The entire model is discretised via a high-order central-FD approach. In particular, the solver allows both uniform and non-uniform structured Cartesian grids discretisation. As said, the peculiarity of the discretisation process stands in the numerical treatment of the viscous terms, which provides strong preservation of viscous stresses in the limit of incompressible or weakly compressible flows. In the following, the authors provide a detailed description of the numerical scheme related to the viscous contributions. Thanks to a simple manipulation of the viscous momentum fluxes of the Navier–Stokes equations, a formulation can be obtained in which the incompressible and the compressible contributions are highlighted.
$$\begin{aligned} \frac{\partial \bar{\tau }_{ij} - T_{ij}^{SGS}}{\partial x_{j}} = \underbrace{\frac{\partial }{\partial x_{j}}\left( \bar{\mu }_{tot} \frac{\partial \tilde{u}_i}{\partial x_{j}}\right) }_{\text {Incompressible contribution}} +\underbrace{\frac{\partial }{\partial x_{j}} \left( \bar{\mu }_{tot} \frac{\partial \tilde{u}_{j}}{\partial x_i} - \frac{2}{3}\bar{\mu }_{tot} \frac{\partial \tilde{u}_s}{\partial x_s} \delta _{ij}\right) }_{\text {Compressible contribution}} \end{aligned}.$$
(8)
In particular, the former acts in both compressible and incompressible flow conditions, while the latter is expected to take large values just near shocks and discontinuities or in highly expanding/deleting regions.
\(\bar{\mu }_{tot} = \bar{\mu } + \mu _{SGS}\) denotes the overall viscosity. A numerical discretisation which accounts for the variability of the overall viscosity usually takes advantage of the
standard Laplacian formulation:
$$\begin{aligned} \frac{\partial }{\partial x} \left( \bar{\mu }_{tot} \frac{\partial \tilde{u}}{\partial x}\right)&= \frac{\partial \bar{\mu }_{tot}}{\partial x} \frac{\partial \tilde{u}}{\partial x} + \bar{\mu }_{tot} \frac{\partial ^{2} \tilde{u}}{\partial x^{2}}, \end{aligned}$$
(9.1)
$$\begin{aligned}&\simeq \frac{1}{\Delta x^{2}} \sum _{l = - L}^{L} \alpha _{l}^{(1)} \bar{\mu }^{tot}_{i+l} \tilde{u}_{i+l} + \bar{\mu }^{tot}_i \frac{1}{\Delta x^{2}} \sum _{l = - L}^{L} \alpha _{l}^{(2)} \tilde{u}_{i+l}.\end{aligned}$$
(9.2)
The method expands all the mixed derivatives and accounts for the viscosity and the velocity gradients separately in a co-located fashion way (see, e.g. [
28,
39]). Here
\(\alpha _{l}^{(1)}\) and
\(\alpha _{l}^{(2)}\) maximise the formal order of accuracy of a central approximation of 2
L-size stencils for the first and the second derivative, respectively. Supposing the grid resolution in the range of the scheme convergence and the overall viscosity vary smoothly, the technique allows to resolve the viscous stresses accurately. However, especially in the case of poorly resolved flows or where the total viscosity experiences highly local variations, the method can provide erroneous behaviours of the diffusive contributions resulting in a wrong prediction of the entire flow field. In these situations, a numerical approximation which strongly conserves the shear stress components represents a more suitable description of incompressible viscous contributions. Let us describe the method for uniform Cartesian grids. The extension to non-uniform meshes is straightforward, and the interested reader is addressed to look at Appendix
6 for the details and the implementation issues. Thus, recalling the definition of a conservative method [
27], the incompressible viscous stress components can be cast as:
$$\begin{aligned} \frac{\partial }{\partial x} \left( \bar{ \mu }_{tot} \frac{\partial \tilde{u}}{\partial x}\right) \simeq \frac{1}{\Delta x}\left( \hat{\tau }_{i+1/2} - \hat{\tau }_{i-1/2}\right). \end{aligned}$$
(10)
Here
\(\hat{\tau }_{i+1/2}\) is a high-order representation of the viscous stresses at the cell interface. The latter can be expressed as a linear combination which accounts for both the viscosity and the velocity gradient interpolations at the
\(i+1/2\) node in a high-order path:
$$\begin{aligned} \hat{\tau }_{i+1/2} =\frac{1}{\Delta x} \sum _{l = m}^{n} \beta _{l}\bar{\mu }_{i+l} \cdot \gamma _{l} \tilde{u}_{i+l} \end{aligned}.$$
(11)
The interpolation coefficients
\(\{\beta _{l}, \gamma _{l}\}_{l = n}^{m}\) depend on the desired order of accuracy, and the authors derived their values as listed in Table
1. In particular, the
\(\{\beta _{l}\}_{l=n}^{m}\) coefficients provide an
\((m-n+1)\)-order explicit interpolation for
\(\mu _{i+1/2}\) at the cell bound, while the
\(\{\gamma _{l}\}_{l=n}^{m}\) allow for the interpolation of the velocity gradient in the intermediate node maximising the formal order of accuracy of a standard 2
L-size stencils second derivative approximation, therefore respecting the following constraint:
$$\begin{aligned} \sum _{l = n}^{m} \gamma _{l} \left( \tilde{u}_{i+l} - \tilde{u}_{i-1+l}\right) = \sum _{l = -L}^{L} \alpha ^{(2)}_{l} \tilde{u}_{i+l} \end{aligned}.$$
(12)
Here it is not worthless to be mentioned that, apart from 2
nd-order of accuracy, the proposed
\(\{\gamma _{l}\}_{l=n}^{m}\) coefficients do not represent the only possible choice for the interpolation of cell-bound gradients. In particular, according to [
40], these coefficients do not maximise the formal order of accuracy of a first cell-bound derivative
\((df/dx)_{i+1/2}\). However, if the
\((df/dx)_{i+1/2}\) is interpolated according to [
40], the process leads to a cell-centred second derivative
\((d^{2}f/dx^{2})_i\) with 2
nd-order of accuracy if this is resembled in a conservative formulation, according to Eq. (
10). Thus, the proposed
\(\{\gamma _{l}\}_{l=n}^{m}\) coefficients are the only which provide a
\((df/dx)_{i+1/2}\) interpolation able to fall back to standard co-located FD second derivatives, with corresponding order of accuracy, if uniform viscosity/diffusivity distributions are taken into account. In the following, the method is addressed as
conservative discretisation. As said, the method recovers the standard Laplacian formulation (i.e. Eq. (
9.1)), with corresponding order of accuracy, in the case of uniform viscosity fields. Besides, due to the inherent characteristic of conservative schemes, in the limit of incompressible or weakly compressible flows, the conservative approach provides the high-order conservation of the viscous terms independently to the resolution and grid stretching (see Appendix
6 for details). Thus, the method relies on a more robust strategy of diffusive terms in any situation, even if the flow embedded shocks.