🌊 Water Hammer Calculator
Solve hydraulic water hammer transients using the 1D Method of Characteristics (MOC). Compute wave speeds, closure times, peak surge pressures, hoop stresses, and pipe support forces.
⚡ Fortran 90 Engine
Double Precision (IEEE 754)
✓ ISO / ASME Validated
Fluid Mechanics
📊 Solver Telemetry
● ACTIVE
👁️ Views
831
⚡ Solves
652
💾 Downloads
401
📦 Fortran Code
4.5 KB
📅 Released
Jun 2026
⏱️ Latency
< 1 ms
📝 Configuration
📊 Simulation Results & Wave Plots
📊 Results Summary
Wave Speed ($a$)
1,342.9 m/s
Joukowsky relation
Critical Time ($2L/a$)
0.2234 s
Round-trip reflection
Closure Regime
Gradual
Regime classification
Pressure Surge ($\Delta P$)
1,411.9 kPa
14.12 bar / 204.8 psi
Peak Pressure ($P_{max}$)
1,711.9 kPa
17.12 bar
Min Pressure ($P_{min}$)
2.34 kPa
Suction limit
Cavitation Risk
⚠️ CAVITATION RISK
Column separation limit
Force on Support ($F$)
11.09 kN
2,492.8 lbf
Pipe Hoop Stress ($\sigma$)
17.12 MPa
Safety limit check
📈 Pressure History at Valve (P in kPa vs Time in s)
🛡️ Transient Wave Propagation Demo
Watch the pressure wave bounce between the reservoir (fixed boundary) and the valve.
Time step:
0.00s
🖨️ Raw Fortran Output
Wave Speed = 1342.86 Critical Time = 0.2234 Closure Type = Gradual DP Inst = 2685.72 DP Grad = 600.00 Pressure Surge = 1411.92 Max Pressure = 1711.92 Min Pressure = 2.34 Cavitation Risk = Yes Support Force = 11.089 Hoop Stress = 17.12 --- TIMELINE DATA --- 0.00000, 244.09 0.00559, 295.24 0.01117, 300.76 0.01676, 306.45 0.02234, 312.27 0.02793, 318.27 0.03351, 324.41 0.03910, 330.74 0.04468, 337.22 0.05027, 343.91 0.05585, 350.76 0.06144, 357.82 0.06702, 365.07 0.07261, 372.54 0.07819, 380.20 0.08378, 388.11 0.08936, 396.23 0.09495, 404.61 0.10053, 413.21 0.10612, 422.09 0.11170, 431.21 0.11729, 440.63 0.12287, 450.31 0.12846, 460.31 0.13404, 470.59 0.13963, 481.22 0.14521, 492.14 0.15080, 503.44 0.15638, 515.06 0.16197, 527.08 0.16755, 539.45 0.17314, 552.24 0.17872, 565.42 0.18431, 579.05 0.18989, 593.10 0.19548, 607.64 0.20106, 622.62 0.20665, 638.13 0.21223, 654.13 0.21782, 670.69 0.22340, 687.78 0.22899, 667.71 0.23457, 681.06 0.24016, 694.81 0.24574, 708.82 0.25133, 723.24 0.25691, 737.94 0.26250, 753.07 0.26808, 768.49 0.27367, 784.36 0.27925, 800.53 0.28484, 817.16 0.29042, 834.11 0.29601, 851.54 0.30159, 869.30 0.30718, 887.54 0.31276, 906.13 0.31835, 925.21 0.32393, 944.65 0.32952, 964.59 0.33511, 984.90 0.34069, 1005.72 0.34628, 1026.90 0.35186, 1048.61 0.35745, 1070.68 0.36303, 1093.27 0.36862, 1116.22 0.37420, 1139.69 0.37979, 1163.51 0.38537, 1187.84 0.39096, 1212.50 0.39654, 1237.67 0.40213, 1263.13 0.40771, 1289.07 0.41330, 1315.29 0.41888, 1341.94 0.42447, 1368.82 0.43005, 1396.10 0.43564, 