Quasi-3D Flow Calculation Method

Why quasi-3D calculation is needed

Cabernet2D’s governing equations are depth-averaged 2D shallow-water equations that assume a uniform velocity profile over the water depth. In real flows, however, there are strongly three-dimensional structures that vary significantly over depth, such as the helical secondary flow in meandering channels (where the near-bed flow turns toward the inner bank) and the horseshoe vortex around obstacles such as bridge piers and spur dikes. These structures make the near-bed flow direction and magnitude different from the depth-averaged velocity, and they govern local scour and deposition patterns.

The quasi-3D flow calculation method is a simplified way of evaluating this kind of depth-wise flow structure in order to correct the near-bed velocity, which directly determines the direction and magnitude of bedload sediment transport. Three methods are available in the GUI calculation condition “Quasi-3D calculation method (j_q3d).”

The three options in the GUI

GUI label	Name	Overview
	Engelund equilibrium secondary flow	Directly computes the near-bed transverse velocity from the streamline curvature.
	Nays2DH \(A_n\) transport equation	Evolves the secondary flow intensity \(A_n\) through a transport equation, representing the delay in development and decay at bend entrances and exits.
	BVC + dynamic pressure correction (Uchida & Fukuoka, 2012)	Directly estimates the near-bed velocity vector from the transport equation of horizontal vorticity, and adds dispersion-stress and non-hydrostatic bottom-pressure correction terms to the depth-averaged momentum equations.

Note

Method (2) BVC is very effective for local scour calculations where the flow changes abruptly over a short distance, such as around bridge piers and impermeable spur dikes. For straight channels or gently meandering channels, however, the simpler methods (0) Engelund and (1) Nays2DH may reproduce the observed bed variation just as well or better. Before adopting (2) as your final result, we recommend comparing it against (0)/(1) and choosing the method that best matches your site conditions and objectives.

(0) Engelund equilibrium secondary flow

The secondary flow intensity is computed directly as an equilibrium value from the local streamline curvature, water depth and velocity (Engelund, 1974).

\[A_n = V \cdot h \cdot \frac{1}{r_s}\]

\[u_{n,\mathrm{bed}} = \beta_g \cdot N^* \cdot V \cdot \frac{h}{r_s}\]

where \(V\) is the depth-averaged velocity, \(h\) is the water depth, \(r_s\) is the radius of curvature of the streamline, \(\beta_g\) is a coefficient of the bedload transport formula, and \(N^*\) is the secondary flow intensity coefficient.

\(N^*\) is set with the GUI parameter “Secondary flow intensity coefficient N* (snst)” (default 7.0, the theoretical value from Engelund). This parameter is only active when this method is selected; for the other methods it stays fixed at its default and is not shown.

Because the calculation is instantaneous (an equilibrium assumption), this is the fastest and most stable method, but it cannot represent the delayed development of secondary flow at a bend entrance.

(1) Nays2DH \(A_n\) transport equation

This method evolves the secondary flow intensity \(A_n\) in space and time via a transport equation, removing the equilibrium assumption (Johannesson & Parker, 1989; Hosoda et al., 2000; Iwasaki, Shimizu & Kimura, 2013).

\[\lambda_v \frac{\partial(A_n/J)}{\partial t} + \nabla \cdot \left[\boldsymbol{u}_{\mathrm{surf}} \cdot \frac{u_{n,\mathrm{surf}}}{J}\right] - \nabla \cdot \left[\boldsymbol{u}_{\mathrm{bed}} \cdot \frac{u_{n,\mathrm{bed}}}{J}\right] = \frac{1}{J}\frac{u_{s,\mathrm{surf}}^2 - u_{s,\mathrm{bed}}^2}{r_s} + \frac{A_n}{J}\cdot\frac{V(\chi^2 + \frac{7}{12}\chi + \frac{1}{12})}{h\,\chi_1^3}\]

\(J\) is the Jacobian of the computational grid, \(\lambda_v\) is a time-scale coefficient, \(\chi,\,\chi_1\) are Engelund parameters, and the subscripts surf and bed denote values near the water surface and near the bed, respectively.

This method can reproduce the delayed development and decay of secondary flow at the entrance and exit of a bend, making it suitable for meandering channels. It also feeds the dispersion stress back into the depth-averaged momentum equations directly: this stress arises from depth-integrating the 3D velocity profile (increased bed shear on the outer bank due to helical flow).

