§14 One-dimensional similarity flow

In this section we will discuss the one-dimensional non-steady gas flow under the assumption that there are characteristic velocities in the flow, but not characteristic lengths. To simplify the discussion we assume that relativistic effects are not taken into account.

The state of the flow at any time is defined by the characteristic velocity parameter and by some other parameters which describe the state of the gas, for example the pressure and density at an initial instant. With these parameters alone it is not possible to find a combination which has the dimensions of length or time. By applying the $\Pi$-Theorem of dimensional analysis (Sedov, 1993) it follows that the distribution of the different hydrodynamical quantities can only depend on the position $x$ and time $t$ through the ratio $x / t \! \equiv \! \xi$ only. In other words, if the lengths are measured in a unit that increases proportional with time, the pattern produced by the flow remains unchanged. Such a flow is called a similarity flow (Landau & Lifshitz, 1995).

Using the fact that all quantities in the problem depend only on the single variable $\xi$, for which $\partial / \partial x = ( 1 / t ) \mathrm{d} / \mathrm{d} \xi$    and $\partial / \partial t = - ( \xi / t ) \mathrm{d} / \mathrm{d} \xi$, we obtain from the equation of conservation of the entropy, eq.(10.4), $( \ensuremath{v}_x - \xi ) s' = 0$. The prime denotes differentiation with respect to $\xi$. From this equation it follows that $s' \! = \! 0$, otherwise as it is obvious from the equations presented below in eq.(14.1) a contradiction will be obtained. This implies that the entropy $s \! =$   const$$ and so, similarity flow in one dimension which is adiabatic must also be isentropic. In exactly the same way, the $x$    and $y$ components of Euler's equation imply that the velocity in the $x$    and $y$ components are constant and we can take them as zero by an appropriate choice of the system of reference.

The $x$ component of Euler's equation, eq.(10.3), and the continuity equation eq.(10.2) can be written as:

$$( \ensuremath{v}- \xi ) \rho' + \rho \ensuremath{v}' = 0,$$ (14.1)

$$( \ensuremath{v}- \xi ) \ensuremath{v}' = -p' / \rho = - c^2 \rho' / \rho.$$ (14.2)

Here $\ensuremath{v}$ denotes the velocity in the $x$ direction and we will use that convention in what follows. The trivial solution of this last set of equations is that of a uniform flow with $\rho \! = \!$   const$, \ensuremath{v}\! = \!$   const$$. The non-trivial solution is found by eliminating $\rho'$    and $\ensuremath{v}'$ from the equations, giving $( \ensuremath{v}- \xi )^2 = c^2$, so that $\xi = \ensuremath{v}\pm c$. We take the plus sign in the following discussion, which means that we have taken the direction of the positive $x$ axis in a definite manner:

$$x / t = \ensuremath{v}+ c$$ (14.3)

Substituting eq.(14.3) into eq.(14.1) we obtain $\rho \mathrm{d} \ensuremath{v}= c \mathrm{d} \rho$. In order to integrate this relation we recall that the velocity of sound is a function of the thermodynamical state of the gas. We can thus take this velocity as a function of the entropy and the mass density. Since the entropy is constant it follows that the velocity of sound can be considered as a function of the density only. In other words (Landau & Lifshitz, 1995):

$$\ensuremath{v}= \int{ c(\rho) \mathrm{d} \rho / \rho } = \int{ \mathrm{d} p / c(\rho) \rho },$$ (14.4)
which can be rewritten as:

$$\ensuremath{v}= \int{ \sqrt{ - \mathrm{d} p \mathrm{d} \ensuremath{\mathit{V}}} },$$ (14.5)

where the choice of the independent variable remains open.

