Hydromechanics

I. Yu. Braidovskaya

A Method for Calculating Flows with Strong Viscous Interaction

(Presented by Academician G. I. Petrov on December 7, 1964)

1. In calculating the flow of a supersonic stream of a viscous gas past bodies, in a number of cases it is impossible to confine oneself to the equations of the boundary layer or of an inviscid gas. The method proposed in the present paper makes it possible to obtain an approximate solution of the system of Navier—Stokes equations in such a way that the complete system has to be solved only in small regions in which the dissipative terms are large.

2. For the flows of interest to us the Reynolds number is a quantity of order (10^4—10^6); therefore the Navier—Stokes equations are equations with a small parameter (\varepsilon) multiplying the highest derivatives ((\varepsilon = 1/\mathrm{Re})). Further, the system of equations of an ideal gas, obtained by discarding the highest derivatives, requires for its solution a smaller number of boundary conditions than the complete system, both on the body being flowed around and on the outer boundary. Therefore the model of aerodynamic problems is the following boundary-value problem (A_\varepsilon) for an ordinary differential equation with a small parameter (\varepsilon) multiplying the highest derivatives.

On the interval ([0,1]) it is required to solve the equation

[
L_\varepsilon u = (L_0 + \varepsilon L_1)u \equiv
\sum_{j=0}^{k} a_j(x)\frac{d^j u}{dx^j}
+
\sum_{r=1}^{l} \varepsilon^r a_{k+r}(x)\frac{d^{k+r}u}{dx^{k+r}}
= h(x);
\tag{1}
]

[
k = k_1 + k_2;\qquad l = l_1 + l_2,
]

with boundary conditions

[
\left.\frac{d^i u}{dx^i}\right|_{x=0} = 0;\qquad
i = 0,1,\ldots,k_1+l_1-1;
\tag{2}
]

[
\left.\frac{d^j u}{dx^j}\right|_{x=1} = 0;\qquad
j = 0,1,\ldots,k_2+l_2-1.
\tag{3}
]

We shall call the solution of the equation (L_0u=h), with the smaller number of conditions of the form (2)—(3) at the points (x=0), (x=1), the degenerate problem (A_0): (k_1) conditions at the left end and (k_2) at the right end of the interval.

The approximate solution of the problem (A_\varepsilon) is a function (w), which is determined as follows: (w=w_1) for (x\in[0,\bar{x}1]), where (w_1) is the solution of the equation (L\varepsilon w_1=h) on the interval ([0,\bar{x}_1]) with conditions (2) at (x=0) and conditions (d^i w_1/dx^i=a_i), (i=0,1,\ldots,k_2+l_2-1), at (x=\bar{x}_1).

(w=w_2) for (x\in[\bar{x}1,\bar{x}_2]), where (w_2) is the solution of the equation (L_0w_2=h) on the interval ([\bar{x}_1,\bar{x}_2]) with conditions (\left.d^j w_2/dx^j\right|=c_r), (r=0,1,\ldots,k_2-1).}=b_j), (j=0,1,\ldots,k_1-1); (\left.d^r w_r/dx^r\right|_{x=\bar{x}_2

(w=w_3) for (x\in[\bar{x}2,1]), where (w_3) is the solution of the equation (L\varepsilon w_3=h) with conditions (3) at (x=1) and conditions (\left.d^l w_3/dx^l\right|_{x=\bar{x}_2}=d_l), (l=0,1,\ldots,\ldots,k_1+l_1-1).

The constants (a_i, b_j, c_r, d_l) (their number is (2(k_2+k_1)+l_2+l_1)) are found from the smoothness matching conditions (4)—(5) at the points (\bar{x}_1,\bar{x}_2) ((\bar{x}_1>x_1,\ \bar{x}_2<x_2,\ x_1<x_2)); the position of the points (x_1) and (x_2) is determined by fulfilling the requi-

condition 2) of Theorem 1),

[
\left. d^j w_1 / dx^j \right|{x=\bar{x}_1}
=
\left. d^j w_2 / dx^j \right|,}_1
\qquad
j=0,1,\ldots,k_1+k_2+l_2-1;
\tag{4}
]

[
\left. d^i w_2 / dx^i \right|{x=\bar{x}_2}
=
\left. d^i w_3 / dx^i \right|,}_2
\qquad
i=0,1,\ldots,k_1+k_2+l_1-1.
\tag{5}
]

Denote by (\lambda_i(x)) the roots of the equation (\sum_{r=0}^{l} a_{k+r}(x)\lambda^r=0), considered with respect to the unknown (\lambda).

