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UDC 517.946.51
GROUP CLASSIFICATION OF DIFFERENTIAL EQUATIONS DESCRIBING ONE-DIMENSIONAL UNSTEADY FLUID MOTION
M. T. GLADYSHEV
To investigate the existence of particular classes of partial solutions of a quasilinear system of two equations of hyperbolic type, the group-theoretic method developed by L. V. Ovsyannikov [1] is applied.
The basic equations of unsteady fluid motion in open channels, first obtained in a somewhat different form by Saint-Venant, are expressions of the laws of conservation of mass and momentum and have the form [2, 3]
\[ \frac{\partial \omega}{\partial t}+\frac{\partial(v\omega)}{\partial x}=0,\qquad \frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x}+g\frac{\partial h}{\partial x} = g\left(i-\frac{v|v|}{C^{2}R}\right), \]
where \(v\) is the mean flow velocity; \(h\) is the depth; \(\omega\) is the cross-sectional area; \(g\) is the acceleration due to gravity; \(i\) is the bed slope; \(C\) is the Chezy coefficient; \(R\) is the hydraulic radius; \(x\) is the coordinate of the cross section; \(t\) is time.
Taking into account the practical features of the problem, it is expedient to pass to other variables, namely to replace the depth \(h\) by the cross-sectional area \(\omega\), and the velocity \(v\) by the discharge \(Q=v\omega\). We have
\[ \frac{\partial v}{\partial t} = \frac{1}{\omega}\frac{\partial Q}{\partial t} - \frac{Q}{\omega^{2}}\frac{\partial \omega}{\partial t}, \qquad \frac{\partial v}{\partial x} = \frac{1}{\omega}\frac{\partial Q}{\partial x} - \frac{Q}{\omega^{2}}\frac{\partial \omega}{\partial x}, \qquad \frac{\partial h}{\partial x} = \frac{1}{B}\frac{\partial \omega}{\partial x} - \frac{1}{B}\left(\frac{\partial \omega}{\partial x}\right)_{h}, \]
where \(B=\dfrac{\partial \omega}{\partial h}\) is the width of the river surface. Introduce the discharge modulus \(K=\omega C\sqrt{R}\).
Then, in the new variables, the equations take the form
\[ \frac{\partial \omega}{\partial t} + \frac{\partial Q}{\partial x} = 0, \qquad \frac{\partial Q}{\partial t} + \left(\frac{g\omega}{B}-\frac{Q^{2}}{\omega^{2}}\right) \frac{\partial \omega}{\partial x} + 2\frac{Q}{\omega}\frac{\partial Q}{\partial x} = \]
\[ = g\omega \left( i+\frac{1}{B}\left(\frac{\partial \omega}{\partial x}\right)_{h} - \frac{Q|Q|}{K^{2}} \right) \]
or, for a prismatic channel,
\[ \frac{\partial \omega}{\partial t}+\frac{\partial Q}{\partial x}=0, \tag{1} \]
\[ \frac{\partial Q}{\partial t}+\left(\frac{dP}{d\omega}-\frac{Q^2}{\omega^2}\right)\frac{\partial \omega}{\partial x} +2\frac{Q}{\omega}\frac{\partial Q}{\partial x} =\omega i-FQ^2 . \tag{1} \]
Here a) in the case of open flows: \(\omega(x,t)\) is the cross-sectional area of the flow; \(Q(x,t)\) is the discharge of the flow; \(P(\omega)=g\int_0^{h(\omega)}\omega(\xi)\,d\xi\) is the force of fluid pressure in the cross section of the flow; \(i(x)\) is the bottom slope; \(F(\omega,x)\) is the coefficient of friction;
b) in the case of pressure pipelines (the equations of hydraulic shock): \(\omega(x,t)\) is the density of the fluid; \(Q(x,t)\) is the discharge, \(P(\omega)-P(\omega_0)=a^2(\omega-\omega_0)\), \(F(\omega,x)=\dfrac{\lambda}{2D\omega}\operatorname{sign} Q\); \(a\) is the speed of sound, determined by the well-known formula of N. E. Zhukovskii; \(\lambda(x)\) is the coefficient of friction; \(D\) is the diameter of the pipeline.
