \(f_2(\mu)\) are arbitrary functions determined from the boundary conditions.
3. The Goursat problem. Along the characteristics \(\lambda=\lambda_0\) and \(\mu=\mu_0\) there are prescribed, respectively, \(\varphi=\varphi_1(\mu)\) and \(\varphi=\varphi_2(\lambda)\). We shall assume that \(\varphi_1(\mu_0)=\varphi_2(\lambda_0)=0\). Introduce the notation
\[ \lambda^*=\lambda+\mu_0-c,\qquad \mu^*=\lambda_0+\mu-c. \tag{3.1} \]
Then the solution of the Goursat problem can be written in the following form:
\[ f_1(\lambda)=\frac{1}{2}\lambda^{*3} \int_{\lambda_0}^{\lambda}\int_{\lambda_0}^{\lambda}\varphi_2(\lambda)\,d\lambda\,d\lambda . \tag{3.2} \]
The turning of a given flow about a corner point is obtained as a special case of the Goursat problem if one of the functions \(\varphi_1\) or \(\varphi_2\) is set equal to zero.
4. The Cauchy problem. Along a known curve
\[ \lambda=\lambda_*(\mu)\quad \text{or}\quad \mu=\mu_*(\lambda) \tag{4.1} \]
there are prescribed
\[ \varphi=\varphi_1(\lambda)=\varphi_2(\mu),\qquad \psi=\psi_1(\lambda)=\psi_2(\mu). \tag{4.2} \]
Using the basic equations (1.1) and the solution (3.0), we obtain that
\[ \frac{d^5 f_1}{d\lambda^5} =30\varphi_1(\lambda) +\frac{3}{2}\sigma_1\left[(\mu_*'^2+6\mu_*'+15)\varphi_1'(\lambda) -2\mu_*'^2\sqrt{K(\sigma_1)}\psi_1'(\lambda)\right]+ \]
\[ +\frac{3}{4}\sigma_1^2\left\{ 2(\mu_*'+3)\varphi_1''(\lambda) +2\mu_*'\sqrt{K(\sigma_1)}\psi_1''(\lambda) +\mu_*''\left[\varphi_1'(\lambda)+\sqrt{K(\sigma_1)}\psi_1'(\lambda)\right] \right\}+ \]
\[ +\frac{\sigma_1^3}{4} \left[\varphi_1'''(\lambda)-\sqrt{K(\sigma_1)}\psi_1'''(\lambda)\right], \tag{4.3} \]
where \(\sigma_1=\lambda+\mu_*(\lambda)-c\).
The formula for \(d^5f_2/d\mu^5\) is obtained from (4.3) by replacing in the right-hand side \(\varphi_1\), \(\psi_1\), \(\lambda\), and \(\sigma_1\), respectively, by \(\varphi_2\), \(-\psi_2\), \(\mu\), and \(\sigma_2=\lambda_*(\mu)+\mu-c\). In the case when the initial data are specified along the line \(p=\mathrm{const}\) or along a rectilinear wall, the functions \(f_1(\lambda)\) and \(f_2(\mu)\) are expressed through multiple integrals of the prescribed functions \(\varphi_1(\lambda)\), \(\varphi_2(\mu)\), \(\psi_1(\lambda)\), and \(\psi_2(\mu)\) in the first case and of \(\varphi_1(\lambda)\) and \(\varphi_2(\mu)\) in the second.
5. The problem with prescribed conditions on a characteristic and a free surface. Along the free surface \(\sigma=\lambda+\mu-c\) assumes the constant value equal to \(\sigma_1\). In addition, along it \(\psi=0\). Along the characteristic \(\lambda=\lambda_0\) the function is prescribed:
\[ \varphi=\varphi_1(\mu). \tag{5.1} \]
At the point of intersection of the characteristic with the free surface \((\lambda=\lambda_0,\ \mu=\mu_0)\) we shall assume that \(\varphi_1(\mu_0)=0\).
