Mass Conservation Momentum Conservation Turbulence Modeling Wind Surface Stress
Mass Conservation
Assuming that the flow is incompressible, the unsteady differential form of the mass conservation (continuity) equation is:
144)
where is time, and are the velocity components in the x- and y- direction respectively and q is a source/sink flux term. Following the starndard HEC-RAS sign conventions, sinks are negative and sources are positive.
In vector form, the continuity equation takes the form:
145)
where is the velocity vector and is the gradient operator given by .
Integrating over a horizontal region with boundary normal vector n and using Gauss' Divergence theorem, the integral form of the equation is obtained:
146)
The volumetric region represents the three-dimensional space occupied by the fluid, and is the unit vector normal to the side boundaries . It is assumed that represents any flow that crosses the bottom surface (infiltration) or the top water surface of (evaporation or rain). The source/sink flow term is also convenient to represent other conditions that transfer mass into, within or out of the system, such as pumps.
This integral form of the continuity equation will be appropriate in order to follow a sub-grid bathymetry approach in subsequent sections. In this context, the volume Ω will represent a finite volume cell and the integrals will be computed using information about the fine underlying topography.
Momentum Conservation
When the horizontal length scales are much larger than the vertical length scale, volume conservation implies that the vertical velocity is small. The Navier-Stokes vertical momentum equation can be used to justify that pressure is nearly hydrostatic. In the absence of baroclinic pressure gradients (variable density), strong wind forcing and non-hydrostatic pressure, a vertically- averaged version of the momentum equation is adequate. Vertical velocity and vertical derivative terms can be safely neglected (in both mass and momentum equations). The shallow water equations are obtained:
147)
148) where
, : Velocities in the Cartesian directions [L/T]
: Gravitational acceleration [L/T2] : Water surface elevation [L]
, : horizontal eddy viscosity coefficients in the x and y directions [L2/T]
, : Bottom shear stresses om the x and y directions [M/L/T2] : Hydraulic radius [L]
, : Surface wind stresses in the x and y directions, respectively [M/L/T2] : Water depth [L]
: Coriolis parameter [1/T]
: Atmospheric pressure [M/L/T2]
The left-hand side of the equation contains the acceleration terms. The right-hand side represents the internal or external forces acting on the fluid. The left- and right-hand side term are typically organized in such a way as to be in accordance with Newton's second law, from which the momentum equations are ultimately derived.
The momentum equations can also be written in vector notation. The advantage of this form of the equation is that it becomes more compact and easily readable. The vector form of the momentum equation is:
149)
where here the velocity vector is , is the eddy viscosity tensor, is the gradient operator, is the unit vector in the vertical direction, and is the wind surface stress vector. It is noted that the notation for the Coriolis term is not strictly correct due to the inconsistent length of vectors. However, this notation is used for shorthand notation and simplicity.
Every term of the momentum equation has a clear physical counterpart. From left to right the terms are the unsteady acceleration, convective acceleration, Coriolis term, barotropic pressure term, momentum diffusion, bottom friction, and wind forcing.
A dimensional analysis shows that when the water depth is very small the bottom friction term dominates the equation. As a consequence, (149) for dry cells takes the limit form V = 0. As before, dry cells are computationally treated as a special case, but the result is continuous and physically consistent during the process of wetting or drying.
Because the conservation of momentum is directionally invariant, the momentum equation may be in any direction. In HEC-RAS, momentum is computed normal to each face.
Acceleration
The Eulerian acceleration terms on the left, can be condensed into a Lagrangian derivative acceleration term taken along the path moving with the velocity term:
151)
Other names usually given to this term are substantial, material and total derivative. The use of the Lagrangian derivative will become evident in subsequent sections when it will be seen that its discretization reduces Courant number constraints and yields a more robust solution method.
Bottom Friction
The bottom shear stress is given by 152)
where is the water density and is the drag coefficient computed using the Manning’s roughness coefficient as
153) where
= Manning's roughness coefficient [T/L1/3]
= hydraulic radius [L]
= gravitational acceleration [L/T2]
The drag coefficient, , is related to the nonlinear friction coefficient, , by 154)
The shear velocity is given by:
155)
Coriolis Effect
The last term of the momentum equation relates to the Coriolis Effect. It accounts for the fact that the frame of reference of the equation is attached to the Earth, which is rotating around its axis. The vertical component of the Coriolis term is disregarded in agreement with the shallow water
assumptions. The apparent horizontal force felt by any object in the rotating frame is proportional to the Coriolis parameter given by:
156)
where = 0.00007292115855306587 1/s is the sidereal angular velocity of the Earth and is the latitude.