1423.55 0.44122, 1451.32 0.44681, 1479.21 0.45239, 1488.67 0.45798, 1515.76 0.46356, 1543.16 0.46915, 1570.78 0.47473, 1598.68 0.48032, 1626.74 0.48590, 1655.01 0.49149, 1683.40 0.49707, 1711.92 0.50266, 1706.10 0.50824, 1661.59 0.51383, 1615.87 0.51941, 1568.95 0.52500, 1520.78 0.53058, 1471.35 0.53617, 1420.62 0.54175, 1368.56 0.54734, 1315.16 0.55292, 1260.37 0.55851, 1204.19 0.56409, 1146.56 0.56968, 1087.50 0.57526, 1026.94 0.58085, 964.90 0.58643, 901.32 0.59202, 836.21 0.59760, 769.53 0.60319, 701.30 0.60877, 631.45 0.61436, 560.04 0.61994, 486.98 0.62553, 412.35 0.63111, 336.06 0.63670, 258.22 0.64228, 178.72 0.64787, 97.69 0.65346, 15.03 0.65904, 2.34 0.66463, 2.34 0.67021, 2.34 0.67580, 2.34 0.68138, 2.34 0.68697, 2.34 0.69255, 2.34 0.69814, 2.34 0.70372, 2.34 0.70931, 2.34 0.71489, 2.34 0.72048, 2.34 0.72606, 2.34 0.73165, 2.34 0.73723, 2.34 0.74282, 2.34 0.74840, 2.34 0.75399, 2.34 0.75957, 2.34 0.76516, 2.34 0.77074, 2.34 0.77633, 2.34 0.78191, 2.34 0.78750, 2.34 0.79308, 2.34 0.79867, 2.34 0.80425, 2.34 0.80984, 2.34 0.81542, 2.34 0.82101, 2.34 0.82659, 2.34 0.83218, 2.34 0.83776, 42.22 0.84335, 114.99 0.84893, 189.32 0.85452, 265.30 0.86010, 342.85 0.86569, 422.03 0.87127, 502.74 0.87686, 585.07 0.88244, 597.71 0.88803, 597.70 0.89361, 597.70 0.89920, 597.69 0.90478, 597.69 0.91037, 597.67 0.91595, 597.65 0.92154, 597.62 0.92712, 597.59 0.93271, 597.55 0.93829, 597.50 0.94388, 597.44 0.94946, 597.39 0.95505, 597.33 0.96063, 597.29 0.96622, 597.24 0.97180, 597.21 0.97739, 597.17 0.98298, 597.14 0.98856, 597.12 0.99415, 597.10 0.99973, 597.07 1.00532, 597.06 1.01090, 597.05 1.01649, 597.04 1.02207, 597.03 1.02766, 597.03 1.03324, 597.02 1.03883, 597.02 1.04441, 597.02 1.05000, 597.02 1.05558, 597.02 1.06117, 557.23 1.06675, 484.62 1.07234, 410.45 1.07792, 334.62 1.08351, 257.24 1.08909, 178.22 1.09468, 97.68 1.10026, 15.54 1.10585, 2.93 1.11143, 2.94 1.11702, 2.94 1.12260, 2.95 1.12819, 2.95 1.13377, 2.97 1.13936, 2.99 1.14494, 3.02 1.15053, 3.05 1.15611, 3.09 1.16170, 3.14 1.16728, 3.20 1.17287, 3.25 1.17845, 3.31 1.18404, 3.35 1.18962, 3.40 1.19521, 3.43 1.20079, 3.47 1.20638, 3.49 1.21196, 3.52 1.21755, 3.54 1.22313, 3.57 1.22872, 3.58 1.23430, 3.59 1.23989, 3.60 1.24547, 3.61 1.25106, 3.61 1.25664, 3.62 1.26223, 3.62 1.26781, 3.62 1.27340, 3.62 1.27898, 3.62 1.28457, 43.32 1.29015, 115.77 1.29574, 189.78 1.30132, 265.45 1.30691, 342.67 1.31250, 421.52 1.31808, 501.89 1.32367, 583.84 1.32925, 596.43 1.33484, 596.42 1.34042, 596.42 1.34601, 596.41 1.35159, 596.41 1.35718, 596.39 1.36276, 596.38 1.36835, 596.34 1.37393, 596.32 1.37952, 596.27 1.38510, 596.23 1.39069, 596.16 1.39627, 596.11 1.40186, 596.06 1.40744, 596.01 1.41303, 595.97 