1. Linear Algebra for CFD Applications¶
1.1. 2D Vector Notation¶
- Divergence of a vector gives a scalar (via the scalar product)
- Divergence of a scalar doesn’t exist
- Gradient of a vector gives a vector
- Gradient of a scalar gives a scalar
1.1.1. Continuity Equation¶
The 2D Continuity Equation:
(1)\[\nabla \cdot \mathbf{u} = 0\]
| Expression 1 | Expression 2 | Component 1a | Component 1b | Operation | Expansion |
|---|---|---|---|---|---|
| \(\nabla \cdot \mathbf{u}(x,y)\) | \(\text{div } \mathbf{u}\) | \(\begin{bmatrix} \mathbf{i} \ {\partial \over {\partial x}} \\ \mathbf{j} \ {\partial \over {\partial y}} \end{bmatrix}\) | \(\begin{bmatrix} {u} \\ {v} \end{bmatrix}\) | Scalar Product | \({{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}}\) |
1.1.2. Momentum Equation¶
The 2D Momentum Equation:
(2)\[{{\partial \mathbf{u}} \over {\partial t}} + \mathbf{u} \cdot \nabla \mathbf{u} =
{-{1 \over \rho} \nabla p} + {\nu \nabla^2 \mathbf{u}}\]
| Expression 1 | Expression 2 | Component 1a | Component 1b | Operation | Expansion-x | Expansion-y |
|---|---|---|---|---|---|---|
| \({\partial \mathbf{u}(x,y)} \over t\) | N/A | \({\partial \over {\partial t}}\) | \(u\) or \(v\) | Time Derivative | \({\partial u} \over {\partial t}\) | \({\partial v} \over {\partial t}\) |
| \(\nabla \mathbf{u}(x,y)\) | \(\text{grad } \mathbf{u}\) | \(\begin{bmatrix} \mathbf{i} \ {\partial \over {\partial x}} \\ \mathbf{j} \ {\partial \over {\partial y}} \end{bmatrix}\) | \({u}\) or \({v}\) | Vector Gradient | \(\begin{bmatrix} {{\partial u} \over {\partial x}} \mathbf{i} \\ {{\partial u} \over {\partial y}} \mathbf{j} \end{bmatrix}\) | \(\begin{bmatrix} {{\partial v} \over {\partial x}} \mathbf{i} \\ {{\partial v} \over {\partial y}} \mathbf{j} \end{bmatrix}\) |
| \(\mathbf{u} \cdot \nabla \mathbf{u}(x,y)\) | \(\mathbf{u} \cdot \text{grad } \mathbf{u}\) | \(\begin{bmatrix} {u} \\ {v} \end{bmatrix}\) | \(\begin{bmatrix} {{\partial u} \over {\partial x}} \mathbf{i} \\ {{\partial u} \over {\partial y}} \mathbf{j} \end{bmatrix}\) or \(\begin{bmatrix} {{\partial v} \over {\partial x}} \mathbf{i} \\ {{\partial v} \over {\partial y}} \mathbf{j} \end{bmatrix}\) | Scalar Product | \(u{{\partial u} \over {\partial x}} + v {{\partial u} \over {\partial y}}\) | \(u{{\partial v} \over {\partial x}} + v {{\partial v} \over {\partial y}}\) |
| \(\nabla p(x)\) or \(\nabla p(y)\) | \(\text{grad } p\) | \({\partial \over {\partial x}}\) or \({\partial \over {\partial y}}\) | \({p}\) | Scalar Gradient | \({{\partial p} \over {\partial x}}\) | \({{\partial p} \over {\partial y}}\) |
| \(\nabla^2 \mathbf{u}(x,y)\) | \(\nabla \cdot \nabla \mathbf{u}(x,y)\) | \(\begin{bmatrix} \mathbf{i} {\partial \over {\partial x}} \\ \mathbf{j} {\partial \over {\partial y}} \end{bmatrix}\) | \(\begin{bmatrix} {{\partial u} \over {\partial x}} \mathbf{i} \\ {{\partial u} \over {\partial y}} \mathbf{j} \end{bmatrix}\) or \(\begin{bmatrix} {{\partial v} \over {\partial x}} \mathbf{i} \\ {{\partial v} \over {\partial y}} \mathbf{j} \end{bmatrix}\) | Laplacian | \({{\partial^2 u} \over {\partial x^2}} + {{\partial^2 u} \over {\partial y^2}}\) | \({{\partial^2 v} \over {\partial x^2}} + {{\partial^2 v} \over {\partial y^2}}\) |