Turbulence Modeling
Eddy Viscosity
Turbulence is a complex phenomenon of chaotic (turbulent) fluid motion and eddies spanning a wide range of length scales. Many of the length scales are too small to be feasibly resolved by a discrete numerical model, so turbulent flow mixing is modeled as a gradient diffusion process. In this approach, the diffusion rate is cast as the eddy viscosity . The eddy viscosity is computed as follows,
157)
where the tensor is the mixing coefficient tensor, is the shear velocity, is the water depth, is the Smagorinsky coefficient (approximately between 0.05 and 0.2), is the filter width equal to local grid resolution, and is the strain rate. The first term on the right-hand-side represents the turbulence produced by vertical shear, and more specifically bottom shear in longitudinal direction and secondary flows in the transverse direction. The mixing coefficients also represent the mixing due to momentum dispersion and not just turbulence. The second term on the right-hand-side of equation2-136 represents the turbulence produced by horizontal shear in the flow. The second term in equation 2-136 is the Smagorinsky-Lilly eddy viscosity model (Smagorinsky 1963; Deardorff 1970). The Smagorinsky-Lilly model assumes that the turbulent energy production and dissipation at small scales are in equilibrium. The Smagorinsky-Lilly model is somewhat expensive to compute because it requires computing the velocity gradients. However, it is more physically accurate, especially in regions of high shear such as close to solid/dry boundaries. It is noted that the velocity gradients are computed at cells using the Green-Gauss divergence theorem and then interpolated at the faces with the weighting coefficients and . The strain rate is given as:
158)
The diffusion coefficient tensor is given by:
159) and
The parameters and are user-specified mixing coefficients in the longitudinal and
Table 1. Longitudinal Coefficients.
Mixing Intensity Geometry and surface
0.1 to 0.3 Little longitudinal mixing Straight channel
Smooth surface
0.3 to 1 Moderate longitudinal mixing Gentle meanders
Moderate surface irregularities
1 to 3 Strong longitudinal mixing Strong meanders
Rough surface Table 2. Transverse Mixing Coefficients.
Mixing Intensity Geometry and surface
0.05 to 0.1 Little transversal mixing Straight channel
Smooth surface
0.1 to 0.3 Moderate transversal mixing Gentle meanders
Moderate surface irregularities
0.3 to 1 Strong transversal mixing Strong meanders
Rough surface
Wind Surface Stress
The wind surface stress vector is calculated as:
160)
where is the air density at sea level (~1.29 kg/m3), is the wind drag coefficient, is the 10-m height wind speed vector, is the 10-m wind velocity magnitude. The wind speed is calculated using either an Eulerian or Lagrangian reference frame as:
161)
where is the 10-m atmospheric wind speed relative to the solid earth (Eulerian wind speed), is equal to zero for the Eulerian reference frame or one for the Lagrangian reference frame, and = current velocity vector.
Winds are specified in an Eulerian reference frame with respect to the solid Earth. The Lagrangian reference frame is with respect to the moving surface water. Using the Lagrangian reference frame (see figure below), or relative wind speed, is more accurate and realistic for field applications (Bye 1985; Pacanowski 1987; Dawe and Thompson 2006), however, the option to use the Eulerian wind speed is provided for idealized cases. In addition, the Lagrangian reference frame is more stable since it introduces a drag or friction term. When the wind in the same direction of the currents the wind shear stress is lowered. When the wind and currents are in opposing directions, the wind shear stress is increased. For example, in the case of a current velocity of 1 m/s, with an opposing wind speed of 5 m/s, the Eulerian reference frame will give a surface stress proportional to (5 m/s)2 = 25 m2
/s2, while the Lagrangian reference frame will produce a surface proportional to (5-(-1) m/s)2 = 36 m2 /s2, which is an increase of 44%.
Drag Coefficient
There are a wide variety of drag coefficient formulas in literature. Within HEC-RAS, four drag coefficient formulations are available for use that provide a reasonable range of options.