1.41861, 595.93 1.42420, 595.89 1.42978, 595.87 1.43537, 595.84 1.44095, 595.82 1.44654, 595.80 1.45212, 595.79 1.45771, 595.77 1.46329, 595.77 1.46888, 595.75 1.47446, 595.75 1.48005, 595.74 1.48563, 595.75 1.49122, 595.74 1.49680, 595.75 1.50239, 595.74 1.50797, 556.13 1.51356, 483.85 1.51914, 409.99 1.52473, 334.48 1.53031, 257.42 1.53590, 178.73 1.54148, 98.54 1.54707, 16.76 1.55265, 4.21 1.55824, 4.22 1.56382, 4.21 1.56941, 4.22 1.57499, 4.23 1.58058, 4.25 1.58616, 4.26 1.59175, 4.29 1.59733, 4.32 1.60292, 4.37 1.60850, 4.41 1.61409, 4.47 1.61967, 4.52 1.62526, 4.58 1.63085, 4.62 1.63643, 4.67 1.64202, 4.70 1.64760, 4.74 1.65319, 4.76 1.65877, 4.80 1.66436, 4.81 1.66994, 4.84 1.67553, 4.84 1.68111, 4.87 1.68670, 4.87 1.69228, 4.88 1.69787, 4.88 1.70345, 4.89 1.70904, 4.88 1.71462, 4.89 1.72021, 4.89 1.72579, 4.89 1.73138, 44.41 1.73696, 116.54 1.74255, 190.24 1.74813, 265.59 1.75372, 342.49 1.75930, 421.01 1.76489, 501.04 1.77047, 582.63 1.77606, 595.16 1.78164, 595.15 1.78723, 595.16 1.79281, 595.14 1.79840, 595.14 1.80398, 595.12 1.80957, 595.11 1.81515, 595.07 1.82074, 595.05 1.82632, 595.00 1.83191, 594.96 1.83749, 594.89 1.84308, 594.85 1.84866, 594.79 1.85425, 594.75 1.85983, 594.70 1.86542, 594.67 1.87100, 594.63 1.87659, 594.61 1.88217, 594.57 1.88776, 594.56 1.89334, 594.53 1.89893, 594.53 1.90451, 594.50 1.91010, 594.50 1.91568, 594.49 1.92127, 594.49 1.92685, 594.48 1.93244, 594.49 1.93802, 594.48 1.94361, 594.49 1.94919, 594.48 1.95478, 555.05 1.96037, 483.08 1.96595, 409.54 1.97154, 334.34 1.97712, 257.60 1.98271, 179.24 1.98829, 99.38 1.99388, 17.97 1.99946, 5.47 2.00505, 5.48 2.01063, 5.47 2.01622, 5.49 2.02180, 5.49 2.02739, 5.51 2.03297, 5.52 2.03856, 5.56 2.04414, 5.58 2.04973, 5.63 2.05531, 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📘 Calculation Methodology
Mathematical Model & Theory
Water hammer is a pressure surge wave created when a fluid in motion is forced to stop suddenly (e.g., valve closure). The speed of the elastic shock wave $a$ is derived from the **Joukowsky equation** including pipe wall elasticity parameters:
$$a = \frac{\sqrt{K_f / \rho}}{\sqrt{1.0 + \frac{K_f \cdot D}{E_{pipe} \cdot e}}}$$
The maximum theoretical pressure rise for an instantaneous valve closure ($t_c < 2L/a$) is given by:
$$\Delta P_{inst} = \rho \cdot a \cdot V_0$$
If the valve closure is slow or gradual ($t_c \ge 2L/a$), pressure reflections return from the reservoir before the valve is fully closed, mitigating the surge. Rigid column theory estimates this maximum gradual surge as:
$$\Delta P_{grad} = \frac{\rho \cdot L \cdot V_0}{t_c}$$
Academic References
- Wylie, E. B. & Streeter, V. L.: Fluid Transients in Systems, Prentice Hall.
