Gradient limiters, also known as slope limiters, are used on the second-order upwind (SOU) scheme to prevent spurious oscillations, which would otherwise appear in the solution flow field near shocks, discontinuities, or near rapid local changes in the flow field. The gradient limiter attempts to invoke and enforce the monotonicity principle by prohibiting the linearly reconstructed field variable on the cell faces to exceed the maximum or minimum values of the neighboring cells.
There are three gradient limiters in the Ansys Fluent solvers:
Standard limiter
Multidimensional limiter
Differentiable limiter
Gradient limiters can be categorized into two general groups: non-differentiable limiters and differentiable limiters. Both, the standard limiter and multidimensional limiter are of the non-differentiable form, since they use minimum and maximum types of functions for limiting the solution variables. The third limiter in Ansys Fluent, as the name indicates, is a differentiable type of limiter, which uses a smooth function to impose the monotonicity principle.
For each of the above mentioned limiter methods, Ansys Fluent provides two limiting directions:
cell to face limiting, where the limited value of the reconstruction gradient is determined at cell face centers. This is the default method.
cell to cell limiting, where the limited value of the reconstruction gradient is determined along a scaled line between two adjacent cell centroids. On an orthogonal mesh (or when cell-to-cell direction is parallel to face area direction) this method becomes equivalent to the default cell to face method. For smooth field variation, cell to cell limiting may provide less numerical dissipation on meshes with skewed cells.
cell to node limiting, where the limited value of the reconstruction gradient is determined at the vertices of a cell. This limiting direction is more conservative than the default cell to face strategy, leading to improved monotonicity and robustness when computing strong gradients on unstructured grids. It is particularly advantageous for meshes containing stretched, skewed, or high aspect-ratio cells.
For more information about how to access the limiter functions in Ansys Fluent through the GUI or TUI, see Selecting Gradient Limiters in the User's Guide.
Important: On unstructured meshes, Ansys Fluent uses the scalar form of the gradient limiter given by the following equation:
 | (23–38) |
Where 
is a scalar value that limits the gradient
.
The standard limiter is the default limiter function in Ansys Fluent and is derived from the work of Barth and Jespersen [44]. This limiter is of a non-differentiable type and uses the Minmod function (Minimum Modulus) to limit and clip the reconstructed solution overshoots and undershoots on the cell faces.
The multidimensional limiter in Ansys Fluent [305] has a similar form to the standard limiter. Since the multidimensional limiter uses a Minmod function for limiting the gradient, it is also classified as a non-differentiable type of limiter. However, in the standard limiter formulation, if limiting took place on any face of the cell, then this will cause the cell gradient to be clipped in an equal manner, in all directions, regardless of whether or not limiting is needed on the other cell faces. This limiting method is rather severe and adds unnecessary dissipation to the numerical scheme. The multidimensional limiter, on the other hand, attempts to lessen the severity of the gradient limiting by carefully examining the gradient on each cell and clipping only the normal components of the gradient to the cell faces. For this procedure to work on a scalar form limiter, the normal components of gradients on cell faces are first sorted out in ascending order of their magnitude so that only the necessary clipping can be applied. The multidimensional limiter is therefore less dissipative than the standard limiter.
One disadvantage with non-differentiable limiters is that they tend to stall the apparent residual’s convergence after a few orders of reduction in residual magnitude. Note that this does not mean that the solution is not converging, but rather the solution continues to converge while the residuals are stalling. This annoying behavior can be directly traced to the non-differentiable nature of the limiting functions. Therefore, the differentiable limiter uses a smooth function to impose the monotonicity condition while allowing the residuals to converge. The differentiable limiter used in Ansys Fluent is a modified form [692] of a limiter which was originally proposed by Venkatakrishnan [669].
Important: Ansys Fluent uses gradient or slope limiters and not flux limiters. Gradient limiters are applied to the gradients of the variable field being linearly reconstructed at the cell faces, while flux limiters are used on the system fluxes.