Finite non-linear waves: Difference between revisions
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Starting point: the governing flow equations on partial differential form | Starting point: the governing flow equations on partial differential form | ||
Continuity equation: | Continuity equation: | ||
<math | {{NumEqn|<math> | ||
\frac{\partial \rho}{\partial t}+u\frac{\partial \rho}{\partial x}+\rho\frac{\partial u}{\partial x}=0 | \frac{\partial \rho}{\partial t}+u\frac{\partial \rho}{\partial x}+\rho\frac{\partial u}{\partial x}=0 | ||
</math> | </math>}} | ||
Momentum equation: | Momentum equation: | ||
<math | {{NumEqn|<math> | ||
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{1}{\rho}\frac{\partial p}{\partial x}=0 | \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{1}{\rho}\frac{\partial p}{\partial x}=0 | ||
</math> | </math>}} | ||
Any thermodynamic property can be expressed as a function of two other thermodynamic properties. This means that we can get density as a function of pressure and entropy: <math>\rho=\rho(p,s)</math> and therefore | Any thermodynamic property can be expressed as a function of two other thermodynamic properties. This means that we can get density as a function of pressure and entropy: <math>\rho=\rho(p,s)</math> and therefore | ||
<math | {{NumEqn|<math> | ||
d\rho=\left(\frac{\partial \rho}{\partial p}\right)_s dp+\left(\frac{\partial \rho}{\partial s}\right)_p ds | d\rho=\left(\frac{\partial \rho}{\partial p}\right)_s dp+\left(\frac{\partial \rho}{\partial s}\right)_p ds | ||
</math> | </math>}} | ||
Assuming isentropic flow <math>ds=0</math> gives | Assuming isentropic flow <math>ds=0</math> gives | ||
<math | {{NumEqn|<math> | ||
d\rho=\left(\frac{\partial \rho}{\partial p}\right)_s dp | d\rho=\left(\frac{\partial \rho}{\partial p}\right)_s dp | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
\begin{aligned} | \begin{aligned} | ||
&\frac{\partial \rho}{\partial t}=\left(\frac{\partial \rho}{\partial p}\right)_s\frac{\partial p}{\partial t}=\frac{1}{a^2}\frac{\partial p}{\partial t}\\ | &\frac{\partial \rho}{\partial t}=\left(\frac{\partial \rho}{\partial p}\right)_s\frac{\partial p}{\partial t}=\frac{1}{a^2}\frac{\partial p}{\partial t}\\ | ||
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&\frac{\partial \rho}{\partial x}=\left(\frac{\partial \rho}{\partial p}\right)_s\frac{\partial p}{\partial x}=\frac{1}{a^2}\frac{\partial p}{\partial x} | &\frac{\partial \rho}{\partial x}=\left(\frac{\partial \rho}{\partial p}\right)_s\frac{\partial p}{\partial x}=\frac{1}{a^2}\frac{\partial p}{\partial x} | ||
\end{aligned} | \end{aligned} | ||
</math> | </math>}} | ||
Now, insert \ref{eq:rhotop} in \ref{eq:pde:cont} gives | Now, insert \ref{eq:rhotop} in \ref{eq:pde:cont} gives | ||
<math | {{NumEqn|<math> | ||
\frac{\partial p}{\partial t}+u\frac{\partial p}{\partial x}+\rho a^2\frac{\partial u}{\partial x}=0 | \frac{\partial p}{\partial t}+u\frac{\partial p}{\partial x}+\rho a^2\frac{\partial u}{\partial x}=0 | ||
</math> | </math>}} | ||
Dividing \ref{eq:pde:cont:b} by <math>\rho a</math> gives | Dividing \ref{eq:pde:cont:b} by <math>\rho a</math> gives | ||
<math | {{NumEqn|<math> | ||
\frac{1}{\rho a}\left(\frac{\partial p}{\partial t}+u\frac{\partial p}{\partial x}\right)+a\frac{\partial u}{\partial x}=0 | \frac{1}{\rho a}\left(\frac{\partial p}{\partial t}+u\frac{\partial p}{\partial x}\right)+a\frac{\partial u}{\partial x}=0 | ||
</math> | </math>}} | ||
A slightly modified form of the momentum equation is obtained by multiplying and dividing the last term by <math>a</math> | A slightly modified form of the momentum equation is obtained by multiplying and dividing the last term by <math>a</math> | ||