We shall consider \(P(\omega)\), \(i(x)\), and \(F(\omega,x)\) as three arbitrary functions whose arbitrariness may affect the breadth of the group admitted by system (1).
In what follows it is assumed that \(\dfrac{dP}{d\omega}>0\). Equations (1) preserve their form under the equivalence transformation
\[ \omega_1=a\omega,\qquad Q_1=aQ,\qquad P_1(\omega)=P(a\omega),\qquad F_1(\omega,x)=\frac1a F(a\omega,x). \tag{2} \]
We seek an operator admitted by system (1) in the form
\[ X=\xi\frac{\partial}{\partial t}+\eta\frac{\partial}{\partial x} +\sigma\frac{\partial}{\partial \omega}+\tau\frac{\partial}{\partial Q}, \]
where \(\xi,\eta,\sigma,\tau\) are the sought functions of \(t,x,\omega,Q\). The problem reduces to solving the following system of determining equations:
\[ \frac{\partial\eta}{\partial Q}=-\frac{\partial\xi}{\partial\omega},\qquad \frac{\partial\eta}{\partial\omega} =2\frac{Q}{\omega}\frac{\partial\xi}{\partial\omega} -\left(P_\omega-\frac{Q^2}{\omega^2}\right)\frac{\partial\xi}{\partial Q}, \]
\[ \frac{\partial\eta}{\partial x} =2\frac{Q}{\omega}\frac{\partial\xi}{\partial x} +\frac{\partial\xi}{\partial t} +(\omega i-FQ^2)\frac{\partial\xi}{\partial Q} +\frac{P_{\omega\omega}}{2P_\omega}\sigma, \tag{3} \]
\[ \frac{\partial\eta}{\partial t} =(\omega i-FQ^2)\frac{\partial\xi}{\partial\omega} +\left(P_\omega-\frac{Q^2}{\omega^2}\right)\frac{\partial\xi}{\partial x} +\frac{\tau}{\omega} -\frac{Q}{\omega}\left(\frac1\omega+\frac{P_{\omega\omega}}{2P_\omega}\right)\sigma; \]
\[ \frac{\partial\tau}{\partial Q} =2\frac{Q}{\omega}\frac{\partial\sigma}{\partial Q} +\frac{\partial\sigma}{\partial\omega} +(\omega i-FQ^2)\frac{\partial\xi}{\partial Q} +\frac{P_{\omega\omega}}{2P_\omega}\sigma, \]
\[ \frac{\partial\tau}{\partial\omega} =\left(P_\omega-\frac{Q^2}{\omega^2}\right)\frac{\partial\sigma}{\partial Q} +(\omega i-FQ^2)\frac{\partial\xi}{\partial\omega} +\frac{\tau}{\omega} -\frac{Q}{\omega}\left(\frac1\omega+\frac{P_{\omega\omega}}{2P_\omega}\right)\sigma, \]
\[ \frac{\partial\tau}{\partial x} =-\frac{\partial\sigma}{\partial t} +(\omega i-FQ^2)\left(\frac{\partial\xi}{\partial x} -\frac{\partial\sigma}{\partial Q}\right). \tag{4} \]
\[
\frac{\partial \tau}{\partial t}
= 2 \frac{Q}{\omega}\frac{\partial \sigma}{\partial t}
+(\omega i-FQ^2)\left(\frac{\partial \xi}{\partial t}-\frac{\partial \sigma}{\partial \omega}\right)
-\left(P_\omega-\frac{Q^2}{\omega^2}\right)\frac{\partial \sigma}{\partial x}
\]
\[
-\left[(\omega i-FQ^2)\frac{P_{\omega\omega}}{2P_\omega}-i+F_\omega Q^2\right]\sigma
+(\omega i_x-F_xQ^2)\eta-2FQ\tau .
