Formulas

The majority of the formulas are sourced from the book Toegepaste Vloeistofmechanica by Nortier (available in Dutch only).

The table of contents is currently under revision. Additionally, the variable nomenclature requires further verification.

The highlighted formulas are implemented in the current version of Unreal Fluid Dynamics.

Table of Contents

Mathematics

$\displaystyle ax^2 + bx + c = 0$

$\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4 \cdot a \cdot c}}{2 \cdot a}$

Partially full pipes

$\displaystyle A_p(h) = \frac{1}{4} \cdot D^2 \cdot \arccos \left( 1 - \frac{2 \cdot h}{D} \right) - \left( \frac{D}{2} - h \right) \cdot \sqrt{h \cdot D - h^2}$

$\displaystyle R_p(h) = \frac{\frac{1}{4} \cdot D^2 \cdot \arccos \left( 1 - \frac{2 \cdot h}{D} \right) - \left( \frac{D}{2} - h \right) \cdot \sqrt{h \cdot D - h^2}}{D \cdot \arccos \left( 1 - \frac{2 \cdot h}{D} \right)}$

Variable Description Unit
$A_p$ Wetted Area partially filled pipe $[\text{m}^2]$
$R_p$ Hydraulic radius partially filled pipe $[\text{m}]$
$h$ water level partially filled pipe $[\text{m}]$
$D$ Diameter pipe $[\text{m}]$

Pressure and pressure head

$\displaystyle p = \frac{F}{A}$

Variable Description Unit
$p$ pressure $[\text{Pa} = \text{N/m}^2]$
$F$ force $[\text{N}]$
$A$ area on which pressure acts $[\text{m}^2]$

$\displaystyle F = m \cdot g$

Variable Description Unit
$F$ force $[\text{N}]$
$m$ mass $[\text{kg}]$
$g$ acceleration due to gravity $[\text{m/s}^2]$

$\displaystyle F = \rho \cdot g \cdot A \cdot h$

$\displaystyle p = \rho \cdot g \cdot h$

$\displaystyle h = \frac{p}{\rho \cdot g}$

Variable Description Unit
$p$ pressure $[\text{Pa}=\text{N/m}^2]$
$\rho$ density of the liquid $[\text{kg/m}^3]$
$g$ acceleration due to gravity $[\text{m/s}^2]$
$h$ pressure head $[\text{m}]$
$\rho_{fresh\ water}$ $1000$ $[\text{kg/m}^3]$
$\rho_{sea\ water}$ $1025$ $[\text{kg/m}^3]$

Continuity Law

$\displaystyle Q = v \cdot A$

Variable Description Unit
$Q$ flow rate or discharge $[\text{m}^3/\text{s}]$
$v$ flow velocity $[\text{m/s}]$
$A$ wetted area (cross-section) $[\text{m}^2]$

$\displaystyle A = 1/4 \cdot \pi \cdot D^2$

Variable Description Unit
$A$ wetted area $[\text{m}^2]$
$D$ pipe diameter $[\text{m}]$

Uniform, non-uniform flow, steady flow and unsteady flow

  • Uniform flow: The water has the same velocity at all points. $Q$ and $A$ are the same.
  • Non-uniform flow: The velocity of the water changes with location. $Q$ and $A$ differ.

Streamlines and flow patterns

  • Streamline is a line where the tangent at every point of the line gives the direction of the fluid flow, as it exists at a certain instant.

Normal Equation

$\displaystyle z_1 + h_1 + \int_1^2 \frac{v^2}{g \cdot r} \cdot dr = z_2 + h_2$

$\displaystyle \Delta h \approx \frac{v_{avg}^2}{g \cdot r_{avg}} \cdot b$

$\displaystyle \int_1^2 \frac{v^2}{g \cdot r_A} \cdot dr$

$\displaystyle \int_1^2 \frac{v^2}{g \cdot r_B} \cdot dr$

Centrifugal force

$\displaystyle F_c = \frac{m \cdot v^2}{r}$

Hydraulic radius

$\displaystyle R = \frac{A}{O}$

Variable Description Unit
$R$ hydraulic radius $[\text{m}]$
$A$ wetted area $[\text{m}^2]$
$O$ wetted perimeter $[\text{m}]$

Hydraulic radius of a fully or half-filled pipe

$\displaystyle R = \frac{A}{O} = \frac{\frac{1}{4} \cdot \pi \cdot D^2}{\pi \cdot D} = \frac{1}{4} \cdot D \quad [m]$

Hydraulic radius wide river

$\displaystyle R \approx \frac{b \cdot h}{b} = h$

Hydraulic diameter

$\displaystyle D = 4 \cdot R$

Variable Description Unit
$R$ Hydraulic Radius $[\text{m}]$
$D$ Hydraulic Diameter $[\text{m}]$

Reynolds Number

$\displaystyle \upsilon = \frac{\mu}{\rho}$

Variable Description Unit
$\mu$ absolute viscosity $[\text{kg/ms}]$
$\upsilon$ kinematic viscosity $[\text{m}^2/\text{s}]$
water, 20°C = $1.00 \cdot 10^{-6}$
$\rho$ density $[\text{kg/m}^3]$