<math | {{NumEqn|<math> | ||
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{1}{\rho a}\left(a\frac{\partial p}{\partial x}\right)=0 | \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{1}{\rho a}\left(a\frac{\partial p}{\partial x}\right)=0 | ||
</math> | </math>}} | ||
If the continuity equation on the form \ref{eq:pde:cont:c} is added to the momentum equation on the form \ref{eq:pde:mom:c}, we get | If the continuity equation on the form \ref{eq:pde:cont:c} is added to the momentum equation on the form \ref{eq:pde:mom:c}, we get | ||
<math | {{NumEqn|<math> | ||
\left[\frac{\partial u}{\partial t}+(u+a)\frac{\partial u}{\partial x}\right]+\frac{1}{\rho a}\left[\frac{\partial p}{\partial t}+(u+a)\frac{\partial p}{\partial x}\right]=0 | \left[\frac{\partial u}{\partial t}+(u+a)\frac{\partial u}{\partial x}\right]+\frac{1}{\rho a}\left[\frac{\partial p}{\partial t}+(u+a)\frac{\partial p}{\partial x}\right]=0 | ||
</math> | </math>}} | ||
If, instead, the continuity equation on the form \ref{eq:pde:cont:c} is subtracted from the momentum equation on the form \ref{eq:pde:mom:c}, we get | If, instead, the continuity equation on the form \ref{eq:pde:cont:c} is subtracted from the momentum equation on the form \ref{eq:pde:mom:c}, we get | ||
<math | {{NumEqn|<math> | ||
\left[\frac{\partial u}{\partial t}+(u-a)\frac{\partial u}{\partial x}\right]+\frac{1}{\rho a}\left[\frac{\partial p}{\partial t}+(u-a)\frac{\partial p}{\partial x}\right]=0 | \left[\frac{\partial u}{\partial t}+(u-a)\frac{\partial u}{\partial x}\right]+\frac{1}{\rho a}\left[\frac{\partial p}{\partial t}+(u-a)\frac{\partial p}{\partial x}\right]=0 | ||
</math> | </math>}} | ||
Since <math>u=u(x,t)</math>, we have from the definition of a differential | Since <math>u=u(x,t)</math>, we have from the definition of a differential | ||
<math | {{NumEqn|<math> | ||
du=\frac{\partial u}{\partial t}dt+\frac{\partial u}{\partial x}dx=\frac{\partial u}{\partial t}dt+\frac{\partial u}{\partial x}\frac{dx}{dt}dt | du=\frac{\partial u}{\partial t}dt+\frac{\partial u}{\partial x}dx=\frac{\partial u}{\partial t}dt+\frac{\partial u}{\partial x}\frac{dx}{dt}dt | ||
</math> | </math>}} | ||
Now, let <math>dx/dt=u+a</math> | Now, let <math>dx/dt=u+a</math> | ||
<math | {{NumEqn|<math> | ||
du=\frac{\partial u}{\partial t}dt+(u+a)\frac{\partial u}{\partial x}dt=\left[\frac{\partial u}{\partial t}+(u+a)\frac{\partial u}{\partial x}\right]dt | du=\frac{\partial u}{\partial t}dt+(u+a)\frac{\partial u}{\partial x}dt=\left[\frac{\partial u}{\partial t}+(u+a)\frac{\partial u}{\partial x}\right]dt | ||
</math> | </math>}} | ||
which is the change of <math>u</math> in the direction <math>dx/dt=u+a</math> | which is the change of <math>u</math> in the direction <math>dx/dt=u+a</math> | ||
| Line 88: | Line 96: | ||
In the same way | In the same way | ||
<math | {{NumEqn|<math> | ||
dp=\frac{\partial p}{\partial t}dt+\frac{\partial p}{\partial x}dx=\frac{\partial p}{\partial t}dt+\frac{\partial p}{\partial x}\frac{dx}{dt}dt | dp=\frac{\partial p}{\partial t}dt+\frac{\partial p}{\partial x}dx=\frac{\partial p}{\partial t}dt+\frac{\partial p}{\partial x}\frac{dx}{dt}dt | ||
</math> | </math>}} | ||
and thus, in the direction <math>dx/dt=u+a</math> | and thus, in the direction <math>dx/dt=u+a</math> | ||
<math | {{NumEqn|<math> | ||
dp=\frac{\partial p}{\partial t}dt+(u+a)\frac{\partial p}{\partial x}dt=\left[\frac{\partial p}{\partial t}+(u+a)\frac{\partial p}{\partial x}\right]dt | dp=\frac{\partial p}{\partial t}dt+(u+a)\frac{\partial p}{\partial x}dt=\left[\frac{\partial p}{\partial t}+(u+a)\frac{\partial p}{\partial x}\right]dt | ||