\]
From the compatibility conditions of equations (3) it follows that, for \(F\ne 0\),
\[ \frac{\partial \xi}{\partial \omega}=\frac{\partial \xi}{\partial Q}=0; \tag{5} \]
\[ \left(\frac{P_{\omega\omega}}{2}-\frac{P_\omega}{\omega}\right) \left(2\frac{\partial \xi}{\partial x}-\frac{\partial \sigma}{\partial Q}\right)=0, \tag{6} \]
\[ \left(\frac{P_{\omega\omega}}{2P_\omega}-\frac{1}{\omega}\right) \left(\frac{Q}{\omega}\frac{\partial \sigma}{\partial Q}+\frac{\partial \sigma}{\partial \omega}\right) +\left(\frac{1}{\omega^2}+\frac{P_{\omega\omega\omega}}{2P_\omega} -\frac{P_{\omega\omega}^2}{2P_\omega^2}\right)\sigma=0, \tag{7} \]
and for \(F=0\) we have, respectively, (6) and (7).
Let us consider the case \(F\ne 0\). The compatibility conditions of equations (3), taking (5) into account, in particular give the equation
\[ 2\frac{\partial \xi}{\partial x} +\frac{\omega P_{\omega\omega}}{2P_\omega}\frac{\partial \sigma}{\partial Q}=0, \tag{8} \]
which, together with (6), shows that two principal cases should be distinguished according as \(P(\omega)=C\omega^k\) or not.
For an arbitrary function \(P(\omega)\), from (6) and (8) we have
\[ \frac{\partial \xi}{\partial x}=\frac{\partial \sigma}{\partial Q}=0. \tag{9} \]
Solving the system (3), (4) obtained after simplifications, we finally obtain the following general solution:
\[ \xi=at+b,\qquad \eta=ax+c,\qquad \sigma=0,\qquad \tau=0, \tag{10} \]
where \(a,b,c\) are arbitrary constants, with \(a\ne 0\) for \(F=\dfrac{\Phi(\omega)}{x}\) and \(i=\dfrac{a}{x}\), \(c\ne 0\) for \(F=\Phi(\omega)\) and \(i=\mathrm{const}\). Here \(\Phi(\omega)\) denotes an arbitrary function, and \(a\) an arbitrary constant. Thus, the principal group of system (1), for arbitrary \(P(\omega)\), \(i(x)\), and \(F(\omega,x)\), consists of one basis operator \(X=\dfrac{\partial}{\partial t}\). The widest group will be for
\[ F=\frac{\Phi(\omega)}{m+x} \quad \text{and} \quad i=\frac{a}{m+x}, \]
when we have the following additional operators:
\[ X_2=\frac{\partial}{\partial x},\qquad X_3=t\frac{\partial}{\partial t}+x\frac{\partial}{\partial x},\qquad X_4=t\frac{\partial}{\partial t}+(m+x)\frac{\partial}{\partial x}. \]
For \(P=C\omega^k\), equation (7), taking (9) into account, assumes the form
\[ \frac{\partial \sigma}{\partial \omega}-\frac{1}{\omega}\sigma=0, \]
whose solution is \(\sigma=f(x,t)\omega\). Returning again to (3),
(4), we find that the general solution will have, in contrast to (10), the following form:
\[ \xi=at+b,\quad \eta=\left(a+\frac{k-1}{2}\,d\right)x+c,\quad \sigma=d\omega,\quad \tau=\frac{k+1}{2}\,dQ. \]
The last equation (4) contains additional restrictions on the constants \(a\) and \(c\), as above in (10). An extension of the group occurs
\[ \text{for }\quad F=x^{-\frac{k+1}{k-1}}\Phi\left(x\omega^{-\frac{k-1}{2}}\right),\quad i=\alpha x, \quad \text{and for}\quad F=x^{-\frac{1}{k-1}}\Phi(x\omega^{-k+1}),\quad i=\mathrm{const}, \]
when we arrive, respectively, at the following additional operators:
\[
X_5=\frac{k-1}{2}x\frac{\partial}{\partial x}
+\omega\frac{\partial}{\partial\omega}
+\frac{k+1}{2}Q\frac{\partial}{\partial Q}
\quad\text{and}\quad
X_6=\frac{k-1}{2}t\frac{\partial}{\partial t}
\]
\[
+(k-1)x\frac{\partial}{\partial x}
+\omega\frac{\partial}{\partial\omega}
+\frac{k+1}{2}Q\frac{\partial}{\partial Q}.