Let us briefly discuss some general properties of the solution. Differentiating eq.(14.3) with respect to $x$ gives:

$$t \frac{ \partial \rho }{ \partial x } \frac{ \mathrm{d} ( \ensuremath{v}+ c ) }{ \mathrm{d} \rho } = 1.$$ (14.6)
The derivative of tex2html_wrap_inline tex2html_wrap_inlinev+ c can be calculated from eq.(14.4) which gives:

$$\frac{ \mathrm{d} ( \ensuremath{v}+ c ) }{ \mathrm{d} \rho } = \f... ...thrm{d} \rho } = \frac{ 1 }{ \rho }{ \mathrm{d}( \rho c ) }{ \mathrm{d} \rho },$$
and

$$\rho c = \rho \sqrt{ ( \partial p / \partial \rho ) } = 1 / \sqrt{ - \partial \ensuremath{\mathit{V}}/ \partial p }.$$
Differentiation of this relation results in:

$$\mathrm{d} ( \rho c ) / \mathrm{d} \rho = c^2 \mathrm{d} ( \rho c... ...rm{d} p = \rho^3 c^5 ( \partial^2 \ensuremath{\mathit{V}}/ \partial p^2)_s / 2,$$
so that eq.(14.6) takes the form:

$$\mathrm{d} ( \rho c ) / \mathrm{d} \rho = c^2 \mathrm{d} ( \rho c... ...rho = \rho^2 c^5 ( \partial^2 \ensuremath{\mathit{V}}/ \partial p^2)_s / 2 > 0.$$ (14.7)

Substitution of eq.(14.7) into eq.(14.6) implies that $\partial \rho / \partial x > 0$    for $t > 0$. Because $\partial p / \partial x = c^2 \partial p / \partial x$, it follows that $\partial p / \partial x > 0$. Also, since $\partial \ensuremath{v}/ \partial x = ( c / \rho ) \partial \rho / \partial x$ according to eq.(14.4), the inequality $\partial \ensuremath{v}/ \partial x > 0$ holds. In other words, we have proved that the following relations are valid in a flow for which eq.(14.3) holds:

$$\partial \rho / \partial x > 0, \qquad \partial p / \partial x > 0, \qquad \partial \ensuremath{v}/ \partial x > 0.$$ (14.8)

In order to understand the meaning of these inequalities, let us rewrite them not as variations of the position $x$ but as variations of the of time for a given fluid element as it moves. This variation is given by the total derivative of the corresponding quantity with respect to time. Thus, for the density it follows from the equation of continuity and eq.(14.8) that: $\mathrm{d} \rho / \mathrm{d} t = \partial \rho / \partial t + \ensuremath{v}\partial \rho / \partial x = - \rho \partial \ensuremath{v}/ \partial x < 0$. This implies that $\mathrm{d} p / \mathrm{d} t = c^2 \mathrm{d} \rho / \mathrm{d} t < 0$ also. On the other hand, Euler's equation implies that $\mathrm{d} \ensuremath{v}/ \mathrm{d} t < 0$. It is important to note that this last inequality does not mean that the magnitude of the velocity decreases as the fluid moves about, since $\ensuremath{v}$ can be negative. We have proved that, as the fluid moves, the following inequalities are satisfied:

$$\mathrm{d} \rho / \mathrm{d} t < 0, \qquad \mathrm{d} p / \mathrm{d} t < 0 , \qquad \mathrm{d} \ensuremath{v}/ \mathrm{d} t < 0.$$ (14.9)

This means that, as the fluid moves, its pressure and density decreases. To put it differently, the gas is continuously rarefied as the fluid moves. Such a flow is called a rarefaction wave.

A rarefaction wave can only be propagated a finite distance along the $x$ axis. This follows from the fact that $\ensuremath{v}\rightarrow \pm \infty$    as $x \rightarrow \pm \infty$. As a result, we can apply eq.(14.3) at the boundaries of a rarefaction wave. For this case, the ratio $x / t$ is the velocity of the boundary relative to a fixed system of coordinates. The velocity relative to the flow itself is $x / t - \ensuremath{v}$ which is equal to the local velocity of sound $c$ according to eq.(14.3). This result implies that the boundaries of a rarefaction wave are weak discontinuities.