</math> | </math>}} | ||
If we go back and examine Eqn. \ref{eq:nonlin:a}, we see that Eqns. \ref{eq:du:b} and \ref{eq:dp:b} appear in the equation and thus it can now be rewritten as follows | If we go back and examine Eqn. \ref{eq:nonlin:a}, we see that Eqns. \ref{eq:du:b} and \ref{eq:dp:b} appear in the equation and thus it can now be rewritten as follows | ||
<math | {{NumEqn|<math> | ||
\frac{du}{dt}+\frac{1}{\rho a}\frac{dp}{dt}=0\Rightarrow du+\frac{dp}{\rho a}=0 | \frac{du}{dt}+\frac{1}{\rho a}\frac{dp}{dt}=0\Rightarrow du+\frac{dp}{\rho a}=0 | ||
</math> | </math>}} | ||
Eqn. \ref{eq:nonlin:a:ode} applies along a <math>C^+</math> characteristic, i.e., a line in the direction <math>dx/dt=u+a</math> in <math>xt</math>-space and is called the compatibility equation along the <math>C^+</math> characteristic. If we instead chose a <math>C^-</math> characteristic, i.e., a line in the direction <math>dx/dt=u-a</math> in <math>xt</math>-space, we get | Eqn. \ref{eq:nonlin:a:ode} applies along a <math>C^+</math> characteristic, i.e., a line in the direction <math>dx/dt=u+a</math> in <math>xt</math>-space and is called the compatibility equation along the <math>C^+</math> characteristic. If we instead chose a <math>C^-</math> characteristic, i.e., a line in the direction <math>dx/dt=u-a</math> in <math>xt</math>-space, we get | ||
<math | {{NumEqn|<math> | ||
du=\left[\frac{\partial u}{\partial t}+(u-a)\frac{\partial u}{\partial x}\right]dt | du=\left[\frac{\partial u}{\partial t}+(u-a)\frac{\partial u}{\partial x}\right]dt | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
dp=\left[\frac{\partial p}{\partial t}+(u-a)\frac{\partial p}{\partial x}\right]dt | dp=\left[\frac{\partial p}{\partial t}+(u-a)\frac{\partial p}{\partial x}\right]dt | ||
</math> | </math>}} | ||
which can be identified as subsets of Eqn. \ref{eq:nonlin:b} and thus | which can be identified as subsets of Eqn. \ref{eq:nonlin:b} and thus | ||
<math | {{NumEqn|<math> | ||
\frac{du}{dt}-\frac{1}{\rho a}\frac{dp}{dt}=0 | \frac{du}{dt}-\frac{1}{\rho a}\frac{dp}{dt}=0 | ||
</math> | </math>}} | ||
In order to fulfil the relation above, either <math>du=dp=0</math> or | In order to fulfil the relation above, either <math>du=dp=0</math> or | ||
<math | {{NumEqn|<math> | ||
du-\frac{dp}{\rho a}=0 | du-\frac{dp}{\rho a}=0 | ||
</math> | </math>}} | ||
Eqn. \ref{eq:nonlin:b:ode} applies along a <math>C^-</math> characteristic, i.e., a line in the direction <math>dx/dt=u-a</math> in <math>xt</math>-space and is called the compatibility equation along the <math>C^-</math> characteristic. | Eqn. \ref{eq:nonlin:b:ode} applies along a <math>C^-</math> characteristic, i.e., a line in the direction <math>dx/dt=u-a</math> in <math>xt</math>-space and is called the compatibility equation along the <math>C^-</math> characteristic. | ||
| Line 144: | Line 152: | ||
If the compatibility equations are integrated along respective characteristic line, i.e., integrate \ref{eq:nonlin:a:ode} along the <math>C^+</math> characteristic and \ref{eq:nonlin:b:ode} along the <math>C^-</math> characteristic, we get the Riemann invariants <math>J^+</math> and <math>J^-</math>. | If the compatibility equations are integrated along respective characteristic line, i.e., integrate \ref{eq:nonlin:a:ode} along the <math>C^+</math> characteristic and \ref{eq:nonlin:b:ode} along the <math>C^-</math> characteristic, we get the Riemann invariants <math>J^+</math> and <math>J^-</math>. | ||
<math | {{NumEqn|<math> | ||
J^+=u+\int\frac{dp}{\rho a}=const | J^+=u+\int\frac{dp}{\rho a}=const | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
