Critical Hydraulic Gradient

Critical hydraulic gradient, abbreviated as \(i_c\), also called critical gradient or critical slope, is used in fluid dynamics, specifically in open channel flow analysis.  It refers to the minimum slope or gradient required for a fluid (liquid) to flow in an open channel without any portion of the flow becoming stagnant or backflow occurring.

In an open channel, such as a river, stream, or canal, the flow of water is affected by the channel's slope and roughness, among other factors.  If the channel slope is too shallow, the water may not have enough energy to overcome frictional resistance, and the flow will slow down or stop at some points.  This condition is known as subcritical flow.  Conversely, if the channel slope is too steep, the water flow velocity may increase to the point where the flow becomes unstable and turbulent, resulting in excessive erosion and other issues.  This condition is known as supercritical flow.

The critical hydraulic gradient marks the transition point between subcritical and supercritical flow.  At this slope, the flow just reaches the threshold of being critical, with the water surface remaining smooth and free from disturbances or standing waves.  The critical slope is a specific value determined by the characteristics of the open channel, such as its cross-sectional shape and roughness.  It's important to note that the critical hydraulic gradient is a critical parameter in open-channel flow analysis as it defines the boundary between different flow regimes and helps engineers and hydrologists design and manage hydraulic structures and water conveyance systems.

 

Critical Hydraulic Gradient Formula
\( i_c \;=\; \dfrac{ \gamma_{sub} }{ \gamma_w }\)     (Critical Hydraulic Gradient) \( \gamma_{sub} \;=\;   i_c  \cdot \gamma_w \) \( \gamma_w \;=\; \dfrac{  \gamma_{sub} }{  i_c }\)
Symbol	English	Metric
\( i_c \) = critical hydraulic gradient	\( dimensionless \)	\( dimensionless \)
\( \gamma_{sub} \)  (Greek symbol gamma) = submerged unit weight	\( lbm \)	\( N \)
\(  \gamma_w \)  (Greek symbol gamma) = unit weight of water	\( lbm \;/\; ft^3\)	\(N \;/\; m^3\)

 

Critical Hydraulic Gradient Formula
\( i_c \;=\;   \dfrac{ SG_s - 1 }{ 1 - e }\)     (Critical Hydraulic Gradient) \( SG_s \;=\;  1 - e  \cdot i_c + 1 \) \( e \;=\;  \dfrac{ 1 - SG_s + i_c }{ i_c }\)
Symbol	English	Metric
\( i_c \) = critical hydraulic gradient	\( dimensionless \)	\( dimensionless \)
\( SG_s \) = specific gravity of soil	\( dimensionless \)	\( dimensionless \)
\( e \) = void ratio	\( dimensionless \)	\( dimensionless \)