Derivation of St. Venant Equation from Navier Stokes Equation for a Water Flow

St. Venant Equations ( one dimensional and two-dimensional) are the equations that can be used to describe the flow characteristics in one or two planes. Navier Stokes Equation describes the forces acting in an infinitesimal element which can be applied to any particle in the world and thus article will describe the derivation of St. Venant equation from the Navier Stokes Equation.

Water is pouring on to the land when it is raining and will flow out. When water is flowing from your tap, have you look at it and wonder how it happens.

For a fluid flowing in streams, canals or in a pipe gutter at our houses there are some governing equations which are were developed for an infinitesimally small fluid element in order to obtain the movement of fluid exactly (Grimm, 2004).

1. conservation of mass (continuity equation)
2. conservation of momentum (Newton's second law)
3. conservation of energy (The first law of thermodynamics)

These equations Conservation of momentum or the Navier -Stokes Equations were developed from the Newton's second law of motion (the rate of increase of momentum on the element is equal to the sum of the forces on the element).

Navier -Stokes Equations of fluid flow for incompressible flow of Newtonian fluids.

in which,

• ρ is the density
• u,v,w are the velocity components in x,y,z directions,
• p is the hydraulic pressure,
• µ is the viscosity
• g_x, g_y, g_z are the uniform gravitational acceleration components in x,y,z directions.

Derivation of St. Venant Equation from Navier Stokes

These are the assumptions for St.Venant Equations. (Brunner, Savant, & Ronald, 2020)

• Water is an incompressible fluid
• Pressure distribution is considered to be hydrostatic
• Vertical acceleration of water is considered negligible
• Bed slope is considered to be mild
• Effects of boundary friction can be taken into account with flow resistance laws derived for steady flows (Manning's equations)
• Boussinesq approximation is valid (ignoring forces caused by differences in density)

Following the assumptions for Navier - Strokes equation 1, for a 1D flow

Assuming friction is taken incorporated into the body forces,

For bed slope, tan tan θ ≈ 0

• Q is the flow discharge
• A is the flow cross-section area
• t represents time, h is the water depth
• g is the gravitational acceleration
• S_{(f )} is the friction slope
• S₀ is the channel bed slope.

References

1. Brunner, G., Savant, G. and Ronald, H. (2020) Modeler Application Guidance for Steady versus Unsteady, and 1D versus 2D versus 3D Hydraulic Modeling. Available at: www.hec.usace.army.mil
2. Grimm, J. P. (2004) An evaluation of alternative methodologies for the numerical simulation of solute transport. University of Sheffield. Available at: http://etheses.whiterose.ac.uk/21777/