Random break

Arbitrary discontinuity is an arbitrary jump in the parameters of a continuous medium , that is, a situation when only the parameters of the state of the medium are set to the left of a certain surface (for example, in gas dynamics - density , temperature and speed - ( $\rho _{1},T_{1},{\vec {v}}_{1}$ ${\ displaystyle \ rho _ {1}, T_ {1}, {\ vec {v}} _ {1}}$ ${\ displaystyle \ rho _ {1}, T_ {1}, {\ vec {v}} _ {1}}$ ), and others on the right ( $\rho _{2},T_{2},{\vec {v}}_{2}$ ${\ displaystyle \ rho _ {2}, T_ {2}, {\ vec {v}} _ {2}}$ ${\ displaystyle \ rho _ {2}, T_ {2}, {\ vec {v}} _ {2}}$ ) During unsteady motion of the medium, the fracture surfaces do not remain motionless, their speed may not coincide with the velocity of the medium.

A physically arbitrary discontinuity cannot exist for a finite time — this would require a violation of the equations of dynamics. For this reason, if in a situation a state described by an arbitrary discontinuity arises, it immediately begins to disintegrate upon occurrence - see the Riemann problem on the disintegration of an arbitrary discontinuity . In this case, depending on the medium in which the phenomenon occurs, and how the values of the state variables on different sides of the discontinuity relate to each other, various combinations of normal discontinuities and rarefaction waves can occur.

Content

Terms

The square brackets below indicate the difference in values on opposite sides of the surface.

On the rupture surfaces, certain relations must be satisfied:

On the surface of the gap must be a continuous flow of matter. The gas flow through the element of the fracture surface per unit area must be the same in magnitude on different sides of the fracture surface, i.e., the condition
$\left[\rho u_{x}\right]=0$ ${\ displaystyle \ left [\ rho u_ {x} \ right] = 0}$
Axis direction $x$ ${\ displaystyle x}$ selected normal to the fracture surface.
There must be a continuous flow of energy, that is, the condition must be met
$\left[\rho u_{x}\left({\frac {u^{2}}{2}}+\varepsilon \right)\right]=0$ ${\ displaystyle \ left [\ rho u_ {x} \ left ({\ frac {u ^ {2}} {2}} + \ varepsilon \ right) \ right] = 0}$
The momentum flow must be continuous, the forces with which the gases act on each other on both sides of the fracture surface must be equal. Since the normal vector is directed along the x axis, the continuity $x$ {\ displaystyle x} -components of the pulse flux leads to the condition
$\left[p+\rho u_{x}^{2}\right]=0$ ${\ displaystyle \ left [p + \ rho u_ {x} ^ {2} \ right] = 0}$
- Continuity $z$ ${\ displaystyle z}$ and $y$ ${\ displaystyle y}$ -component gives
$\left[\rho u_{x}u_{y}\right]=0$ ${\ displaystyle \ left [\ rho u_ {x} u_ {y} \ right] = 0}$ and $\left[\rho u_{x}u_{z}\right]=0$ ${\ displaystyle \ left [\ rho u_ {x} u_ {z} \ right] = 0}$

The equations above represent the complete system of boundary conditions on the discontinuity surface. From them we can conclude that there are two types of discontinuity surfaces.

Tangential Breaks

There is no substance flow through the fracture surface

{\begin{cases}\rho _{1}u_{1x}=\rho _{2}u_{2x}=0\\\rho _{1},\rho _{2}\neq 0\end{cases}}\Rightarrow \qquad u_{1x}=u_{2x}=0\qquad \Rightarrow p_{1}=p_{2}

{\ displaystyle {\ begin {cases} \ rho _ {1} u_ {1x} = \ rho _ {2} u_ {2x} = 0 \\\ rho _ {1}, \ rho _ {2} \ neq 0 \ end {cases}} \ Rightarrow \ qquad u_ {1x} = u_ {2x} = 0 \ qquad \ Rightarrow p_ {1} = p_ {2}}

Thus, in this case, the normal velocity component and gas pressure are continuous on the fracture surface. Tangential speeds $u_{z}$ ${\ displaystyle u_ {z}}$ , $u_{y}$ ${\ displaystyle u_ {y}}$ and density can experience an arbitrary jump. Such gaps are called tangential .

Contact discontinuities are a special case of tangential discontinuities. The speed is continuous. Density experiences a jump, and with it other thermodynamic quantities, with the exception of pressure.