$\displaystyle \Re = \frac{v \cdot R}{\vartheta}$

Variable Description Unit
$\vartheta$ kinematic viscosity $[\text{m}^2/\text{s}]$
water, 20°C = $1.00 \cdot 10^{-6}$
$v$ average flow velocity $[\text{m/s}]$
$R$ hydraulic radius $[\text{m}]$
$\text{Re}$ Reynolds number $[1]$

$\text{Re} > 800$ turbulent flow $\text{Re} < 400$ laminar flow

Permissible flow velocities

$\displaystyle v_{eh} = t \cdot f_t \cdot a \cdot C_k \cdot \sqrt{d_{50}}$

$\displaystyle t = \sqrt[4]{1 - \frac{\sin^2 \alpha}{\sin^2 \varphi}}$

$\displaystyle a = 0,22 \cdot \sqrt{\frac{\rho_{material} - \rho_{water}}{\rho_{water}}}$

$\displaystyle C_k = 18 \log \frac{12 \cdot R}{d_{90}}$

Variable Description Unit
$v_{eh}$ flow velocity at which particles begin to move $[\text{m/s}]$
$t$ slope factor $[1]$
$\alpha$ slope angle $[\text{degrees}]$
$\varphi$ internal friction angle $[\text{degrees}]$
$f_t$ turbulence factor $[1]$
$R$ hydraulic radius $[\text{m}]$
$d_{90}$ grain diameter $[\text{m}]$
$d_{50}$ average grain diameter (median) $[\text{m}]$

Bernoulli's Law

$\displaystyle E_{tot} = E_{potential} + E_{kinetic} = m \cdot g \cdot d + \frac{1}{2} \cdot m \cdot v^2 = constant$

$\displaystyle \frac{E_{tot}}{m \cdot g} = d + \frac{v^2}{2 \cdot g} = \frac{constant}{m \cdot g}$

$\displaystyle H = h + z + \frac{v^2}{2 \cdot g}$

$\displaystyle z_1 + h_1 + \frac{v_1^2}{2 \cdot g} = z_2 + h_2 + \frac{v_2^2}{2 \cdot g} + \Delta H_{1-2}$

$\displaystyle z_a + h_a + \frac{v_a^2}{2 \cdot g} = z_b + h_b + \frac{v_b^2}{2 \cdot g} + \Delta H_{a-b}$

$\displaystyle z_2 + h_2 + \frac{v_2^2}{2 \cdot g} + \Delta H_{pump} = z_3 + h_3 + \frac{v_3^2}{2 \cdot g} + \Delta H_{2-3}$

$\displaystyle z_a + h_a + \frac{v_a^2}{2 \cdot g} + \Delta H_{pump} = z_b + h_b + \frac{v_b^2}{2 \cdot g} + \Delta H_{a-b}$

Variable Description Unit
$H$ energy head $[\text{m}]$
$h = \frac{p}{\rho \cdot g}$ pressure head $[\text{m}]$
$z$ elevation head / position head $[\text{m}]$
$h+z$ piezometric level / pressure level / potential $[\text{m}]$
$\frac{v^2}{2 \cdot g}$ velocity head $[\text{m}]$
$\Delta H_{1-2}$ energy loss between 1 and 2 $[\text{m}]$
$\Delta H_{pump}$ added pump energy (pump head) $[\text{m}]$
$v$ flow velocity $[\text{m/s}]$
$g$ acceleration due to gravity $[\text{m/s}^2]$

Bernoulli's Equation in Pa (pressure)

$\displaystyle p_1 + \rho_1 \cdot g \cdot z_1 + \rho_1 \cdot \frac{v_1^2}{2} = p_2 + \rho_2 \cdot g \cdot z_2 + \rho_2 \cdot \frac{v_2^2}{2} + \Delta H_{1-2}$

Variable Description Unit
$p$ pressure $[\text{Pa}=\text{N/m}^2]$
$\rho$ fluid density $[\text{kg/m}^3]$
$z$ elevation head $[\text{m}]$
$g$ acceleration due to gravity $[\text{m/s}^2]$
$v$ flow velocity $[\text{m/s}]$
$\Delta H_{1-2}$ energy loss between 1 and 2 $[\text{Pa}]$

Specific energy

$\displaystyle E_s = y + \frac{v^2}{2 g}$

$\displaystyle h = \frac{p}{\rho \cdot g}$

Variable Description Unit
$E_s$ Specific energy $[\text{m}]$
$v$ average flow velocity $[\text{m/s}]$
$h$ pressure head $[\text{m}]$
$g$ acceleration due to gravity $[\text{m/s}^2]$

