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