adding documentation Refs idaholab#29846

modules/navier_stokes/doc/content/modules/navier_stokes/index.md

@@ -76,7 +76,7 @@ As Navier-Stokes Finite Volume solvers continue to evolve in MOOSE, many new sol
 | ------------------ | ------------------------- | ---------- | -----------------------------------------------------------  | ------------------------------------------------------------ | ---------------------- |
 | Transient          |       --                  | Yes        | Yes                                                          | Yes                                                           | Yes                                                          |
 | Turbulence         | Mixing length             | Yes        | Yes                                                          | Yes                                                          |                        |
-|                    | $k-\epsilon$              |            | Yes                                                          | Yes                                                          | under development      |
+|                    | $k-\epsilon$              |            | Yes                                                          | Yes                                                          | Yes                    |
 |                    | $k-\omega$ SST            |            |                                                              | in [PR #28151](https://github.com/idaholab/moose/pull/28151) |                        |
 | Two-phase          | Mixture model             | Yes        | Yes                                                          | Yes                                                          | in [PR #29614](https://github.com/idaholab/moose/pull/29614) |
 |                    | Eulerian-Eulerian         |            |                                                              | Yes                                                          |                        |

modules/navier_stokes/doc/content/source/linearfvkernels/LinearFVTKEDSourceSink.md