Power

$\displaystyle P = \rho \cdot g \cdot Q \cdot H$

$\displaystyle P_{pump} = \rho \cdot g \cdot Q \cdot \Delta H$

Variable Description Unit
$P$ power $[\text{Nm/s} = \text{J/s} = \text{W}]$
$\rho$ fluid density $[\text{kg/m}^3]$
$g$ acceleration due to gravity $[\text{m/s}^2]$
$Q$ flow rate $[\text{m}^3/\text{s}]$
$H$ energy head $[\text{m}]$
$\Delta H$ added energy $[\text{m}]$

Torricelli's Law:

$\displaystyle v = \sqrt{2 \cdot g \cdot x}$

Variable Description Unit
$v$ flow velocity $[\text{m/s}]$
$g$ acceleration due to gravity $[\text{m/s}^2]$
$x$ distance from hole to water surface $[\text{m}]$

Momentum Equation

$\displaystyle F_x = \rho \cdot Q \cdot (v_{2,x} - v_{1,x})$

Variable Description Unit
$F_x$ resultant force in x-direction $[\text{N}]$
$\rho$ density $[\text{kg/m}^3]$
$Q$ flow rate / discharge $[\text{m}^3/\text{s}]$
$v_{2,x}$ velocity at 2 in x-direction $[\text{m/s}]$
$v_{1,x}$ velocity at 1 in x-direction $[\text{m/s}]$

Energy losses (turbulent flow)

$\displaystyle \Delta H = \xi \cdot \frac{v^2}{2 \cdot g}$

Variable Description Unit
$\Delta H$ energy loss due to friction or deceleration $[\text{m}]$
$\frac{v^2}{2 \cdot g}$ velocity head $[\text{m}]$
$\xi$ loss coefficient due to friction or deceleration $[1]$

Friction losses Darcy-Weisbach

$\displaystyle \Delta H_w = \lambda \cdot \frac{L}{4 \cdot R} \cdot \frac{v^2}{2 \cdot g} = \xi_w \cdot \frac{v^2}{2 \cdot g}$

$\displaystyle \xi_w = \lambda \cdot \frac{L}{4 \cdot R}$

Variable Description Unit
$\Delta H_w$ energy loss due to friction $[\text{m}]$
$\lambda$ friction factor $[1]$
$\frac{v^2}{2 \cdot g}$ velocity head $[\text{m}]$
$R$ hydraulic radius $[\text{m}]$
$L$ length $[\text{m}]$
$\xi_w$ loss coefficient $[1]$
$g$ acceleration due to gravity $[\text{m/s}^2]$

Energy gradient

$\displaystyle I = \frac{\Delta H}{L}$

Variable Description Unit
$I$ energy gradient / energy slope $[-]$
$L$ length $[\text{m}]$
$\Delta H$ energy loss $[\text{m}]$

Chézy

$\displaystyle v = C \cdot \sqrt{R \cdot I}$

$\displaystyle C = \sqrt{\frac{8 g}{\lambda}}$

Variable Description Unit
$v$ average flow velocity $[\text{m/s}]$
$R$ hydraulic radius $[\text{m}]$
$I$ energy gradient $[1]$
$C$ Chézy coefficient $[\text{m}^{1/2}/\text{s}]$

Chézy coefficient

$\displaystyle \delta = \frac{12 \cdot \vartheta}{\sqrt{g \cdot R \cdot I}}$

$\displaystyle C = 18 \cdot \log \left[ \frac{48 \cdot R}{\delta} \right] \quad if \ \delta > 4 \cdot k \quad \text{hydraulically smooth}$

$\displaystyle C = 18 \cdot \log \left[ \frac{12 \cdot R}{k} \right] \quad if \ k > 6 \cdot \delta \quad \text{hydraulically rough}$

$\displaystyle C = 18 \cdot \log \left[ \frac{12 \cdot R}{k + \frac{1}{4}\delta} \right] \quad \text{technically rough}$

Variable Description Unit
$C$ Chézy coefficient $[\text{m}^{1/2}/\text{s}]$
$\delta$ laminar boundary layer $[\text{m}]$
$\vartheta$ kinematic viscosity $[\text{m}^2/\text{s}]$
$k$ wall roughness $[\text{m}]$
$R$ hydraulic radius $[\text{m}]$
$I$ energy gradient $[1]$

Manning

$\displaystyle v = \frac{R^{\frac{2}{3}} \cdot I^{\frac{1}{2}}}{n}$

$\displaystyle C = \frac{R^{\frac{1}{6}}}{n}$

Variable Description Unit
$v$ average flow velocity $[\text{m/s}]$
$R$ hydraulic radius $[\text{m}]$
$I$ energy gradient $[1]$
$n$ Manning coefficient $[\text{s/m}^{1/3}]$

Equilibrium depth

$\displaystyle h_e = \sqrt[3]{\frac{Q^2}{b^2 \cdot C^2 \cdot I}}$

$\displaystyle b_n = \frac{Q}{C \cdot h_e \cdot \sqrt{h_e \cdot I}}$

Variable Description Unit
$h_e$ equilibrium depth $[\text{m}]$
$b_n$ normal width $[\text{m}]$
$Q$ discharge / flow rate $[\text{m}^3/\text{s}]$
$b$ width $[\text{m}]$
$I$ energy gradient / bed slope $[1]$
$C$ Chézy coefficient $[\text{m}^{1/2}/\text{s}]$

