### 2.6.4. Discretization of the Advection Terms

Following Strelets [58], the zonal DES model switches from a second-order upwind scheme in the RANS region to a central difference scheme in the LES region. The finite volume method approximates the advection term in the transport equation for a generic variable as:

| (2–200)

where is the control volume value, the index denotes the integration points on the control volume faces, and are the mass flow and the transported variable value estimated at the integration point . The value of is obtained by interpolating from the surrounding mesh nodes according to the selected discretization scheme. Blending between the upwind-biased scheme and the central scheme is achieved by combining the corresponding interpolation values and using the blending function :

| (2–201)

A specific form of the blending function is taken from Strelets [58] with only minor changes:

| (2–202)

| (2–203)

| (2–204)

with being the maximum neighboring mesh edge, and the same constant values as in Strelets [58]:

| (2–205)

Note: For the SST-DES model, the blending function is additionally limited in order to prevent the mesh-induced separation:

| (2–206)

where F1 and F2 are the first and second SST blending function and ([129]), respectively. BF1 and BF2 are blend factors and are used in order to select one of the two blending functions. The default and the recommendation is to use the second SST blending function F2, therefore the corresponding default values are BF1 = 0 and BF2 = 1.

A limiter based on the Courant-number has been developed in order to avoid oscillations due to the central difference scheme, which can occur for medium and high Courant-numbers (this limiter is not given in the original formulation of Strelets [58]). The blending function is limited in the following way:

| (2–207)

The default values of and are 5.0 and 1.0, respectively.

The blending function, , is available in CFD-Post and allows you to distinguish between the regions in the flow where the advection-scheme is user-specified (upwind-biased) or the central difference scheme is used. The corresponding variable name in CFD-Post is `Blending Function for DES model`.

According to Equation 2–201, the value of the variable is `1` to denote the user-specified advection scheme and `0` for the central difference scheme. Therefore, a value of `1` typically denotes the RANS region using an upwind-biased scheme and a value of `0` denotes the LES region using the central-difference scheme. However, if the choice of the user-specified advection scheme is central difference, then the `Blending Function for DES model` returns a value of `1` everywhere.

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