--- title: "Characteristic curve; velocity and pressure at P" subject: "Computational Techniques in Civil Engineering" chapter: 6 key: "semester_8/computational_techniques_in_civil_engineering:characteristic-curve-pipe-numerical-b4" source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/characteristic-curve-pipe-numerical-b4 site: BCT Engineering --- # Characteristic curve; velocity and pressure at P Computational Techniques in Civil Engineering · Chapter 6: Method of Characteristics - Asked in: 2074 Bhadra, 2+6 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/characteristic-curve-pipe-numerical-b4 - Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-6 ## Question What do you understand by characteristic curve? Explain. A pipe of diameter 35 cm carrying water has the following data at two points A and B: $V_A = 6$ m/s, $V_B = 6.25$ m/s, $p_A = 102$ kN/m$^2$, $p_B = 124$ kN/m$^2$, $\Delta x = 500$ m, $\Delta t = 0.5$ s, $f = 0.02$, $a = 1000$ m/s ($dx/dt = \pm a$), elevation difference between A to P $= 2.50$ m. By the use of the finite difference form of the characteristics equations, compute the velocity and pressure at point P. [Figure: x-t grid with A and B at time t separated by $\Delta x$ on either side of C, and P at time $t+\Delta t$ above C, reached along the $C^+$ characteristic from A and the $C^-$ characteristic from B] ## Short answer Characteristics are dx/dt=plus or minus a lines in the x-t plane along which the unsteady pipe equations reduce to ODEs; C+ moves downstream and C- upstream. Using the finite-difference C+ and C- equations with uniform slope gives V_P=5.5533 m/s and p_P=9.88 kPa. ## Answer #### Characteristic curve In the $x$-$t$ plane, a characteristic curve is a line $dx/dt=\pm a$ along which the partial differential equations of unsteady pipe flow reduce to ordinary differential equations. It is the path of a small pressure disturbance: the $C^+$ curve ($dx/dt=+a$) carries information downstream and the $C^-$ curve ($dx/dt=-a$) upstream. The solution at a point $P$ is found from the values at the two points A and B where the $C^+$ and $C^-$ curves through $P$ meet the previous time line. ```text t+dt : P / \ C+ / \ C- t : A----C----B ``` #### Numerical part Equations used (velocity and pressure form, $\rho=1000$ kg/m$^3$, $\Delta x=500$ m so $a\Delta t=1000\times0.5=500$ m): $$ C^+:\ p_P=p_A-\rho a(V_P-V_A)-\rho g(z_P-z_A)-\frac{\rho f\Delta x}{2D}V_A|V_A| $$ $$ C^-:\ p_P=p_B+\rho a(V_P-V_B)+\rho g(z_B-z_P)+\frac{\rho f\Delta x}{2D}V_B|V_B| $$ **Assumption:** the pipe has a uniform slope, so P is 2.50 m above A and 2.50 m below B ($z_B-z_A=5.0$ m, $z_P-z_A=2.5$ m, $z_B-z_P=2.5$ m). **Constants** - $\rho a=1000\times1000=1.0\times10^{6}$ kg/(m$^2$s) - $\dfrac{\rho f\Delta x}{2D}=\dfrac{1000\times0.02\times500}{2\times0.35}=14285.71$ kg/m$^3$ (friction coefficient) - $\rho g(z_B-z_A)=9810\times5.0=49050$ N/m$^2$ **Velocity at P** $$ V_P=\frac{(p_A-p_B)+\rho a(V_A+V_B)-\rho g(z_B-z_A)-\frac{\rho f\Delta x}{2D}\left[V_A|V_A|+V_B|V_B|\right]}{2\rho a} $$ $$ V_P=\frac{(102000-124000)+10^6(12.25)-49050-14285.71(36+39.0625)}{2\times10^6} =\frac{11106629}{2\times10^6}=5.5533\ \text{m/s} $$ **Pressure at P** (from $C^+$) $$ p_P=102000-10^6(5.5533-6)-9810(2.5)-14285.71(36)=9875\ \text{N/m}^2=9.88\ \text{kN/m}^2 $$ Check with $C^-$: $p_P=124000+10^6(5.5533-6.25)+9810(2.5)+14285.71(39.0625)=9875$ N/m$^2$ (same). (The same result follows from the head form: $B_p=a/g=101.94$ s, $H_A=10.398$ m, $H_B=17.640$ m, giving $H_P=3.507$ m and $p_P=\gamma(H_P-z_P)$.) **Answer:** $V_P\approx5.5533$ m/s and $p_P\approx9.88$ kN/m$^2$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is a characteristic curve in the x-t plane for unsteady pipe flow? A. A line dx/dt = ±a along which the PDEs reduce to ODEs; it is the path of a pressure disturbance B. A line of constant pressure in the pipe C. A curve joining points of equal velocity D. The pump head-discharge curve Answer: A. Characteristics dx/dt = ±a are the paths of small disturbances and carry the ODE form of the equations. (Key point: Characteristic curve) 2. Which statement about the C+ and C- curves is correct? A. C+ carries information upstream, C- downstream B. Both carry information downstream C. C+ (dx/dt = +a) carries information downstream, C- (dx/dt = -a) upstream D. They are parallel and never meet Answer: C. C+ has slope +a (downstream), C- has slope -a (upstream); they meet at P. (Key point: C+ and C- directions) 3. How is V_P obtained from the C+ and C- pressure equations? A. Add the equations so the pressures cancel B. Equate the two expressions for p_P and solve for V_P C. Assume V_P = (V_A + V_B)/2 D. Use only the C+ equation with p_P known Answer: B. Setting the two p_P expressions equal removes pressure and gives V_P in terms of known values. (Key point: Velocity at P method) 4. Which are the computed velocity and pressure at P? A. V_P = 6.25 m/s, p_P = 124 kN/m^2 B. V_P = 5.5533 m/s, p_P = 9.88 kN/m^2 C. V_P = 5.5533 m/s, p_P = 98.8 kN/m^2 D. V_P = 6.0 m/s, p_P = 102 kN/m^2 Answer: B. V_P = 5.5533 m/s and p_P = 9875 N/m^2, which is 9.88 kN/m^2; the C- check gives the same. 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