Sudden expansion / exit loss

Carnot Equation

$\displaystyle \Delta H_v = \frac{(v_1 - v_2)^2}{2 g}$

Derivation 1 (based on $v_2$) Derivation 2 (based on $v_1$)
$\displaystyle \Delta H_v = \frac{(v_1 - v_2)^2}{2 g}$ $\displaystyle v_1 \cdot A_1 = v_2 \cdot A_2$
$\displaystyle v_1 = v_2 \cdot \frac{A_2}{A_1}$ $\displaystyle v_2 = v_1 \cdot \frac{A_1}{A_2}$
$\displaystyle \xi_v = \left( \frac{A_2}{A_1} - 1 \right)^2$ $\displaystyle \xi_v = \left( 1 - \frac{A_1}{A_2} \right)^2$
$\displaystyle \Delta H_v = \xi_v \cdot \frac{v_2^2}{2 g}$ $\displaystyle \Delta H_v = \xi_v \cdot \frac{v_1^2}{2 g}$

Gradual expansion

$\displaystyle \xi_g = n \cdot \left( \frac{A_2}{A_1} - 1 \right)^2$

Sudden contraction

$\displaystyle \xi_v = \left( \frac{1}{\mu} - 1 \right)^2$

Bend losses

$\displaystyle \Delta H_b = \xi_b \cdot \frac{v^2}{2 g}$

$\displaystyle \xi_{b,90} = \frac{0,44 \cdot D^2}{r^2} + 6 \cdot \lambda$

$\displaystyle \xi_{b,\alpha} = \sin \alpha \cdot \xi_{b,90}$

Fully filled culvert

$\displaystyle \xi_i = \left( \frac{1}{\mu} - 1 \right)^2$

$\displaystyle \xi_e = \left( 1 - \frac{A_1}{A_2} \right)^2$

$\displaystyle \xi_{total} = \xi_{inflow} + \xi_{friction} + \xi_{outflow}$

$\displaystyle \Delta H_{culvert} = \xi_{total} \cdot \frac{v_{culvert}^2}{2 g}$

Total head loss Coefficient composition
$\displaystyle \Delta H_{culvert} = \xi_{total} \cdot \frac{v_{culvert}^2}{2 g}$ $\displaystyle \xi_{total} = \xi_{inflow} + \xi_{friction} + \xi_{outflow}$
$\displaystyle \xi_i = \left( \frac{1}{\mu} - 1 \right)^2$ $\displaystyle \xi_f = \frac{\lambda \cdot l}{4 \cdot R}$ $\displaystyle \xi_e = \left( 1 - \frac{A_1}{A_2} \right)^2$

Discharge coefficient structure

$\displaystyle Q_{duiker} = m \cdot A_{duiker} \cdot \sqrt{2g \cdot \Delta H_{duiker}}$

$\displaystyle m = \frac{1}{\sqrt{\xi_{tot}}}$

Variable Description Unit
$Q$ discharge / flow rate $[\text{m}^3/\text{s}]$
$m$ discharge coefficient $[1]$
$A$ wetted area $[\text{m}^2]$
$\Delta H_{duiker}$ energy loss culvert $[\text{m}]$
$\xi_{tot}$ sum of loss coefficients $[1]$
$g$ acceleration due to gravity $[\text{m/s}^2]$

Laminar flow

$\displaystyle v_{avg\ open} = \frac{g \cdot h^2}{3 \cdot \vartheta} \cdot I$

$\displaystyle v_{avg\ pipe} = \frac{g \cdot R^2}{2 \cdot \vartheta} \cdot I$

Variable Description Unit
$v$ average flow velocity $[\text{m/s}]$
$R$ hydraulic radius $[\text{m}]$
$h$ water depth $[\text{m}]$
$I$ energy gradient $[1]$
$\vartheta$ kinematic viscosity $[\text{m}^2/\text{s}]$
water, 20°C = $1.00 \cdot 10^{-6}$

Long weirs / partially filled culvert

$\displaystyle h_3 > \frac{2}{3}H \quad \text{submerged / imperfect}$

$\displaystyle Q = c_{ol} \cdot b \cdot h_3 \cdot \sqrt{2g \cdot (H - h_3)}$

$\displaystyle c_{ol} \approx \frac{1}{\sqrt{\xi_{tot}}}$

$\displaystyle h_3 \le \frac{2}{3}H \quad \text{free flow / perfect}$

$\displaystyle Q = c_{vl} \cdot b \cdot H^{\frac{3}{2}}$

Variable Description Unit
$Q$ discharge $[\text{m}^3/\text{s}]$
$b$ weir width $[\text{m}]$
$c_{ol}$ discharge coefficient (submerged) $[1]$
$c_{vl}$ discharge coefficient (free flow) $[\text{m}^{1/2}/\text{s}]$
$H$ upstream energy head measured relative to crest $[\text{m}]$
$h_3$ downstream water level measured relative to crest $[\text{m}]$