Shockwaves

In the second case, the flow of matter, and with it the quantities are nonzero. Then from the conditions:

\left[\rho u_{x}\right]=0;\qquad \left[\rho u_{x}u_{y}\right]=0;\qquad \left[\rho u_{x}u_{z}\right]=0

{\ displaystyle \ left [\ rho u_ {x} \ right] = 0; \ qquad \ left [\ rho u_ {x} u_ {y} \ right] = 0; \ qquad \ left [\ rho u_ {x} u_ {z} \ right] = 0}

we have:

\left[u_{y}\right]=0\quad

{\ displaystyle \ left [u_ {y} \ right] = 0 \ quad}

and

\quad \left[u_{z}\right]=0

{\ displaystyle \ quad \ left [u_ {z} \ right] = 0}

tangential velocity is continuous on the fracture surface. Density, pressure, and with them other thermodynamic quantities, also experience a jump, and the jumps of these quantities are related by relations — conditions of discontinuity.

Of

\left[\rho u_{x}\left({\frac {u^{2}}{2}}+\varepsilon \right)\right];

{\ displaystyle \ left [\ rho u_ {x} \ left ({\ frac {u ^ {2}} {2}} + \ varepsilon \ right) \ right];}

\left[u_{y}\right]=0;

{\ displaystyle \ left [u_ {y} \ right] = 0;}

\left[u_{z}\right]=0

{\ displaystyle \ left [u_ {z} \ right] = 0}

we get

\left[\rho u_{x}\right]=0;\qquad \left[{\frac {u_{x}^{2}}{2}}+\varepsilon \right]=0;\qquad \left[p+\rho u_{x}^{2}\right]=0

{\ displaystyle \ left [\ rho u_ {x} \ right] = 0; \ qquad \ left [{\ frac {u_ {x} ^ {2}} {2}} + \ varepsilon \ right] = 0; \ qquad \ left [p + \ rho u_ {x} ^ {2} \ right] = 0}

Gaps of this type are called shock waves .

Gap propagation speed

To derive the relations on moving discontinuities, we can use the equations

{\begin{cases}{\begin{array}{lll}\oint \limits _{\partial \Omega }(\rho \;d\,x-\rho u\;d\,t)&=&0\\\oint \limits _{\partial \Omega }(\rho u\;d\,x-(p+\rho u^{2})\;d\,t)&=&0\\\oint \limits _{\partial \Omega }(E\;d\,x-(p+E)\;d\,t)&=&0\\\end{array}}\end{cases}}

{\ displaystyle {\ begin {cases} {\ begin {array} {lll} \ oint \ limits _ {\ partial \ Omega} (\ rho \; d \, x- \ rho u \; d \, t) & = & 0 \\\ oint \ limits _ {\ partial \ Omega} (\ rho u \; d \, x- (p + \ rho u ^ {2}) \; d \, t) & = & 0 \\\ oint \ limits _ {\ partial \ Omega} (E \; d \, x- (p + E) \; d \, t) & = & 0 \\\ end {array}} \ end {cases}}}

,

obtained using the Godunov method . She is:

\oint \limits _{\partial \Omega }(qdx-fdt)=0

{\ displaystyle \ oint \ limits _ {\ partial \ Omega} (qdx-fdt) = 0}

The gas-dynamic discontinuity in the one-dimensional non-stationary case is geometrically a curve in the plane. We construct the control volume near the gap so that the two sides of the contour covering this volume are parallel to the gap on both sides of the gap, and the other two sides are perpendicular to the gap. Writing down the system for a given control volume, then contracting the sides to zero and neglecting the integral on these sides, we obtain, taking into account the direction of the contour bypass and the signs of increments of coordinates and along the sides adjacent to the discontinuity:

\int \limits _{1-2}(qdx-fdt)-\int \limits _{3-4}(qdx-fdt)=0

{\ displaystyle \ int \ limits _ {1-2} (qdx-fdt) - \ int \ limits _ {3-4} (qdx-fdt) = 0}

Means

\int \limits _{1-2}(q{\frac {dx}{dt}}-f)-\int \limits _{3-4}(q{\frac {dx}{dt}}-f)=0

{\ displaystyle \ int \ limits _ {1-2} (q {\ frac {dx} {dt}} - f) - \ int \ limits _ {3-4} (q {\ frac {dx} {dt} } -f) = 0}

Value $D={\frac {dx}{dt}}$ ${\ displaystyle D = {\ frac {dx} {dt}}}$ - burst propagation velocity

Break Ratios

Passing to approximations of the integrals by the method of rectangles and using the notation for jumps of quantities at a break, we obtain a system of relations:

\left[\rho \right]D-\left[\rho u\right]=0;

{\ displaystyle \ left [\ rho \ right] D- \ left [\ rho u \ right] = 0;}

\left[\rho u\right]D-\left[p+\rho u^{2}\right]=0;

{\ displaystyle \ left [\ rho u \ right] D- \ left [p + \ rho u ^ {2} \ right] = 0;}

\left[E\right]D-\left[u(E+p)\right]=0;

{\ displaystyle \ left [E \ right] D- \ left [u (E + p) \ right] = 0;}

Examples

The boundary between two colliding bodies at the moment of collision, hereinafter, due to instability, an arbitrary discontinuity breaks up into two normal discontinuities moving in opposite directions.