(2) BVC + dynamic pressure correction

The general bottom-velocity analysis method (Uchida & Fukuoka, 2011, 2012) directly estimates the near-bed velocity vector from the depth integral of the surface velocity and horizontal vorticity.

\[u_{b,x} = u_{s,x} - \Omega_y \cdot h - \frac{\partial W_h}{\partial x} - w_s\frac{\partial z_s}{\partial x}, \qquad u_{b,y} = u_{s,y} + \Omega_x \cdot h - \frac{\partial W_h}{\partial y} - w_s\frac{\partial z_s}{\partial y}\]

\(\Omega_i\) is the depth integral of horizontal vorticity, \(W_h\) is the depth integral of vertical velocity, \(w_s\) is the vertical velocity at the water surface, and \(z_s\) is the water surface elevation. Through the rise and “blow-up” of the water surface on the upstream face of an obstacle, the reversal of near-bed velocity associated with a horseshoe vortex emerges naturally. The shallow-water approximation omits this effect, so it cannot reproduce the correct near-bed velocity pattern upstream of an obstacle.

In method (2), the near-bed velocity estimate above is always combined with two additional correction terms that are built into the depth-averaged momentum equations (in the GUI these are offered as a single option, and operate internally with fixed default settings):

Dispersion tensor: evaluates the additional momentum transport that arises from depth-integrating the 3D velocity profile (an effect of helical flow in meandering channels and horseshoe vortices around obstacles), computed from the difference between the BVC surface and bottom velocities, and adds it as an extra force in the momentum equations.
Non-hydrostatic bottom pressure correction: adds the non-hydrostatic bottom pressure deviation, obtained by integrating the quasi-steady vertical momentum equation, to the pressure gradient term of the depth-averaged momentum equations.

This method is especially powerful in situations such as local scour around bridge piers and spur dikes, where the flow changes abruptly over a short distance. On the other hand, it has the highest computational cost of the three methods, and in regions with sharp curvature changes, internal numerical stabilization (an upper clamp on the vorticity, and exclusion of evaluation in cells adjacent to obstacles) is applied.

Optional output of the 3D velocity profile

Whichever method you choose, the “3D Velocity Profile” group in the GUI lets you output the vertical velocity profile. This is purely a visualization option and does not affect the flow, sediment transport or bed variation calculations themselves.

Parameter	Description
3D velocity computation (j_3d)	Whether to compute the vertical velocity profile. When method (2) BVC is selected, this is always computed regardless of this checkbox.
Number of vertical layers (nz)	The number of vertical subdivisions (default 10).
Vertical velocity calculation (j_3dv)	Whether to compute the vertical velocity component by integrating the 3D continuity equation.

Methods (0) and (1) use the Engelund (1974) polynomial profile (second-order in the streamwise direction, sixth-order for the secondary flow direction); method (2) uses the BVC third-order polynomial profile.

Guidance for choosing a method

If you want a fast, stable computation, or only need a simple estimate of transverse sediment transport → (0) Engelund
If you want to reproduce the delayed development of secondary flow at the entrance and exit of bends in a meandering channel → (1) Nays2DH
If you want to accurately reproduce local scour around a localized obstacle such as a bridge pier or spur dike → (2) BVC + dynamic pressure correction (but for straight or gently meandering channels, (0)/(1) may give better results, so always compare)

References

Engelund, F. (1974): Flow and bed topography in channel bends. J. Hydraul. Div., 100(11), 1631-1648.
Johannesson, H. & Parker, G. (1989): Secondary flow in mildly sinuous channel. J. Hydraul. Eng., 115(3), 289-308.
Hosoda, T., Nagata, N., Iwata, M. & Kimura, I. (2000): A depth averaged model of open channel flows with lag between main flow and secondary flow in a generalized curvilinear coordinate system. J. Hydraul. Eng. JSCE, 44, 587-592.
Iwasaki, T., Shimizu, Y. & Kimura, I. (2013): Effects of a secondary flow model on bed variation in a mildly meandering channel. J. JSCE, Ser. B1 (Hydraulic Engineering), 69(3), 147-163.
Uchida, T. & Fukuoka, S. (2011): A new depth-averaged quasi-3D bed variation calculation method based on a general bottom velocity analysis method. J. JSCE, Ser. B1 (Hydraulic Engineering), 67(1), 16-29.
Uchida, T. & Fukuoka, S. (2012): A quasi-3D bed variation calculation method considering non-hydrostatic bottom pressure. Annual Journal of Hydraulic Engineering, JSCE, 56, I_1225-I_1230.