\]
Now let us consider the case \(F\equiv 0\). If \(P(\omega)\) is an arbitrary function, then it is sufficient to consider the following cases:
-
\(i_x\ne0,\quad \xi=at+b,\quad \eta=ax,\quad \sigma=0,\quad \tau=0.\)
-
\(i=\mathrm{const}\ne0,\quad \xi=a,\quad \eta=bt+c,\quad \sigma=0,\quad \tau=b\omega.\)
-
\(i\equiv0,\quad \xi=bt+c+\xi^0(\omega,Q),\quad \eta=at+bx+d+\eta^0(\omega,Q),\)
\[ \sigma=0,\quad \tau=a\omega. \]
Here \(\xi^0,\eta^0\) is any solution of the system of the first two equations of system (3). The fact that the principal group of system (1) for \(F\equiv0\) and \(i\equiv0\) has turned out to be infinite means that system (1) in this case is linearized if the quantities \(\omega,Q\) are taken as independent variables.
If \(P=C\omega^k\), then we have the following cases:
-
\(i_x\ne0,\quad \xi=\left(-\frac{k+1}{2}a+b\right)t+c,\quad \eta=(b-a)x,\quad \sigma=a\omega,\quad \tau=\frac{k+1}{2}aQ.\)
-
\(i=\mathrm{const}\ne0,\quad \xi=-\frac{k+1}{2}at+c,\quad \eta=-ax+et+d,\)
\[ \sigma=a\omega,\quad \tau=\frac{k+1}{2}aQ+e\omega. \]
- \(i\equiv0,\quad \xi=\left(-\frac{k+1}{2}a+b\right)t+c+\xi^0(\omega,Q),\quad \eta=(b-a)x+\)
\[ +et+d+\eta^0(\omega,Q),\quad \sigma=a\omega,\quad \tau=\frac{k+1}{2}aQ+e\omega. \]
It remains to investigate the case \(P=C\omega^3\), for which (6) and (7) are satisfied identically and (9) no longer follows from (8). It is easy to verify that we have a difference from \(P=C\omega^k\) only for \(F\equiv0\) and \(i\equiv0\), when from the compatibility conditions of equations (3), (4) we do not obtain additional relations. In this case the principal group is extended:
\[
\xi = ax + (c - 2b)t + d + \xi^0(\omega,Q), \quad
\eta = (c - 2b)x + ft + e + \eta^0(\omega,Q),
\]
\[
\sigma = b\omega - 2aQ, \quad
\tau = 2bQ - 3aQ^2\omega^{-1} - 3a\omega^3 + f\omega.
\]
Let us note that \(P = C\omega^k\) \((k > 1)\), in particular, for channels with transverse section \(\omega = Ah^{\frac{1}{k-1}}\), where \(h\) is the depth of the flow; \(A\) is a certain constant.
The solution of the group classification for the system of equations (1) showed that the form of equations (1) usually used for \(\omega = Ah^{\frac{1}{k-1}}\) [2], whose aim was to simplify them for the possibility of solving boundary-value problems of hydraulics in a simple analytic form, reduces to the following: the equations are taken in such a form as permits the broadest possible group. Moreover, among all values of the exponent \(k\) there is one exceptional value, analogous to the case of gas dynamics [1], namely \(k = 3\), for which the principal group is broader in comparison with the case of arbitrary \(k\). In this case equations (1) are reduced to the simplest form
\[ \frac{\partial r}{\partial t} + r\frac{\partial r}{\partial x} = f_1(x,r,s), \quad \frac{\partial s}{\partial t} + s\frac{\partial s}{\partial x} = f_2(x,r,s), \]
where
\[ r = \frac{Q}{\omega} + \sqrt{2gh}, \quad s = \frac{Q}{\omega} - \sqrt{2gh}. \]
If we have a concrete form of system (1), i.e., the functions \(P(\omega)\), \(i(x)\), and \(F(\omega,x)\) are known, then, using the results of the group classification, one can find all particular solutions.