Measuring weirs

$\displaystyle Q = c_{ok} \cdot b \cdot h_3 \cdot \sqrt{2g \cdot (H - h_3)}$

$\displaystyle Q = c_{vk} \cdot b \cdot H^{\frac{3}{2}}$

Parshall flume

$\displaystyle Q = c \cdot h^{1,55}$

V-notch weir

$\displaystyle Q = c \cdot \tan (0,5 \cdot \varphi) \cdot H^{2,5}$

Froude Number, subcritical and supercritical flow

$\displaystyle h_c = \sqrt[3]{\frac{Q^2}{g \cdot B^2}}$

$\displaystyle v_c = \sqrt[2]{g \cdot h_c}$

$\displaystyle H_{min} = \frac{3}{2} \cdot h_c$

$\displaystyle Fr = \frac{v}{\sqrt[2]{g \cdot h_c}} = \frac{v_{actual}}{v_c}$

Variable Description Unit
$h_c$ critical depth $[\text{m}]$
$Q$ discharge / flow rate $[\text{m}^3/\text{s}]$
$B$ width $[\text{m}]$
$v_c$ critical flow velocity $[\text{m/s}]$
$v$ actual flow velocity $[\text{m/s}]$
$Fr$ Froude number $[-]$
$g$ acceleration due to gravity $[\text{m/s}^2]$

Flow regimes

  • Subcritical (tranquil): $Fr < 1$ and $v < v_c$
  • Supercritical (rapid): $Fr > 1$ and $v > v_c$

Darcy's Law

$\displaystyle q = k \cdot A \cdot \frac{\Delta H}{l} = k \cdot A \cdot I$

$\displaystyle v = k \cdot I$

Variable Description Unit
$q$ discharge / flow rate $[\text{m}^3/\text{s}]$
$k$ hydraulic conductivity / permeability coefficient $[\text{m/s}]$
$A$ cross-section of soil sample $[\text{m}^2]$
$\Delta H$ hydraulic head difference $[\text{m}]$
$L$ length $[\text{m}]$
$v$ filter velocity $[\text{m/s}]$

$\displaystyle \text{Re}_k = \frac{v \cdot d}{\vartheta}$

Variable Description Unit
$\vartheta$ kinematic viscosity $[\text{m}^2/\text{s}]$
water, 20°C = $1.00 \cdot 10^{-6}$
$v$ filter velocity according to Darcy $[\text{m/s}]$
$d$ average grain diameter $[\text{m}]$
$\text{Re}_k$ Reynolds number related to grain $[1]$

$\text{Re}_k < 5$ laminar flow

Groundwater mounding

$\displaystyle H_0 = \sqrt{\frac{v_y \cdot l^2}{k} + H_1^2}$

Variable Description Unit
$H_1$ groundwater level at drain/ditch measured from impermeable layer $[\text{m}]$
$H_0$ groundwater level in the middle measured from impermeable layer $[\text{m}]$
$v_y$ vertical flow velocity $[\text{m/s}]$
$l$ distance drain/ditch to middle $[\text{m}]$
$k$ hydraulic conductivity $[\text{m/s}]$

Groundwater lowering (Well point dewatering)

Dupuit

$\displaystyle q_v = \frac{(H_b^2 - H^2) \cdot \pi \cdot k}{\ln R - \ln r}$

Variable Description Unit
$q_v$ extraction flow rate $[\text{m}^3/\text{s}]$
$H_b$ groundwater level without dewatering measured from impermeable layer $[\text{m}]$
$H$ groundwater level at distance $r$ measured from impermeable layer $[\text{m}]$
$r$ distance to well $[\text{m}]$
$R$ radius of influence $[\text{m}]$
$k$ hydraulic conductivity $[\text{m/s}]$

Sichardt

$\displaystyle R = 3000 \cdot d_0 \cdot \sqrt{k}$

Variable Description Unit
$d_0$ max lowering of groundwater level at well $[\text{m}]$
$R$ radius of influence $[\text{m}]$
$k$ hydraulic conductivity $[\text{m/s}]$

Maximum filter discharge

$\displaystyle q_{v,max} = 0.42 \cdot r_0 \cdot H_0 \cdot \sqrt{k}$

Variable Description Unit
$q_v$ max extraction flow rate $[\text{m}^3/\text{s}]$
$H_0$ filter length $[\text{m}]$
$r_0$ filter radius $[\text{m}]$
$R$ radius of influence $[\text{m}]$
$k$ hydraulic conductivity $[\text{m/s}]$

Multiple wells

$\displaystyle q_{v,tot} = \frac{(2 \cdot H_b \cdot d - d^2) \cdot \pi \cdot k}{\ln R - \frac{1}{n} \cdot \ln (r_1 \cdot r_2 \cdot \dots \cdot r_n)}$

