4.2.1. Linear Algebra for CFD Applications¶

2D Vector Notation

4.2.1.1. 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

4.2.1.1.1. 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	\({{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}}\)

4.2.1.1.2. 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	\({{\partial^2 u} \over {\partial x^2}} + {{\partial^2 u} \over {\partial y^2}}\)	\({{\partial^2 v} \over {\partial x^2}} + {{\partial^2 v} \over {\partial y^2}}\)