As was to be expected, the general equations of hydraulics with a right-hand side have few particular solutions. One can say in advance only that any system (1) has a solution of the form \(\omega = \omega(x)\) and \(Q = Q(x)\), which physically corresponds to steady flow. For example, a monoclinal wave [3], i.e., a solution of the form \(\omega = \omega(x-at)\) and \(Q = Q(x-at)\) \((a=\mathrm{const})\), will occur when \(F(\omega,x)=\Phi(\omega)\) and \(i(x)=\mathrm{const}\).
A self-similar solution in the combination \(\lambda = \frac{x}{t}\), describing simple centered waves, will occur for \(F(\omega,x)=\frac{\Phi(\omega)}{x}\) and \(i(x)=\frac{a}{x}\).
In conclusion, let us consider the system of differential equations for an open channel with \(\omega = Ah^{\frac{1}{k-1}}\):
\[ \frac{\partial h}{\partial t} + v\frac{\partial h}{\partial x} + (k-1)h\frac{\partial v}{\partial x} =0, \quad \frac{\partial v}{\partial t} + v\frac{\partial v}{\partial x} + g\frac{\partial h}{\partial x} =0. \tag{11} \]
From the group properties of system (11) it follows that it has the following invariant solutions, whose determination in the general case reduces to the solution of ordinary differential equations:
1) \(v = \mathrm{const}, \quad h = \mathrm{const}\)—trivial solutions;
2)
\[
v = \frac{2}{k+1}\frac{x}{t} + a, \quad
h = \frac{1}{(k-1)g}
\left(\frac{k-1}{k+1}\frac{x}{t} - a\right)^2,
\]
where \(a\) is an arbitrary constant. These are simple waves.
3)
\[
v=x\left(\frac{k-1}{k+1}t+a\right)^{-1},\qquad
h=\frac{(k-1)x^2}{4g}\left(\frac{k-1}{k+1}t+a\right)^{-2},
\]
4)
\[
v=\frac{x}{t}+V(t),\qquad h=H(t).
\]
5)
\[
v=x-\exp t\cdot V(\lambda),\qquad h=\exp(2t)\cdot H(\lambda),\qquad
\lambda=x\exp(-t).
\]
6)
\[
v=t^{-1}V(\lambda),\qquad h=t^{-2}H(\lambda),\qquad
\lambda=t\exp(-x).
\]
7)
\[
v=\frac{x}{t}+V(\lambda),\qquad h=H(\lambda),\qquad
\lambda=t\exp\left(-\frac{x}{t}\right).
\]
8)
\[
v=\frac{x}{t}V(\lambda),\qquad h=x^{2-2a}H(\lambda),\qquad
\lambda=tx^{-a}.
\]
9)
\[
v=\frac{x}{t}V(\lambda),\qquad h=x^2t^{-2}H(\lambda),\qquad
\lambda=xt^{-a}.
\]
10)
\[
v=t+V(\lambda),\qquad h=H(\lambda),\qquad
\lambda=x-\frac{1}{2}t^2.
\]
If in system (11) we make a transformation of the dependent and independent variables, namely, if we regard \(v,h\) as the independent variables and \(t,x\) as the dependent variables, then we obtain the linear system of equations
\[ \frac{\partial x}{\partial v} -v\frac{\partial t}{\partial v} +(k-1)h\frac{\partial t}{\partial h}=0,\qquad \frac{\partial x}{\partial h} -v\frac{\partial t}{\partial h} +g\frac{\partial t}{\partial v}=0. \]
I thank L. V. Ovsyannikov for discussion of the work.
References
- Ovsyannikov L. V. Group properties of differential equations. Publishing House of the Siberian Branch of the Academy of Sciences of the USSR, Novosibirsk, 1962.
- Khristianovich S. A., Mikhlin S. G., Devison B. B. Some new problems of continuum mechanics. Publishing House of the Academy of Sciences of the USSR, Moscow, 1938.
- Stoker J. Water Waves. Foreign Literature Publishing House, Moscow, 1959.
Received by the editors
January 26, 1965
Institute of Hydrodynamics, Siberian Branch of the Academy of Sciences of the USSR