The choice of sign in eq.(14.3) is now clear. It must be such that the weak discontinuities, bounding the rarefaction wave, are assumed to be moving in the positive $x$ direction relative to the gas. By an appropriate choice of the system of reference we can choose the region, region I, to the right of the rarefaction wave to be at rest, as it is shown in fig.(II.4). Region II is the rarefaction wave and region III has gas moving with constant velocity. The arrows in the figure represent the direction of the flow and of the weak discontinuities. Due to the fact that the weak discontinuities move to the right relative to the gas, the weak discontinuity at the right of the diagram moves to the right. However, the weak discontinuity on the left might move in either direction. The direction depends on the value of the velocity reached in the rarefaction wave (Landau & Lifshitz, 1995).

Figure II.4: Rarefaction wave (region II) bounded by two tangential discontinuities (dashed lines). The gas to the right of the rarefaction wave in region I has been chosen to be at rest. The arrows in the figure represent direction of motion of the flow and of the weak discontinuities.

For a polytropic gas eq.(14.4) gives:

$$\ensuremath{v}= \frac{ 2 }{ \gamma - 1 } \int{ \mathrm{d} c } = \frac{ 2 }{ \gamma - 1 } ( c - c_0 ),$$ (14.10)

in which the constant of integration $c_0$ corresponds to the value of the velocity of sound when the velocity of the rarefaction wave vanishes. Here and in what follows we denote by the suffix $0$ the region of the flow where the gas is at rest. Using the Poisson adiabatic for a polytropic gas, and rewriting eq.(14.10), we find that:

$$c = c_0 - \frac{ 1 }{ 2 } ( \gamma - 1 ) \vert \ensuremath{v}\vert,$$ (14.11)

$$\rho = \rho_0 \left\{ 1 - \frac{ 1 }{ 2 } ( \gamma - 1 ) \vert \ensuremath{v}\vert / c_0 \right\}^{ 2 / ( \gamma - 1 ) },$$ (14.12)

$$p = p_0 \left\{ 1 - \frac{ 1 }{ 2 } ( \gamma - 1 ) \vert \ensuremath{v}\vert / c_0 \right\}^{ 2 \gamma / ( \gamma - 1 ) }$$ (14.13)
The value of the velocity as a function of the ratio tex2html_wrap_inline x / t is obtained by substituting eq.(14.11) in eq.(14.3):

$$\vert \ensuremath{v}\vert = \frac{ 2 }{ \gamma + 1 } \left( c_0 - x/t \right).$$ (14.14)

Lastly we mention that from eq.(14.11), since the velocity of sound is non negative, the following inequality has to be satisfied:

$$\vert \ensuremath{v}\vert \le 2 c_0 / ( \gamma - 1 ).$$ (14.15)

When the velocity reaches this limiting value, the pressure, density and velocity of sound in the gas become zero.

Footnotes

... discontinuities.

Weak discontinuities are surfaces of discontinuities for which the hydrodynamical quantities, which are continuous across this surface, are not regular functions of the coordinates. This irregularity might be of various forms, for example the first spatial derivatives (or any other higher derivative) might be discontinuous across the surface or have an infinite value on it. These discontinuities are called weak as opposed to the strong ones (such as shock waves and tangential discontinuities) in which the hydrodynamical quantities themselves are discontinuous. Since the values of each hydrodynamical quantity are continuous across the surface of discontinuity, they can be ``smoothed'' by modifying them only near this surface and by very small amounts, in such a way that the smoothed functions lack of singularity. The true distribution of the pressure, say, can be represented as a superposition of a completely smoothed function $p_0$ and a very small perturbation $p'$ near the weak discontinuity which contains the singularity. This perturbation, like any other perturbation in the flow, moves with the velocity of sound with respect to the gas.