Theorem 1. Suppose: 1) the problem (A_0) is solvable; 2) inside the interval ([0,1]) there exist points (x=x_1), (x=x_2), (x_1<x_2), such that for (x\in[x_1,x_2]), among the (\lambda_i(x)) there are (l_1) functions (-\lambda_1,\ldots,-\lambda_{l_1}) with negative real parts, and (l_2=l-l_1) functions (\lambda_{l_1+1},\ldots,\lambda_l) with positive real parts, which coincides with the number of conditions additional to the conditions of the problem (A_0), respectively at the left and right ends of the interval; 3) the problem (A_\varepsilon), (L_\varepsilon u=g), under conditions (2)—(3) is solvable, its solution is unique, and for any (\varepsilon)

[
|u|<k|g|,
]

where (k) does not depend on (\varepsilon).

Then there exist (\bar{x}_1>x_1), (\bar{x}_2<x_2), such that the approximate solution (w) exists, is unique, and differs from the exact one by no more than a quantity of order (o(\varepsilon)).

The approximate solution (w), constructed by the method described, coincides with the first approximation of [1] in the case when the problem (A_\varepsilon) degenerates regularly into (A_0) in the sense of [1]. However, the method described makes it possible to obtain an approximate solution also in those cases when there is no regular degeneration in the sense of [1], for example, in a problem with an initial jump (initial conditions of the form (1/\varepsilon^k)), and also in all those cases when the coefficients of the equation are such that its solution behaves in a complicated way near the ends of the interval ([0,1]) and only inside the interval ([\bar{x}_1,\bar{x}_2]), (\bar{x}_1>x_1), (\bar{x}_2<x_2), tends, as (\bar{x}_1/\varepsilon\to\infty), ((-\bar{x}_2)/\varepsilon\to\infty), to some smooth function, and this smooth function may differ from the solution of the degenerate problem (A_0) by a finite amount. Such behavior of the solution occurs if, for (x\in[0,x_1)) and (x\in(x_2,1]), the signs of the real parts of the functions (\lambda_i(x)) do not satisfy requirement 2) of Theorem 1. This is a model of such gas-dynamic problems as, for example, the calculation of the flow near the nose of a sharpened plate; the calculation of the flow in the region of separation of the boundary layer arising as a result of the interaction of a shock wave with the boundary layer; the calculation of flow past a step. In the cases listed, the flow parameters are functions that behave in a complicated manner in some region near the body (separation), while in the outer region they differ by a finite amount from the solution of the degenerate problem, i.e., from the parameters of simply inviscid flow (strong viscous interaction).

§ 3. The following model is a partial differential equation of (n)-th order with a small parameter (\varepsilon) at the highest derivatives.

Let the problem (A_\varepsilon) consist in solving an equation of ((k+l))-th order

[
L_\varepsilon u = L_0 u + \sum_{s=1}^{l}\varepsilon^s L_{k+s}u = 0
\tag{6}
]

in a domain (Q) (by (L_s) is denoted a linear differential operator of (s)-th order). On different parts (\Gamma_i) of the boundary (\Gamma) of the domain (Q) there are prescribed, generally speaking, different nonhomogeneous boundary conditions. Suppose that the domain (Q) can be divided into (m) parts (Q_1,Q_2,\ldots,Q_m) in such a way that in certain (Q_i) ((1\le i\le m)) the problem (A_\varepsilon) is expressed in terms of simpler problems (A_{\varepsilon i}), in the sense that the exact solution (u) of the problem (A_\varepsilon) in the domains (Q_i) satisfies, with accuracy up to quantities of order (o(\varepsilon)), certain simpler equations

(L_{\varepsilon i}u=0) with the corresponding boundary conditions for them. Suppose that (u) in the remaining subdomains (Q_j) does not satisfy any simplified equation and can be found only as the solution of equation (6) with conditions analogous to the conditions on (\Gamma) in problem (A_\varepsilon).