Variable Description Unit
$q_v$ total extraction flow rate (all wells) $[\text{m}^3/\text{s}]$
$H_b$ groundwater level without dewatering measured from impermeable layer $[\text{m}]$
$d$ groundwater lowering at considered point $[\text{m}]$
$r$ distance to point per well $[\text{m}]$
$n$ number of wells $[1]$
$R$ radius of influence $[\text{m}]$
$k$ hydraulic conductivity $[\text{m/s}]$

Confined water (Confined aquifer)

$\displaystyle s = H - h = \frac{Q}{2 \cdot \pi \cdot k \cdot D} (\ln R - \ln r_0)$

Variable Description Unit
$s$ drawdown / lowering of hydraulic head $[\text{m}]$
$H$ hydraulic head without dewatering $[\text{m}]$
$h$ desired hydraulic head with dewatering $[\text{m}]$
$Q$ extraction flow rate $[\text{m}^3/\text{s}]$
$D$ thickness of water-bearing layer (aquifer) $[\text{m}]$
$r_0$ filter radius $[\text{m}]$
$R$ radius of influence $[\text{m}]$
$k$ hydraulic conductivity $[\text{m/s}]$

Waves

$\displaystyle L = c \cdot T$

Variable Description Unit
$L$ wavelength $[\text{m}]$
$c$ wave propagation speed / celerity $[\text{m/s}]$
$T$ wave period $[\text{s}]$

$\displaystyle T = (3.5 \dots 4) \cdot \sqrt{H}$

Variable Description Unit
$H$ wave height $[\text{m}]$
$T$ wave period $[\text{s}]$

$\displaystyle u_{surf} = \frac{\pi H_0}{T}$

Variable Description Unit
$u_{surf}$ particle velocity at surface $[\text{m/s}]$
$H_0$ deep water wave height $[\text{m}]$
$T$ wave period $[\text{s}]$

$\displaystyle u_y = \frac{\pi H}{T} \cdot \frac{\cosh \frac{2 \pi y}{L}}{\sinh \frac{2 \pi h}{L}}$

Variable Description Unit
$u_y$ velocity at $y$ from bottom $[\text{m/s}]$
$H$ wave height $[\text{m}]$
$T$ wave period $[\text{s}]$
$L$ wavelength $[\text{m}]$
$y$ distance measured from bottom $[\text{m}]$
$h$ water depth $[\text{m}]$

$\displaystyle u_{bottom} = \frac{\pi H}{T} \cdot \frac{1}{\sinh \frac{2 \pi h}{L}}$

$\displaystyle u_{surf} = \frac{\pi H}{T} \cdot \coth \frac{2 \pi h}{L}$

$\displaystyle c = \sqrt{\frac{g \cdot L}{2 \pi} \cdot \tanh \frac{2 \pi h}{L}}$

Variable Description Unit
$c$ wind wave propagation speed $[\text{m/s}]$
$L$ wavelength $[\text{m}]$
$h$ water depth $[\text{m}]$

$\displaystyle c = \sqrt{g \cdot h} \quad \text{very shallow water}$

$\displaystyle E_{tot} = \frac{1}{8} \cdot \rho \cdot g \cdot b \cdot L \cdot H^2$

$\displaystyle E_{m^2} = \frac{1}{8} \cdot \rho \cdot g \cdot H^2$

Variable Description Unit
$E_{tot}$ total energy in one wave $[\text{Nm}]$
$E_{m^2}$ energy per m$^2$ wave $[\text{Nm}]$
$\rho$ fluid density $[\text{kg/m}^3]$
$g$ acceleration due to gravity $[\text{m/s}^2]$
$L$ wavelength $[\text{m}]$
$H$ wave height $[\text{m}]$
$b$ wave width $[\text{m}]$

$\displaystyle L_0 = 1.56 \cdot T^2$

Variable Description Unit
$L_0$ deep water wavelength $[\text{m}]$
$T$ wave period $[\text{s}]$

$\displaystyle H_D = 0.5 \cdot h$

$\displaystyle H_{tot} = \sqrt{H_D^2 + H_n^2}$

Variable Description Unit
$H_D$ wave height transmission (depth limited) $[\text{m}]$
$h$ water depth foreshore $[\text{m}]$
$H_n$ wave generated over foreshore $[\text{m}]$

$\displaystyle s_u \approx 0.15 \cdot H_B$

Variable Description Unit
$s_u$ set-up $[\text{m}]$
$H_B$ breaker height $[\text{m}]$

$\displaystyle \xi = \frac{\tan \alpha}{\sqrt{\frac{H_0}{L_0}}}$

Variable Description Unit
$\xi$ wave breaking parameter (surf similarity) $[-]$
$H_0$ deep water wave height $[\text{m}]$
$L_0$ deep water wavelength $[\text{m}]$
$\tan \alpha$ coastal slope $[-]$

Bretschneider Formula

$\displaystyle \frac{g \cdot H_s}{U^2} = 0.283 \tanh \left[ 0.530 \left( \frac{gd}{U^2} \right)^{0.75} \right] \cdot \tanh \left[ \frac{0.0125 \left( \frac{gF}{U^2} \right)^{0.42}}{\tanh \left( 0.530 \left( \frac{gd}{U^2} \right)^{0.75} \right)} \right]$