- Chaudhry, M. H.: Applied Hydraulic Transients, Springer.
- Joukowsky, N.: Uber den hydraulischen Stoss in Wasserleitungsrohren, 1898.
Worked Engineering Example
Problem Statement:
Water ($\rho = 1000\text{ kg/m³}, K_f = 2.2\text{ GPa}$) flows through a $L = 150\text{ m}$, $D = 100\text{ mm}$ steel pipe ($E = 200\text{ GPa}$, thickness $e = 5\text{ mm}$) at initial velocity $V_0 = 2\text{ m/s}$. The valve closes in $t_c = 0.5\text{ s}$. Find the wave speed and maximum pressure surge.
Step-by-step Solution:
1. Calculate wave speed $a$:
$$a = \frac{\sqrt{2.2 \times 10^9 / 1000}}{\sqrt{1.0 + \frac{2.2 \times 10^9 \times 0.100}{200 \times 10^9 \times 0.005}}} = \frac{1483.24}{\sqrt{1.0 + 0.22}} = 1342.86\text{ m/s}$$ 2. Calculate critical closure time $t_{crit}$:
$$t_{crit} = \frac{2 L}{a} = \frac{300}{1342.86} = 0.2234\text{ s}$$ 3. Evaluate closure type:
Since $t_c = 0.5\text{ s} > 0.2234\text{ s}$, closure is **Gradual**.
4. Run Method of Characteristics simulation:
Steady state friction results in pressure drops during closing. The simulation solves MOC and yields a maximum pressure surge of **1411.9 kPa** (at $t = t_c = 0.5\text{ s}$), which is higher than the rigid-column approximation (600 kPa) but smaller than the full Joukowsky surge (2685.7 kPa).
Water ($\rho = 1000\text{ kg/m³}, K_f = 2.2\text{ GPa}$) flows through a $L = 150\text{ m}$, $D = 100\text{ mm}$ steel pipe ($E = 200\text{ GPa}$, thickness $e = 5\text{ mm}$) at initial velocity $V_0 = 2\text{ m/s}$. The valve closes in $t_c = 0.5\text{ s}$. Find the wave speed and maximum pressure surge.
Step-by-step Solution:
1. Calculate wave speed $a$:
$$a = \frac{\sqrt{2.2 \times 10^9 / 1000}}{\sqrt{1.0 + \frac{2.2 \times 10^9 \times 0.100}{200 \times 10^9 \times 0.005}}} = \frac{1483.24}{\sqrt{1.0 + 0.22}} = 1342.86\text{ m/s}$$ 2. Calculate critical closure time $t_{crit}$:
$$t_{crit} = \frac{2 L}{a} = \frac{300}{1342.86} = 0.2234\text{ s}$$ 3. Evaluate closure type:
Since $t_c = 0.5\text{ s} > 0.2234\text{ s}$, closure is **Gradual**.
4. Run Method of Characteristics simulation:
Steady state friction results in pressure drops during closing. The simulation solves MOC and yields a maximum pressure surge of **1411.9 kPa** (at $t = t_c = 0.5\text{ s}$), which is higher than the rigid-column approximation (600 kPa) but smaller than the full Joukowsky surge (2685.7 kPa).