The posed problem is solved with the aid of the functions
\[ f_1(\lambda)=c_1(\lambda)e^{r\lambda} +e^{\alpha\lambda}\left[\cos(\beta\lambda)c_2(\lambda) +\sin(\beta\lambda)c_3(\lambda)\right], \]
\[ f_2(\mu)=\frac12(\lambda_0+\mu-c)^3 \int_{\mu_0}^{\mu}\!\!\int \varphi_1(\mu)\,d\mu\,d\mu, \tag{5,2} \]
where
\[ c_1(\lambda)=A\beta\int_{\lambda_0}^{\lambda} e^{-r\lambda}F(\lambda)\,d\lambda, \]
\[ c_2(\lambda)=-A\int_{\lambda_0}^{\lambda} e^{-\alpha\lambda}\left[(\alpha-r)\sin(\beta\lambda)+\beta\cos(\beta\lambda)\right] F(\lambda)\,d\lambda, \]
\[ c_3(\lambda)=A\int_{\lambda_0}^{\lambda} e^{-\alpha\lambda}\left[(\alpha-r)\cos(\beta\lambda)-\beta\sin(\beta\lambda)\right] F(\lambda)\,d\lambda, \]
\[ F(\lambda)=-f_2'''(\nu_0-\lambda) +\frac{12}{\sigma_1}f_2''(\nu_0-\lambda) -\frac{60}{\sigma_1^2}f_2'(\nu_0-\lambda) +\frac{120}{\sigma_1^3}f_2(\nu_0-\lambda), \]
\[ A=\frac{\sigma_1^3}{12\sqrt{15}},\qquad \nu_0=\lambda_0+\mu_0,\qquad r=\frac{12}{\sigma_1}-2\alpha, \]
\[ \alpha=\frac{1}{2\sigma_1}\left\{ 8-\sqrt[3]{4}\left[\sqrt[3]{\sqrt5+1}-\sqrt[3]{\sqrt5-1}\right]\right\}, \]
\[ \beta=\frac{\sqrt3}{2\sigma_1}\sqrt[3]{4} \left[\sqrt[3]{\sqrt5+1}+\sqrt[3]{\sqrt5-1}\right]. \]
6. Problem with prescribed conditions on a characteristic and a rectilinear wall
Along a rectilinear wall \(\theta=\mathrm{const}\). We shall assume that the wall is situated so that \(\theta=0\). Then along the wall \(\lambda=\mu\). Along the characteristic \(\lambda=\lambda_0\) the function is prescribed:
\[ \varphi=\varphi_1(\mu). \tag{6,1} \]
Then the problem is solved with the aid of the functions
\[ f_1(\lambda)=(2\lambda-c)^4c_1(\lambda) +(2\lambda-c)^2c_2(\lambda)+c_3(\lambda), \]
\[ f_2(\mu)=\frac12(\lambda_0+\mu-c)^3 \int_{\mu_0}^{\mu}\!\!\int \varphi_1(\mu)\,d\mu\,d\mu, \tag{6,2} \]
where
\[ c_1(\lambda)=\frac14\int_{\lambda_0}^{\lambda} (2\lambda-c)^{-2}F(\lambda)\,d\lambda,\qquad c_2(\lambda)=-\frac12\int_{\lambda_0}^{\lambda}F(\lambda)\,d\lambda, \]
\[ c_3(\lambda)=\frac14\int_{\lambda_0}^{\lambda} (2\lambda-c)^2F(\lambda)\,d\lambda, \]
\[ F(\lambda)=f_2'''(\lambda) -\frac{6}{2\lambda-c}f_2''(\lambda) +\frac{12}{(2\lambda-c)^2}f_2'(\lambda). \]
Received
16 V 1958
CITED LITERATURE
- L. I. Sedov, Plane Problems of Hydrodynamics and Aerodynamics, 1950.
- S. A. Khristianovich, Prikl. matem. i mekh., 11, 2 (1947).
- G. A. Dombrovskii, Collection of articles No. 11, Theoretical Hydromechanics, ed. L. I. Sedov, 1953.
- G. A. Dombrovskii, Collection of articles No. 12, Theoretical Hydromechanics, ed. L. I. Sedov, 1954.