$\displaystyle \frac{g \cdot T}{U} = 2 \pi \cdot 1.2 \tanh \left[ 0.833 \left( \frac{gd}{U^2} \right)^{0.375} \right] \cdot \tanh \left[ \frac{0.077 \left( \frac{gF}{U^2} \right)^{0.25}}{\tanh \left( 0.833 \left( \frac{gd}{U^2} \right)^{0.375} \right)} \right]$

Variable Description Unit
$H_s$ significant wave height $[\text{m}]$
$U$ wind speed $[\text{m/s}]$
$F$ fetch length < 5 km $[\text{m}]$
$d$ water depth $[\text{m}]$
$T$ period $[\text{s}]$

Fetch length (Bretschneider)

$\displaystyle F_e = \frac{\sum (l_\alpha \cdot \cos^2 \alpha)}{\sum (\cos \alpha)}$

Variable Description Unit
$F_e$ effective fetch length $[\text{m}]$
$\alpha$ angle with the normal $[\text{degrees}]$
$l_\alpha$ length at angle $\alpha$ $[\text{m}]$

Translatory waves De Saint-Venant

$\displaystyle c = 3 \cdot \sqrt{g \cdot (h+z)} - 2 \cdot \sqrt{g \cdot h}$

$\displaystyle z = \frac{Q}{b \cdot c} \quad \text{or} \quad c = \frac{Q}{b \cdot z}$

Variable Description Unit
$c$ propagation speed / celerity $[\text{m/s}]$
$h$ initial water depth $[\text{m}]$
$z$ height of the translatory wave (surge) $[\text{m}]$
$Q$ discharge / flow rate $[\text{m}^3/\text{s}]$

Hydraulic Engineering / Revetment

Leakage length

$\displaystyle \Lambda = \sqrt{\frac{b \cdot D \cdot k}{k'}}$

Variable Description Unit
$\Lambda$ leakage length $[\text{m}]$
$b$ thickness of the filter layer $[\text{m}]$
$D$ thickness of the top layer (cover layer) $[\text{m}]$
$k$ permeability of filter $[\text{m/s}]$
$k'$ permeability of top layer $[\text{m/s}]$

Stability formula "Van der Meer"

Breaking waves $\xi < \xi_m$

$\displaystyle \frac{H_s}{\Delta \cdot D_n} = 6.2 \cdot P^{0.18} \cdot \left( \frac{S}{\sqrt{N}} \right)^{0.2} \cdot \xi^{-0.5}$

Non-breaking waves $\xi > \xi_m$

$\displaystyle \frac{H_s}{\Delta \cdot D_n} = \left( \frac{S}{\sqrt{N}} \right)^{0.2} \cdot \sqrt{\cot \alpha} \cdot \xi^P \cdot P^{-0.13}$

Auxiliary parameters

$\displaystyle \Delta = \frac{\rho_s - \rho_w}{\rho_w}$

$\displaystyle \xi = \frac{\tan \alpha}{\sqrt{\frac{H_s}{L_0}}} = \frac{\tan \alpha}{\sqrt{\frac{2 \cdot \pi \cdot H_s}{g \cdot T^2}}}$

$\displaystyle \xi_m = \left( 6.2 \cdot P^{0.31} \cdot \sqrt{\tan \alpha} \right)^{\frac{1}{P+0.5}}$

Variable Description Unit
$H_s$ significant wave height $[\text{m}]$
$\Delta$ relative density $[-]$
$D_n$ nominal stone diameter ($D_{50}$) $[\text{m}]$
$P$ notional permeability factor $[-]$
permeable $P=0.5$ / impermeable $P=0.1$
$S$ damage level ($A/D_n^2$) $[-]$
$S < 4$ no damage
$A$ eroded area in a cross-section $[\text{m}^2]$
$N$ number of waves (7500) $[-]$
$\xi$ breaker parameter (surf similarity parameter) $[-]$
$\xi_m$ critical breaker parameter $[-]$
$L_0$ deep water wavelength $[\text{m}]$
$\tan \alpha$ slope $[-]$

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Soil Mechanics

Coulomb's Law

$\displaystyle \tau = c + \sigma_n \cdot \tan \varphi$

Variable Description Unit
$\tau$ shear stress $[\text{N/m}^2]$
$c$ cohesion $[\text{N/m}^2]$
$\sigma_n$ normal pressure / normal stress $[\text{N/m}^2]$
$\varphi$ angle of internal friction $[^\circ]$

Grain stress (Effective stress)

$\displaystyle \sigma' = \sigma - \sigma_w$

Variable Description Unit
$\sigma'$ effective stress (grain stress) $[\text{N/m}^2]$
$\sigma$ total soil stress $[\text{N/m}^2]$
$\sigma_w$ pore water pressure $[\text{N/m}^2]$

