Characteristic form FD solution (velocity, pressure)

Governing equations (velocity $V$, pressure $p$)

For a pipe of diameter $D$ inclined at angle $\theta$ to the horizontal (water density $\rho$, wave speed $a$, friction factor $f$), neglecting the small convective terms:

Momentum:

$$L_1=\frac{\partial V}{\partial t}+\frac1\rho\frac{\partial p}{\partial x}+g\sin\theta+\frac{f}{2D}V|V|=0$$

Continuity:

$$L_2=\frac{\partial p}{\partial t}+\rho a^2\frac{\partial V}{\partial x}=0$$

Characteristic equations

Form $L_1+\lambda L_2=0$:

$$\left[\frac{\partial V}{\partial t}+\lambda\rho a^2\frac{\partial V}{\partial x}\right]+\lambda\left[\frac{\partial p}{\partial t}+\frac{1}{\lambda\rho}\frac{\partial p}{\partial x}\right]+g\sin\theta+\frac{f}{2D}V|V|=0$$

For both brackets to be total derivatives, $\dfrac{dx}{dt}=\lambda\rho a^2=\dfrac{1}{\lambda\rho}$, so $\lambda=\pm\dfrac{1}{\rho a}$ and $\dfrac{dx}{dt}=\pm a$:

$$C^+\ \left(\frac{dx}{dt}=+a\right):\quad \frac{dV}{dt}+\frac{1}{\rho a}\frac{dp}{dt}+g\sin\theta+\frac{f}{2D}V|V|=0$$ $$C^-\ \left(\frac{dx}{dt}=-a\right):\quad \frac{dV}{dt}-\frac{1}{\rho a}\frac{dp}{dt}+g\sin\theta+\frac{f}{2D}V|V|=0$$

Finite difference form

Use the rectangular grid with A, C, B at time $t$ and P at $t+\Delta t$, with $\Delta x=a\Delta t$. Multiply by $\rho a$, integrate along each characteristic (friction at the known point), and put $\Delta x\sin\theta=\Delta z$ (rise of the pipe over $\Delta x$):

 t+dt :         P
               / \
         C+  /     \  C-
 t    :    A----C----B

$$C^+:\quad p_P=p_A-\rho a\left(V_P-V_A\right)-\rho g\left(z_P-z_A\right)-\frac{\rho f\Delta x}{2D}V_A|V_A|$$ $$C^-:\quad p_P=p_B+\rho a\left(V_P-V_B\right)+\rho g\left(z_B-z_P\right)+\frac{\rho f\Delta x}{2D}V_B|V_B|$$

Solution

Equating the two expressions for $p_P$:

$$V_P=\frac{(p_A-p_B)+\rho a(V_A+V_B)-\rho g(z_B-z_A)-\dfrac{\rho f\Delta x}{2D}\left[V_A|V_A|+V_B|V_B|\right]}{2\rho a}$$

and $p_P$ is then obtained from either equation. At each time step, the interior points are found this way; boundary points use one characteristic plus the boundary condition (e.g. $p=$ reservoir pressure, or $V=0$ at a closed valve).