@@ -0,0 +1,113 @@
+# LinearFVTKEDSourceSink
+
+This kernel implements a version of [`INSFVTKEDSourceSink`](INSFVTKEDSourceSink.md) for a linear kernel.
+The documentation is repeated down here for completeness.
+
+The object computes the turbulent source and sink term for the turbulent kinetic energy dissipation rate equation.
+
+Two terms are computed: `destruction` and `production` and the term `destruction - production` is
+passed to the residual.
+A different treatment is used for the bulk and the near wall regions.
+
+## Bulk formulation:
+
+The production of turbulent kinetic energy dissipation $G_\epsilon$ is modeled as follows:
+
+\begin{equation}
+G_{\epsilon} = C_{1,\epsilon} \frac{\epsilon}{k} G_k \,,
+\end{equation}
+
+where:
+
+- $C_{1,\epsilon} = 1.44$ is a closure parameter,
+- $G_k$ is the limited turbulent kinetic energy production. For more details please refer to [LinearFVTKESourceSink](LinearFVTKESourceSink.md).
+
+The destruction of the turbulent kinetic energy dissipation rate is modeled as follows:
+
+\begin{equation}
+\epsilon_{\epsilon} = \frac{C_{2,\epsilon} \rho \epsilon}{t_k} \,,
+\end{equation}
+
+where:
+
+- $C_{2,\epsilon} = 1.92$ is a closure parameter,
+- $\epsilon$ is the solution variable, i.e., the dissipation rate of the turbulent kinetic energy,
+- $k$ is the turbulent kinetic energy,
+- $t_k = \frac{k}{\epsilon}$ is the turbulent time scale; if the [!param](/LinearFVKernels/LinearFVTKEDSourceSink/linearized_model) is `true`, this timescale is computed from the previous iteration; if [!param](/LinearFVKernels/LinearFVTKEDSourceSink/linearized_model) is `false`, in a nonlinear solve, this timescale is aded to the Jacobian.
+
+## Wall formulation:
+
+All cells in contact with a boundary identified in the [!param](/LinearFVKernels/LinearFVTKEDSourceSink/walls) list are applied a different
+treatment in which the equilibrium value for the $\epsilon = \epsilon_{eq}$ is set.
+A separate formulation is used for the `sub-laminar` and `logarithmic` boundary layers.
+The determination of whether the near-wall cell lies in the laminar or logarithmic region
+is performed via the non-dimensional wall distance $y^+$.
+The non-dimensional wall distance can be defined differently according to the
+[!param](/LinearFVKernels/LinearFVTKEDSourceSink/wall_treatment) parameter. 
+
+The four formulations are described in more detail in [INSFVTurbulentViscosityWallFunction.md]. 
+
+If an equilibrium [!param](/LinearFVKernels/LinearFVTKEDSourceSink/wall_treatment) is defined, i.e. `eq_newton`,`eq_incremental` or `eq_linearized`, the standard wall function formulations are used in which $y^+$ is found:
+
+\begin{equation}
+y^+ = \frac{\rho y_p u_{\tau}}{\mu} \,,
+\end{equation}
+
+where:
+
+- $\rho$ is the density,
+- $y_p$ is the distance from the wall to the centroid of the next-to-wall cell,
+- $u_{\tau}$ is the friction velocity, defined as $u_{\tau} = \sqrt{\frac{\tau_w}{\rho}}$ with $\tau_w$ the shear stress at the wall for which the condition is applied,
+- $\mu$ is the dynamic molecular viscosity.
+
+If a non-equilibrium [!param](/LinearFVKernels/LinearFVTKEDSourceSink/wall_treatment) is defined, i.e. `neq`,
+the $y^+$ is defined non-iteratively as follows:
+
+\begin{equation}
+y^+ = \frac{y_p \sqrt{\sqrt{C_{\mu}}k}}{\mu} \,,
+\end{equation}
+
+!alert note
+Using non-equilibrium wall functions is recommended for problems with recirculations and boundary layer detachment. However, using non-equilibrium wall functions will deteriorate results for standard problems such as flow developing over walls.
+
+The cells with $y^+ < 11.25$ belong to `sub-laminar` boundary layer.
+The ones belonging to the `logarithmic` boundary layer are those for which $y^+ \ge 11.25$.
+
+A different value is used for $\epsilon_{eq}$ in each of the two regions.
+For the `sub-laminar` boundary layer, the equilibrium value is determined as follows:
+
+\begin{equation}
+\epsilon_{eq} = 2 \frac{k \mu}{y_p^2}\,,
+\end{equation}
+
+where:
+
+- $\mu_t$ is the turbulent dynamic viscosity.
+
+For the `logarithmic` boundary layer, the value is determined as follows:
+
+\begin{equation}
+\epsilon_{eq} = \frac{C_{\mu}^{0.75} \rho k^{1.5}}{\kappa y_p}\,,
+\end{equation}
+
+where:
+
+- $\kappa = 0.4187$ is the von Kármán constant.
+
+!alert note
+When using wall functions, since the equilibrium value for $\epsilon$ is set in the cells near the wall, the user is recommended to deactivate advection and diffusion for those near wall cells.
+
+!alert note
+When the wall treatment is specified in this kernel, any boundary condition for $\epsilon$ will be ignored.
+In other words, there is no need to impose boundary conditions for $\epsilon$ when the wall treatment
+is specified in his kernel.
+
+!alert note
+When using near-wall treatment, we assume that the $\mu_t$ functor is computed by an object
+that performs near-wall treatment. Otherwise, the results obtained won't be physically correct.
+
+!syntax parameters /LinearFVKernels/LinearFVTKEDSourceSink
+
+!syntax inputs /LinearFVKernels/LinearFVTKEDSourceSink
+
+!syntax children /LinearFVKernels/LinearFVTKEDSourceSink

modules/navier_stokes/doc/content/source/linearfvkernels/LinearFVTKESourceSink.md

