In order to compute turbulent flows with the RANS equations, it is necessary to determine the Reynolds stresses and the scalar transport terms in order to close the system of mean flow equations. Turbulence models are commonly used in practice to achieve this closure. Turbulence models may be classified by the number of additional transport equations that need to be solved along with the RANS equations. In practice, two of the most common closure models are the zero-equation mixing length model and the two-equation k-ε model.
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Perhaps the first attempt to develop a turbulence model was by Boussinesq in the 1870’s. As mentioned at the end of section 3.4, the total shear stress, 𝜏 represents the fluid shear tensor, incorporating molecular stresses and those stresses resulting from the
Reynolds averaging process. If u represents the horizontal velocity in the stream-wise direction and w in the vertical direction, equilibrium analysis has shown that the total shear stress may be written in the form (Prasuhn, 1980):
𝜏 = 𝜇𝜕𝑢̅
𝜕𝑧− 𝜌𝑢̅̅̅̅̅̅′𝑤′ (4-16)
Boussinesq proposed that the two stress sources were analogous in a fully turbulent flow by introducing an eddy viscosity term, η:
−𝜌𝑢̅̅̅̅̅̅ = 𝜂′𝑤′ 𝜕𝑢̅
𝜕𝑧 (4-17)
The total shear stress may then be written as:
𝜏 = 𝜇𝜕𝑢̅ 𝜕𝑧 + 𝜂
𝜕𝑢̅
𝜕𝑧 (4-18)
The turbulent mixing contribution far outweighs the molecular viscosity term in the macro- scale, therefore, the total shear stress for estuary considerations may be written as:
𝜏 = 𝜂𝜕𝑢̅
𝜕𝑧 = 𝜌𝐾𝑚 𝜕𝑢̅
𝜕𝑧 (4-19)
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The mixing length model was developed by Ludwig Prandtl in the early 20th century,
and is a method that attempts to describe momentum transfer by turbulence Reynolds stresses within a Newtonian fluid boundary by means of an eddy viscosity. Although the model is considered to give a rough approximation of the turbulent Reynolds stress, it is commonly used in practice today due to its low computational overhead.
The mixing length, L, is defined as the distance a fluid parcel can travel while
conserving its properties before mixing with the surrounding fluid. Prandtl described that the mixing length: “may be considered as the diameter of the masses of fluid moving as a whole in each individual case, or again, as the distance traversed by a mass of this type before it becomes blended in with neighboring masses…”(Prandtl, 1926). The concept is depicted in Figure 4-6; here a fluid parcel of temperature, T, is seen travelling vertically across a temperature gradient. The fluctuation in temperature that the parcel experienced throughout the process is T’. So T’ is the temperature deviation from its surrounding environment after it has moved over the mixing length L’.
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Figure 4-6 Mixing Length concept.
Through Reynolds decomposition, the temperature may be expressed as, = 𝑇̅ + 𝑇′ , where 𝑇̅, is the slowly varying component and T’ is the fluctuating component. T’ can be expressed in terms of the mixing length:
𝑇′ = −𝐿𝜕𝑇̅
𝜕𝑧 (4-20)
Although with a somewhat weaker theoretical justification since the pressure gradient force can alter the fluctuating components, the fluctuating components of velocity, u’ and w’, can also be expressed in a similar fashion:
𝑢’ = −𝐿𝜕𝑢̅
𝜕𝑧 𝑤’ = −𝐿 𝜕𝑤̅
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𝜏 = −𝜌𝑢̅̅̅̅̅̅ ≈ 𝜌𝑤′𝑤′ ′𝐿𝜕𝑢̅
𝜕𝑧 (4-22)
In practice, 𝑤′ is assumed to be roughly equal to 𝑢′, resulting in:
𝜏 = 𝜌𝐿2𝜕𝑢̅
𝜕𝑧| 𝜕𝑢̅
𝜕𝑧| (4-23)
A commonly used turbulence closure model in engineering applications is the two- equation k-ε model (Launder, 1974). The model is known as the k-ε because the mixing length L is determined from transport equations for the turbulent kinetic energy, k and the energy dissipation ε:
𝐿 = 𝐶𝐷𝑘√𝑘
𝜀 (4-24)
In practice, 𝐶𝐷 has been determined to be about 0.1925 through calibration
In application of this turbulence model, it is assumed that the horizontal scale is much larger than the vertical one. It is also assumed that the production, buoyancy, and dissipation terms dominate, because of this assumption, conservation of the turbulent quantities is less important and the transport equation is non-conservative.
The exact k-ε equations contain many un-measurable terms and unknowns. The standard k-ε turbulence model, which is based on our best understanding of the relevant processes, minimizes unknowns and is applicable to a wide range of turbulent applications. The model uses two PDE’s to describe turbulent kinetic energy and turbulent energy
dissipation: 𝜕(𝜌𝑘) 𝜕𝑡 + 𝜕(𝜌𝑘𝑢𝑖) 𝜕𝑥𝑖 = 𝜕 𝜕𝑥𝑗[ 𝜇𝑡 𝜎𝑘 𝜕𝑘 𝜕𝑥𝑗] + 2𝜇𝑡𝐸𝑖𝑗𝐸𝑖𝑗 − 𝜌𝜀 (4-25)
58 𝜕(𝜌𝜀) 𝜕𝑡 + 𝜕(𝜌𝜀𝑢𝑖) 𝜕𝑥𝑖 = 𝜕 𝜕𝑥𝑗[ 𝜇𝑡 𝜎𝜀 𝜕𝜀 𝜕𝑥𝑗] + 𝐶1𝜀 𝜀 𝑘2𝜇𝑡𝐸𝑖𝑗𝐸𝑖𝑗− 𝐶2𝜀𝜌 𝜀2 𝑘 (4-26)
where 𝑢𝑖 represents the velocity in the corresponding direction, 𝐸𝑖𝑗 is the component of the
rate of deformation, and 𝜇𝑡 represents the eddy viscosity:
𝜇𝑡= 𝜌𝐶𝜇𝑘2
𝜀 (4-27)
The equations contain some adjustable constants whose values have been determined through data fitting for a wide range of turbulent applications (Rodi, 1984):
𝐶𝜇 = 0.09 𝜎𝑘 = 1.00 𝜎𝜀 = 1.30 𝐶1𝜀 = 1.44 𝐶2𝜀= 1.92
The k-ε turbulence model is the most widely used and validated turbulence model and is usually available in computer model codes solving the shallow water equations for estuary and river applications. However, it should be used with caution in stratified flow situations, as the turbulent eddy viscosity at the interface would reduce to zero and vertical mixing due to internal gravity waves (Kelvin-Helmholtz instability) would not be
accounted for. Vertical mixing would only occur through molecular diffusion calculations. Therefore, an additional means would be required to account for vertical mixing due to breaking waves along the interface for stratified flow applications.