Merge branch 'main' into feature/license-header-manager

Author: deepanshu-zade

doc/sphinx/source/examples/Navier-Stokes/Cylinder-Flow-2D.md

@@ -0,0 +1,209 @@
+# 2D Channel Flow with a Cylinder
+
+This example solves the two-dimensional incompressible Navier-Stokes equations in a channel using MOLE mimetic operators and a projection (pressure-correction) method.
+
+The cylinder obstacle is introduced as a masked no-slip region inside the channel. To make things easy, the current implementation uses an axis-aligned masked block of cells rather than a fitted curved boundary. The purpose of this example is to show how MOLE operators can be combined to build a transient incompressible flow solver in a direct and compact way.
+
+## Governing Equations
+
+We solve the incompressible Navier-Stokes equations
+
+$$
+\frac{\partial \mathbf{u}}{\partial t}
++ (\mathbf{u}\cdot\nabla)\mathbf{u}
+= -\frac{1}{\rho}\nabla p + \nu \nabla^2 \mathbf{u}
+$$
+
+$$
+\nabla \cdot \mathbf{u} = 0
+$$
+
+where $\mathbf{u} = (u,v)$ is the velocity field, $p$ is the pressure, $\rho$ is the density, and $\nu$ is the kinematic viscosity.
+
+## Spatial Domain
+
+The computational domain is
+
+$$
+x \in [0,8], \qquad y \in [-1,1]
+$$
+
+The default grid is
+
+- $m = 481$ cells in the $x$-direction
+- $n = 121$ cells in the $y$-direction
+
+The Reynolds number is defined through
+
+$$
+\nu = \frac{U_{\mathrm{init}} D_0}{Re}
+$$
+
+with the default values
+
+- $Re = 200$
+- $U_{\mathrm{init}} = 1$
+
+## Initial and Boundary Conditions
+
+### Initial Conditions
+
+At $t = 0$,
+
+$$
+u = U_{\mathrm{init}}, \qquad v = 0
+$$
+
+in the fluid region, and the masked obstacle region is initialized with
+
+$$
+u = 0, \qquad v = 0
+$$
+
+### Velocity Boundary Conditions
+
+- **Inlet (left):** Dirichlet inflow
+
+  $$
+  u = U_{\mathrm{init}}, \qquad v = 0
+  $$
+
+- **Outlet (right):** zero streamwise gradient
+
+  $$
+  \frac{\partial u}{\partial x} = 0, \qquad \frac{\partial v}{\partial x} = 0
+  $$
+
+- **Top and bottom walls:** no-slip
+
+  $$
+  u = 0, \qquad v = 0
+  $$
+
+- **Obstacle mask:** no-slip
+
+  $$
+  u = 0, \qquad v = 0
+  $$
+
+### Pressure Boundary Conditions
+
+During the pressure Poisson step,
+
+- **Outlet (right):** reference pressure
+
+  $$
+  p = 0
+  $$
+
+- **All other boundaries:** homogeneous Neumann
+
+  $$
+  \frac{\partial p}{\partial n} = 0
+  $$
+
+## Numerical Method
+
+A projection method is used to enforce incompressibility.
+
+This is the same overall time-stepping strategy used in both the MATLAB/Octave and C++ implementations.
+
+### Time Integration
+
+The momentum equation is advanced with a mixed time discretization:
+
+- the **convective term** is treated explicitly with **AB2** (Adams-Bashforth 2)
+- the **first time step** uses **AB1**
+- the **diffusive term** is treated implicitly with **Crank-Nicolson**
+
+The convective term is nonlinear, so an explicit AB2 treatment keeps the method simple and avoids solving a nonlinear system at each time step. The diffusive term is linear and is treated with Crank-Nicolson to obtain a stable and second-order accurate semi-implicit update.
+
+As a result, the intermediate velocity solve uses the matrices
+
+$$
+M = I - \frac{1}{2}\Delta t \, \nu L,
+\qquad
+M_p = I + \frac{1}{2}\Delta t \, \nu L
+$$
+
+which are the Crank-Nicolson diffusion matrices used in the Helmholtz-type solves for $u^*$ and $v^*$.
+
+### Intermediate Velocity
+
+An intermediate velocity $\mathbf{u}^*$ is computed first from the momentum equation using the explicit convective update and the semi-implicit diffusive update.
+
+### Pressure Poisson Equation
+
+The pressure is then obtained from
+
+$$
+\nabla^2 p^{n+1}
+=
+\frac{\rho}{\Delta t}\nabla \cdot \mathbf{u}^*
+$$
+
+### Velocity Correction
+
+The velocity is corrected by
+
+$$
+\mathbf{u}^{n+1}
+=
+\mathbf{u}^*
+-
+\frac{\Delta t}{\rho}\nabla p^{n+1}
+$$
+
+### Re-application of Velocity Boundary Conditions and Mask
+
+After the pressure correction step, the code re-applies the velocity boundary values and the obstacle mask.
+
+This is done because the projection step updates the full cell-centered velocity field. Re-applying these values ensures that the final updated velocity satisfies the intended inlet, wall, outlet, corner, and masked-region constraints exactly.
+
+## Role of the MOLE Operators
+
+This example is intended to highlight how MOLE operators are used in an incompressible flow solver.
+
+The main operators are
+
+- **Laplacian $L$:** used for viscous diffusion
+- **Divergence $D$:** used in the convective flux form and in the pressure Poisson right-hand side
+- **Gradient $G$:** used in the pressure correction step
+- **Interpolation operators:** used to map between cell-centered values and face-based quantities for flux evaluation and projection
+
+These operators are combined to build
+
+- the Crank-Nicolson diffusion operators
+- the pressure Poisson operator
+- the interpolation maps needed for face fluxes and pressure-gradient correction
+
+
+## Running the Example
+
+After building the C++ examples, run
+
+```bash
+cd <mole-repo>/examples/cpp
+../../build/examples/cpp/cylinder_flow_2D
+```
+
+## MATLAB and C++ Versions
+
+MATLAB/Octave and C++ versions of this example are provided with the same problem setup and the same overall projection-method structure. In particular, both versions use
+
+- AB2 for the convective term
+- AB1 at the first step
+- Crank-Nicolson for the diffusive term
+- a pressure Poisson solve followed by velocity correction
+
+This makes it easier to compare the two implementations and to see how the same MOLE operators are used in each language.
+
+## Results
+
+### C++ result
+
+![C++ final fields](figures/cylinder_flow_2D_plot_cpp.png)
+
+### MATLAB result
+
+![MATLAB final fields](figures/cylinder_flow_2D_plot_matlab.png)

doc/sphinx/source/examples/Navier-Stokes/figures/cylinder_flow_2D_plot_cpp.png

@@ -7,4 +7,5 @@ The Navier-Stokes equations describe the motion of viscous fluid substances, for
 :caption: Contents
 
 Lock-Exchange
+Cylinder-Flow-2D
 ``` 

doc/sphinx/source/examples/Navier-Stokes/figures/cylinder_flow_2D_plot_matlab.png

@@ -5,6 +5,7 @@ set(EXAMPLE_SOURCES
   backward_euler.cpp
   burgers1D.cpp
   convection_diffusion3D.cpp
+  cylinder_flow_2D.cpp
   elliptic1D.cpp
   elliptic1DHomogeneousDirichlet.cpp
   elliptic1DLeftNeumannRightDirichlet.cpp