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RANS equations

The starting point of the derivation are the classical Reynolds-Averaged Navier-Stokes equations, the mass conservation and the momentum conservation equations.

Conservation of mass can be posed in two different forms, first as a single fluid approach:

in which и is the Reynolds-averaged velocity vector and x, denotes the i-th direction in space.

It is often the case that for free surface applications there is a need to simulate two fluids, therefore the alternative form of the equation, also called Volume Of Fluid (VOF) equation,

is:

where a is the so-called VOF or color function and t denotes time. In the VOF technique (Hirt and Nichols, 1981), a represents the amount of fluid per unit volume. By convention, for systems with air and water phases, a denotes the unit volume of water, while the unit volume of air would simply be (1 — a). In this way, the calculation of fluid properties, as density (p) or kinematic viscosity (v), under the incompressible assumption can be simply achieved by means of a weighted average: p = pv, a + рл (1 — a), where the subscript w denotes water and a denotes air.

Finally, the momentum conservation equations are:

in which the only new variables are p, pressure; g, the acceleration due to gravity; and //, the dynamic viscosity. The last term, the Reynolds stresses, includes the effects of turbulence. Reynolds stresses need to be modelled because they cannot be simulated under the RANS assumptions.

Volume-Averaged RANS equations

By applying the volume-averaging procedure the RANS equations are transformed. Nevertheless, the new set of equations retains a high degree of similarity with respect to the original RANS equations. For example, the volume-averaged mass conservation equation in terms of extended variables is:

A new term appears inside the time derivative. фоу accounts for the variation in dynamic porosity, therefore, it can change in time and must be kept inside the differential operator.

Before introducing the volume-averaged VOF equation, we need to recall that before introducing porosity into the equations a was defined as the amount of fluid per unit volume. However, some portion of the control volume may be blocked by solids. Therefore, the correct definition of a is the amount of fluid per unit of the volume that can be occupied by fluids. With the new definition, 0 < a < 1 still holds and (a)^ = a, as demonstrated in (Higuera, 2015, Appendix A.2). Taking this into consideration, the volume-averaged VOF equation is:

which resembles closely Equation 11, with alpha instead of p. However, only the dynamic component of porosity аY) appears inside the partial derivative with respect to time term in Equation 11, whereas ф appears in Equation 12. The reason behind this is that under the incompressible assumption, = 0, while generally ^ ф 0, hence Щр- = ф%£ + a ■

Finally, the volume-averaged momentum equations involve an increasing complexity. New terms that cannot be simulated appear during the derivation, as a result of filtering out the unknown internal geometry of the porous media. Therefore, a closure model needs to be introduced to retain the contribution of such terms, as follows:

The closure terms are divided in two, corresponding to those derived from static porosity ([CT]st) and dynamic porosity ([CT]dy)> and will be discussed in next subsection.

It is often the case that porosity does not change with time. This means that the porous structure is fixed and will not move due to the wave action. Under this condition, only static porosity is involved (ф = ф3T), and the conservation of mass and VOF equations are reduced to simpler expressions:

The only change in momentum conservation (Equations 13) is that the term [CTJdy = 0.

Closure

Similarly to what occurs with Reynolds stresses in RANS, the volume-averaging operation also yields a number of terms that cannot be simulated and need to be modelled instead. These terms represent the inertial (Polubarinova-Kochina, 1962) and drag forces (Forcheimer, 1901) due to the geometry of solids which have been filtered out. Different closure models can be found in literature, as previously referenced in Section 2. In this work the formulation developed by Engelund (1953), later applied in Burcharth and Andersen (1995), will be used. The inertial force, linear drag force and nonlinear drag force are as follows:

Equation 16a is usually combined with the local derivative in the NS equations (i.e., first term in Equation 13). C is a calibration parameter, which is often taken as 0.34, given that it has been found to produce a negligible contribution when compared to the other two terms (del Jesus, 2011). Equation 16b is a drag force, which is linearly dependent on velocity, a is a calibrat ion factor and DrA) is the mean nominal diameter of the solid elements in the porous medium. Equation 16c is the nonlinear drag force, dependent on the /3 calibration factor. КС is the Keulegan-Carpenter number, defined as where им is the maximum

oscillatory velocity and Tw is the wave period. This element, first introduced in van Gent (1995), increases the friction under oscillatory conditions (i.e., in wave simulations).

The values of a and в are dependent on the case conditions, including the porous material and flow regime. Therefore, experimental data is usually required to obtain suitable values for both parameters. The process involves tuning the calibration factors until the numerical solution matches the experimental results. This procedure will be discussed in Section 3.6.

 
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