J^-=u-\int\frac{dp}{\rho a}=const | J^-=u-\int\frac{dp}{\rho a}=const | ||
</math> | </math>}} | ||
The Riemann invariants are constants along the associated characteristic line. | The Riemann invariants are constants along the associated characteristic line. | ||
| Line 156: | Line 164: | ||
We have assumed isentropic flow and thus we may use the isentropic relations | We have assumed isentropic flow and thus we may use the isentropic relations | ||
<math | {{NumEqn|<math> | ||
p=C_1T^{\gamma/(\gamma-1)}=C_2a^{2\gamma/(\gamma-1)} | p=C_1T^{\gamma/(\gamma-1)}=C_2a^{2\gamma/(\gamma-1)} | ||
</math> | </math>}} | ||
where <math>C_1</math> and <math>C_2</math> are constants. Differentiating Eqn. \ref{eq:isentropic:a} gives | where <math>C_1</math> and <math>C_2</math> are constants. Differentiating Eqn. \ref{eq:isentropic:a} gives | ||
<math | {{NumEqn|<math> | ||
dp=C_2\left(\frac{2\gamma}{\gamma-1}\right)a^{[2\gamma/(\gamma-1)-1]}da | dp=C_2\left(\frac{2\gamma}{\gamma-1}\right)a^{[2\gamma/(\gamma-1)-1]}da | ||
</math> | </math>}} | ||
Now, if we further assume the gas to be calorically perfect | Now, if we further assume the gas to be calorically perfect | ||
<math | {{NumEqn|<math> | ||
a^2=\gamma RT=\frac{\gamma p}{\rho}\Rightarrow \rho=\frac{\gamma p}{a^2} | a^2=\gamma RT=\frac{\gamma p}{\rho}\Rightarrow \rho=\frac{\gamma p}{a^2} | ||
</math> | </math>}} | ||
Eqn. \ref{eq:isentropic:a} in \ref{eq:isentropic:c} gives | Eqn. \ref{eq:isentropic:a} in \ref{eq:isentropic:c} gives | ||
<math | {{NumEqn|<math> | ||
\rho=C_2\gamma a^{[2\gamma/(\gamma-1)-2]} | \rho=C_2\gamma a^{[2\gamma/(\gamma-1)-2]} | ||
</math> | </math>}} | ||
and thus | and thus | ||
<math | {{NumEqn|<math> | ||
J^+=u+\int\frac{C_2\left(\frac{2\gamma}{\gamma-1}\right)a^{[2\gamma/(\gamma-1)-1]}}{C_2\gamma a^{[2\gamma/(\gamma-1)-2]}a}da=u+\left(\frac{2}{\gamma-1}\right)\int da | J^+=u+\int\frac{C_2\left(\frac{2\gamma}{\gamma-1}\right)a^{[2\gamma/(\gamma-1)-1]}}{C_2\gamma a^{[2\gamma/(\gamma-1)-2]}a}da=u+\left(\frac{2}{\gamma-1}\right)\int da | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
J^+=u+\frac{2a}{\gamma-1} | J^+=u+\frac{2a}{\gamma-1} | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
J^-=u-\frac{2a}{\gamma-1} | J^-=u-\frac{2a}{\gamma-1} | ||
</math> | </math>}} | ||
Eqns. \ref{eq:riemann:a:b} and \ref{eq:riemann:b:b} are the Riemann invariants for a calorically perfect gas. The Riemann invariants are constants along <math>C^+</math> and <math>C^-</math> characteristics and if the situation shown in Fig. \ref{fig:characteristics} appears, that fact can be used to calculate the flow velocity and speed of sound in the location (<math>x_1</math>, <math>t_1</math>). | Eqns. \ref{eq:riemann:a:b} and \ref{eq:riemann:b:b} are the Riemann invariants for a calorically perfect gas. The Riemann invariants are constants along <math>C^+</math> and <math>C^-</math> characteristics and if the situation shown in Fig. \ref{fig:characteristics} appears, that fact can be used to calculate the flow velocity and speed of sound in the location (<math>x_1</math>, <math>t_1</math>). | ||
<math | {{NumEqn|<math> | ||
J^++J^-=u+\frac{2a}{\gamma-1}+u-\frac{2a}{\gamma-1}=2u\Rightarrow u=\frac{1}{2}(J^++J^-) | J^++J^-=u+\frac{2a}{\gamma-1}+u-\frac{2a}{\gamma-1}=2u\Rightarrow u=\frac{1}{2}(J^++J^-) | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
J^+=u+\frac{2a}{\gamma-1}=\frac{1}{2}(J^++J^-)+\frac{2a}{\gamma-1}\Rightarrow a=\frac{\gamma-1}{4}(J^+-J^-) | J^+=u+\frac{2a}{\gamma-1}=\frac{1}{2}(J^++J^-)+\frac{2a}{\gamma-1}\Rightarrow a=\frac{\gamma-1}{4}(J^+-J^-) | ||
</math> | </math>}} | ||