The approximate solution is defined as follows: (w=w_i) in the regions (Q_i), where each (w_i) is a solution of the corresponding equation (L_{\varepsilon i}w_i=0). If in the given (Q_j) equation (6) cannot be simplified, then (w_j) is a solution of equation (6). On the boundary of each of the subdomains, conditions are prescribed corresponding to the equation that is being solved in it. On those parts of the boundary that coincide with the boundary of the region (Q), the conditions coincide with the conditions on the corresponding parts (\Gamma_i) in problem (A_\varepsilon). On those parts of the boundaries (\Gamma_j) of the subdomains (Q_i) that lie inside (Q), the right-hand sides of the corresponding inhomogeneous conditions are unknown (obviously, the number and type of conditions on each (\Gamma_j) depend on the form of this line). We set the right-hand sides equal to certain unknown functions (\varphi_j) and prescribe on each (\Gamma_j) as many smooth matching conditions (i.e., continuity of the function (w) and of the necessary number of its normal derivatives to (\Gamma_j)) as are needed for the unique determination of all unknown (\varphi_j) prescribed on (\Gamma_j).

Fig. 1

Fig. 2

Theorem 2. If: 1) the problems (A_\varepsilon), (A_{\varepsilon i}) are uniformly solvable with respect to (\varepsilon); 2) the system obtained for determining (\varphi_j) is uniquely solvable, then the approximate solution (w) exists, is unique, and differs from the exact one by a quantity of order (O(\varepsilon)).

For several examples of partial differential equations, an approximate solution (w) has been constructed according to this algorithm, and it has been proved that it is unique and differs from the exact solution by no more than a quantity (o(\varepsilon)).

Example. In the circle (Q), (\rho\le a), the equation

[
L_\varepsilon u\equiv \varepsilon\Delta u+\frac{\partial u}{\partial x}=h(x,y)
]

is given with the boundary condition (u|_\Gamma=0). Let us divide the region (Q) by the line (\Gamma_3) into two parts (Q_1) and (Q_2) so that everywhere in (Q_2), (|\varepsilon\Delta u|=O(\varepsilon)), if (u) is the exact solution of the problem (the possibility of such a division is proved in (1)).

The approximate solution in (Q_1) (Fig. 1), (w=w_1), is the solution of the equation (L_\varepsilon w_1=h) with the conditions (w_1|{\Gamma_1}=0); (w_1|=\varphi_1); in (Q_2), (w=w_2) is the solution of the equation (\partial w_2/\partial x=h) with the condition (w_2|{\Gamma_2}=0). The unknown function (\varphi_1) is found from the condition of continuity of (w) on (\Gamma_3), i.e. (w_1|). The constructed approximate solution (w) is such that (|u-w|=O(\varepsilon)) everywhere in the circle (\rho\le a).}=w_2|_{\Gamma_3

4. The algorithm for constructing an approximate (with accuracy up to quantities (o(\varepsilon))) solution of the Navier—Stokes equations is analogous to the algorithm for a single partial differential equation.

For example, suppose it is required to find the flow parameters for the supersonic flow of a viscous gas past an angle less than (\pi). The approximate solution of the Navier—Stokes equations is sought in the region (Q=Q_1+Q_2+Q_3+Q_4) (Fig. 2), bounded on the left by a vertical straight line, on which all parameters of the incident flow are known, and above by the characteristic (\Gamma_1) of the system of inviscid-gas equations.

The approximate solution in (Q_4), (w=w_4), is a solution of the system of inviscid-gas equations; in (Q_2) and (Q_3), (w=w_2), (w=w_3) are solutions of the system of equations

boundary layer; in (Q_1), (w = w_1) is the solution of the complete Navier—Stokes system. The nature of the equations solved in the regions (Q_4) and (Q_2) is such that no conditions are required on the boundaries (\Gamma_1) and (\Gamma_2) for finding (w_4) and (w_2).

The Navier—Stokes equations, the boundary-layer equations, and the inviscid-gas equations are solved jointly throughout the entire region (Q), with the necessary number of smooth matching conditions prescribed on the boundaries of the regions (Q_1, Q_2, Q_3), and (Q_4) lying inside (Q). This ensures that the mutual influence of the flows near the body and in the outer region is taken into account. The boundaries of the regions (Q_1, Q_2, Q_3, Q_4) are chosen so that the terms of the Navier—Stokes equations discarded in (Q_2, Q_3, Q_4) are quantities of order (O(\varepsilon)).

Thus, the complete Navier—Stokes system need be solved only in the immediate neighborhood of the corner point. Similarly, in computing the flow past a sharpened plate, the complete Navier—Stokes system need be solved only in the neighborhood of the “nose,” etc. A difference method is proposed for the numerical solution of the resulting boundary-value problems.

Moscow State University
named after M. V. Lomonosov

Received
2 VII 1964

REFERENCES

1. M. I. Vishik, L. A. Lyusternik, UMN, 12, no. 5 (77), 3 (1957).