The Hsu (1988) formula is written as:
162)
where is the 10-m wind speed [m/s]. The Hsu formula was developed by assuming a logarithmic wind velocity profile and substituting an expression for the aerodynamic roughness length based on fully developed ocean waves. Several linear formulas have been proposed in literature for the drag coefficient. Here two are implemented and available in HEC-RAS. Garratt (1977) proposed:
163)
Large and Pond (1981) proposed a similar expression:
164)
Andreas et al. (2012) utilized almost 7,000 measurements over the sea to fit an empirical expression for the water surface shear velocity, which is applicable to both smooth and rough turbulent flow:
165)
where = wind shear velocity [m/s]. The drag coefficient is then calculated as:
166)
A comparison of the four wind drag coefficient formulations is provided in the figure below. The four methods differ significantly at weak wind speeds but especially strong wind speeds. For strong wind speeds above 30 m/s, only the Andreas et al. (2012) method is recommended, since it is the only method which has been compared to measurements at high wind speeds. The other methods were not calibrated for high wind speeds.
Figure 1. Comparison of four wind drag coefficient formulations.
It is important to mention that each of the expressions were derived assuming fully developed sea states. That is, they do not consider the wind/wave variables such as wave height and period or swell direction and height. More importantly, for fetch limited water bodies, such as reservoirs and rivers, the above expressions may overestimate the drag coefficient.
Wind Ratio
The wind ratio is directly multiplied by the input wind velocities. It may be used to scale the wind, to convert the input wind velocity units, or to convert between different wind velocity definitions. Wind measurements are usually obtained as mean or “maximum sustained” wind in a certain time period.
Generally, for hydrodynamic modeling, mean wind speeds should be used (e.g., 10-min or 30-min averages). Maximum sustained winds, may be converted to mean values using the wind ratio.
Diffusion Wave Approximation to the Shallow Water Equations
In the previous section, Manning’s formula was used to estimate the bottom friction. If further constraints are assumed on the physics of the flow, a relation between barotropic pressure gradient and bottom friction is obtained from the diffusion wave form of the momentum equation. This relation is extremely useful due to its simplicity. However it must be noted that this relation can be applied only in a narrower scope than the more general momentum equation studied before. Under
the conditions described in this section, the Diffusion Wave equation can be used in place of the momentum equation. It will be seen in subsequent sections that the corresponding model becomes a one equation model known as the Diffusion Wave Approximation of the Shallow Water equations (DSW).
Up to this point, we have described the hydraulics for momentum. From now on the discussion will gear towards the formulation and numerical methods of the solution. It will be convenient to denote the hydraulic radius and the face cross section areas as a function of the water surface elevation H, so R= R(H), A=A(H).
In shallow frictional and gravity controlled flow; unsteady, advection, turbulence and Coriolis terms of the momentum equation can be disregarded to arrive at a simplified version. Flow movement is driven by a barotropic pressure gradient balanced by bottom friction. Simplifying the momentum equation results in:
167) where
: Velocity vector [L/T]
: Hydraulic radius [L]
: Water surface elevation elevation [L]
: Manning’s roughness coefficient [T/L1/3] : Water density [M/L3]
: Atmospheric pressure [M/L/T2] : Wind shear stress [M/L/T2]
Dividing both sides of the equation by the square root of their norm, the equation can be rearranged into the more classical form
168)
When the velocity is determined by a balance between barotropic pressure gradient and bottom friction, the Diffusion Wave form of the Momentum, can be used in place of the full momentum equation, and the corresponding system of equations can in fact be simplified to a one equation model. Direct substitution of the Diffusion Wave approximation of the momentum equation in the mass conservation equation, yields the classical Diffusion-Wave Equation:
169) where:
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Boundary Conditions
At any given time step, boundary conditions must be given at all the edges of the domain. Within HEC-RAS boundary conditions can be one of three different kinds:
Water surface elevation: The value of the water surface elevation is given at one of the boundary edges.
Normal Depth: The friction slope, , is specified and used to impose a flow boundary condition
computed as .
Flow: The flow that crosses the boundary is provided. In the continuity, this condition is implemented by direct substitution into the flow formula of the corresponding boundary faces.