4.2.1. Linear Algebra for CFD Applications¶

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 $${{\partial^2 u} \over {\partial x^2}} + {{\partial^2 u} \over {\partial y^2}}$$ $${{\partial^2 v} \over {\partial x^2}} + {{\partial^2 v} \over {\partial y^2}}$$