@@ -0,0 +1,126 @@
+# LinearFVTKESourceSink
+
+This kernel implements a version of [`INSFVTKESourceSink`](INSFVTKESourceSink.md) for a linear kernel.
+The documentation is repeated down here for completeness.
+
+The object computes the turbulent source and sink term for the turbulent kinetic energy equation.
+
+Two terms are computed `destruction` = $\epsilon$ and `production` = $G_k$ and the term $\epsilon - G_k$ is
+passed to the residual.
+A different treatment is used for the bulk and the near wall regions.
+
+## Bulk formulation:
+
+The turbulent production $G_k$ is modeled as:
+
+\begin{equation}
+G_k = \mu_t S^2 \,,
+\end{equation}
+
+where:
+
+- $\mu_t$ is the turbulent dynamic viscosity,
+- $S$ is the shear strain tensor internal norm, defined as $S = \sqrt{2\mathbf{S}:\mathbf{S}}$ with the shear strain tensor defined as $\mathbf{S} = \frac{1}{2} [\nabla \vec{u} + (\nabla \vec{u})^T]$.
+
+The turbulent kinetic energy dissipation rate $\epsilon$ is generally coming from a coupled
+transport equation for $\epsilon$.
+However, for canonical or measured cases, e.g., isotropic decaying turbulence,
+the user can utilize predefined fields through functors in MOOSE.
+
+To avoid the overproduction of turbulent kinetic energy in stagnation zones \cite{durbin1996k}, a production limiter is imposed in relation to the dissipation using the formulation in \cite{menter1994two}:
+
+\begin{equation}
+G_k = min \left( G_k , C_{PL} \rho \epsilon \right) \,,
+\end{equation}
+
+where:
+
+- $C_{PL}$ it the limiter constant, and set by default to a recommended value of 10 \cite{durbin1996k}.
+
+## Wall formulation:
+
+All cells in contact with a boundary identified in the [!param](/LinearFVKernels/LinearFVTKESourceSink/walls) list are applied a different
+treatment for production and destruction.
+A different formulation is used for the `sub-laminar` and `logarithmic` boundary layers.
+The determination of whether the near-wall cell lies in the laminar or logarithmic region
+is performed via the non-dimensional wall distance $y^+$.
+The non-dimensional wall distance is defined as
+
+\begin{equation}
+y^+ = \frac{\rho y_p u_{\tau}}{\mu} \,,
+\end{equation}
+
+where:
+
+- $\rho$ is the density,
+- $y_p$ is the distance to the wall to the centroid of the next-to-wall cell,
+- $u_{\tau}$ is the friction velocity, defined as $u_{\tau} = \sqrt{\frac{\tau_w}{\rho}}$ with $\tau_w$ the shear stress at the wall for which the condition is applied,
+- $\mu$ is the dynamic molecular viscosity.
+
+For every next-to-wall cell and every iteration step, $y^+$ is found via an
+incremental fixed-point search algorithm.
+The cells belonging to the `sub-laminar` boundary layers are defined as those
+for which $y^+ < 11.25$.
+The ones belonging to the `logarithmic` boundary layer are those for which $y^+ \ge 11.25$.
+The imposed threshold of $y^+ = 11.25$ is given by the value of $y^+$ at which the `sub-laminar`
+and `logarithmic` boundary profiles intersect.
+
+In the `sub-laminar` region production of turbulent kinetic energy is negligible, therefore, if $y^+ \lt 11.25$:
+
+\begin{equation}
+G_k = 0.0 \,,
+\end{equation}
+
+In the `logarithmic` boundary layers the production term is no longer negligible and is defined as:
+
+\begin{equation}
+G_k = \tau_w ||\nabla \vec{u}|| = \mu_w ||\nabla \vec{u}|| \frac{ C_{\mu}^{0.25} \sqrt(k)}{\kappa y_p} \,,
+\end{equation}
+
+where:
+
+- $C_{\mu} = 0.09$ is a closure parameter,
+- $k$ is the turbulent kinetic energy,
+- $||\nabla \vec{u}||$ is the near wall velocity gradient norm, which is defined as $||\nabla \vec{u}|| = (\nabla \vec{u} \cdot \hat{n}) \cdot \hat{n}$,
+- $\kappa = 0.41$ is the von Kármán constant.
+
+The formulation assumes that the near wall value is already imposed in the $\mu_t$ functor. 
+
+When solving a linear problem, instead of the nonlinear formulation, the production term is formulated as:
+
+\begin{equation}
+G_k =  \mu_w ||\nabla \vec{u}|| \frac{ C_{\mu}^{0.25} k}{\sqrt{k_{old}} \kappa y_p} \,.
+\end{equation}
+
+where:
+
+- $k_{old}$ is the value of the turbulent kinetic energy in the previous iteration.
+
+For the destruction, formulation is different for the `sub-laminar` and `logarithmic` layers.
+For the `sub-laminar` layer, the destruction is defined as follows:
+
+\begin{equation}
+\epsilon = \frac{2 \mu k}{y_p ^2} \,.
+\end{equation}
+
+For the `logarithmic` layer, the destruction is defined as follows:
+
+\begin{equation}
+\epsilon = C_{\mu}^{0.75} \frac{\rho k^{\frac{3}{2}}}{\kappa y_p} \,.
+\end{equation}
+
+!alert note
+When the wall treatment is specified in this kernel, any boundary condition for $k$ will be ignored.
+In other words, there is no need to impose boundary conditions for $k$ when the wall treatment
+is specified in his kernel.
+
+!alert note
+When using near-wall treatment, we assume that the $\mu_t$ functor is computed by an object
+that performs near-wall treatment.
+Otherwise, the results obtained won't not physically correct
+
+!syntax parameters /LinearFVKernels/LinearFVTKESourceSink
+
+!syntax inputs /LinearFVKernels/LinearFVTKESourceSink
+
+!syntax children /LinearFVKernels/LinearFVTKESourceSink