Bishop's Method

$\displaystyle SF = \frac{\tau \cdot \beta \cdot R^2}{G \cdot d}$

$2 \pi \ [\text{rad}] = 360^\circ$

Variable Description Unit
$SF$ stability factor (safety factor) $[-]$
$\tau$ shear stress along circle $[\text{N/m}]$
$\beta$ angle $[\text{rad}]$
$R$ radius $[\text{m}]$
$G$ weight of soil in the slice $[\text{N}]$
$d$ distance to center / arm $[\text{m}]$

$\displaystyle \tau_{cr} = c' + \sigma_n' \cdot \tan \varphi$

$\displaystyle \tau = \frac{1}{SF} \cdot (c' + \sigma_n' \cdot \tan \varphi')$

Stability (Moments and Slices)

$\displaystyle \tau = \text{actual occurring shear stress} \quad [\text{N/m}^2]$

Stability Factor

$\displaystyle \frac{M_{stab}}{M_{drive}} = SF \ge 1$

$\displaystyle M_{stab} = R \cdot \sum_i (\tau_i \cdot l_i)$

$\displaystyle M_{driving} = \sum_i (G_i \cdot a_i)$

Shear stress per slice

$\displaystyle \tau_i = \frac{\sin \varphi' \cdot \cos \alpha_i}{\cos(\varphi' + \alpha_i)} \cdot (\sigma_i' + c_i' \cdot \cot \varphi') = z_i \cdot (\sigma_i' + c_i' \cdot \cot \varphi')$

$\displaystyle z_i = \frac{\sin \varphi' \cdot \cos \alpha_i}{\cos(\varphi' + \alpha_i)}$

Validity range

$\displaystyle |\alpha| \le 45^\circ - \frac{\varphi'}{2}$

Variable Description Unit
$SF$ Stability Factor $[-]$
$M_{stab}$ stabilizing moment $[\text{Nm}]$
$M_{drive}$ driving moment $[\text{Nm}]$
$R$ radius of slip circle $[\text{m}]$
$\tau_i$ allowable shear stress per slice $i$ $[\text{N/m}^2]$
$l_i$ length of slice along slip circle $[\text{m}]$
$G_i$ weight of slice $i$ $[\text{N}]$
$a_i$ moment arm of weight relative to center $[\text{m}]$
$c_i'$ cohesion of slice $i$ $[\text{N/m}^2]$
$\sigma_i'$ vertical effective stress (grain stress) of slice $i$ $[\text{N/m}^2]$
$\alpha_i$ angle between normal to slip surface and vertical $[^\circ]$

Piping

Bligh's Formula

$\displaystyle L_{critical} = C_{Bligh} \cdot \Delta h$

Variable Description Unit
$L_{critical}$ critical creep length $[\text{m}]$
$C_{Bligh}$ Bligh's creep ratio $[-]$
$\Delta h$ head loss / hydraulic head difference $[\text{m}]$

Lane's Formula (Weighted Creep)

$\displaystyle \Delta H_c = \frac{\frac{1}{3} \cdot L_h + L_v}{C_{Lane}}$

Variable Description Unit
$L_h$ length of horizontal segments $[\text{m}]$
$L_v$ length of vertical segments $[\text{m}]$
$C_{Lane}$ Lane's creep ratio $[-]$
$\Delta H_c$ maximum allowable head difference $[\text{m}]$

Sellmeijer

$\displaystyle L_{crit} = \frac{H}{\alpha \cdot C \cdot \Delta \cdot \tan(\theta) \cdot [0.68 - 0.1 \cdot \ln(C)]}$

Sellmeijer Auxiliary parameters

$\displaystyle \Delta = \frac{\rho_g - \rho_w}{\rho_w}$

$\displaystyle \alpha = \left( \frac{D}{L} \right) ^ {\frac{0.28}{(D/L)^{2.8} - 1}}$

$\displaystyle C = \eta \cdot d_{70} \cdot \left[ \frac{1}{\kappa \cdot L} \right]^{1/3}$

$\displaystyle \kappa = \frac{\nu}{g} \cdot k$

Variable Description Unit
$L_{crit}$ critical piping length $[\text{m}]$
$H$ head difference across the structure $[\text{m}]$
$\rho_g$ density of soil (sand) $[\text{kg/m}^3]$
$\rho_w$ density of water $[\text{kg/m}^3]$
$\Delta$ relative density of soil (sand) submerged $[-]$
$D$ thickness of the sand layer $[\text{m}]$
$\kappa$ intrinsic permeability of sand layer $1.35 \cdot 10^{-7} [\text{m}^2]$
$\nu$ kinematic viscosity (groundwater 10°C) $1.33 \cdot 10^{-6} [\text{m}^2/\text{s}]$
$g$ acceleration due to gravity $9.81 [\text{m/s}^2]$
$d_{70}$ 70% value of the grain size distribution $[\text{m}]$
$\theta$ rolling resistance angle / bedding angle $46^\circ - 55^\circ$
$\eta$ White's coefficient / drag force factor $0.31 - 0.37 [-]$