modules/navier_stokes/doc/content/source/linearfvkernels/LinearFVTurbulentLimitedAdvection.md

@@ -0,0 +1,19 @@
+# LinearFVTurbulentLimitedAdvection
+
+This object adds a $\nabla \cdot \vec u \phi$ term for an arbitrary scalar field
+$\phi$, where $\phi$ corresponds to the nonlinear variable that this kernel acts
+on. This kernel acts in a linear solve. 
+The linear `variable` can be of type `MooseLinearVariableFVReal`.
+
+The particularity of this kernel is that it allows us to skip computing advection
+for near-wall elements. The key for this skip are the boundaries identified in
+the [!param](/LinearFVKernels/LinearFVTurbulentLimitedAdvection/walls) list.
+For any element that is in contact with a boundary identified
+in the [!param](/LinearFVKernels/LinearFVTurbulentLimitedAdvection/walls) list,
+advection will be skipped for that element over all faces.
+
+!syntax parameters /LinearFVKernels/LinearFVTurbulentLimitedAdvection
+
+!syntax inputs /LinearFVKernels/LinearFVTurbulentLimitedAdvection
+
+!syntax children /LinearFVKernels/LinearFVTurbulentLimitedAdvection

modules/navier_stokes/doc/content/source/linearfvkernels/LinearFVTurbulentLimitedDiffusion.md

@@ -0,0 +1,17 @@
+# LinearFVTurbulentLimitedDiffusion
+
+This object extends [`LinearFVDiffusion`](LinearFVDiffusion.md) to allow diffusion to be skipped
+at certain boundaries.
+
+The particularity of this kernel is that it allows us to skip computing diffusion
+for near-wall elements. The key for this skip are the boundaries identified in
+the [!param](/LinearFVKernels/LinearFVTurbulentLimitedDiffusion/walls) list.
+For any element that is in contact with a boundary identified
+in the [!param](/LinearFVKernels/LinearFVTurbulentLimitedDiffusion/walls) list,
+diffusion contributions will be skipped for that element over all faces.
+
+!syntax parameters /LinearFVKernels/LinearFVTurbulentLimitedDiffusion
+
+!syntax inputs /LinearFVKernels/LinearFVTurbulentLimitedDiffusion
+
+!syntax children /LinearFVKernels/LinearFVTurbulentLimitedDiffusion

modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated-transient/channel_ERCOFTAC.i

@@ -431,7 +431,7 @@ wall_treatment = 'eq_newton'  # Options: eq_newton, eq_incremental, eq_linearize
   print_fields = false
   continue_on_max_its = true
   dt = 1.0
-  num_steps = 10
+  num_steps = 2
 []
 
 [Outputs]

modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated-transient/tests

@@ -19,7 +19,7 @@
       detail = 'and pass debugging checks.'
       abs_zero = 5e-6
       rel_err = 5e-6
-      cli_args = 'Executioner/dt=1.0 Executioner/num_steps=1 Outputs/exodus/file_base=channel_ERCOFTAC_short'
+      cli_args = 'Executioner/dt=1.0 Executioner/num_steps=1 Outputs/file_base=channel_ERCOFTAC_short'
     []
   []
 []