Computational Fluid Dynamics - Elementary (c)

Curvilinear coordinates

In Cartesian coordinates, there are three mutually perpendicular families of planes: $ x $ = constant, $ y $ = constant, and $ z $ = constant. Imagine that we superimpose on this system three other families of surfaces. Describing the curvilinear coordinate surfaces by $ q_1 $ = constant, $ q_2 $ = constant, and $ q_3 $ = constant, we may identify our point by ($ q_1, q_2, q_3 $):
\begin{equation} \begin{aligned} & x = x(q_1, q_2, q_3) \ \ \ \longleftrightarrow \ \ \ q_1 = q_1(x, y, z), \\ & y = y(q_1, q_2, q_3) \ \ \ \longleftrightarrow \ \ \ q_2 = q_2(x, y, z), \\ & z = z(q_1, q_2, q_3) \ \ \ \longleftrightarrow \ \ \ q_3 = q_3(x, y, z). \\ \end{aligned} \end{equation} As an example, relabelling ($ q_1, q_2, q_3 $) as ($ r, \theta, \phi $), the spherical polar coordinates are introduced.

The scale factors $ h_i $ ($ i = 1,2,3 $) can specify the nature of the coordinate system ($ q_1, q_2, q_3 $) and for orthogonal (mutually perpendicular surfaces) coordinate systems can be derived as:
\begin{equation} h_i^2 = \left(\frac{\partial x}{\partial q_i}\right)^2 + \left(\frac{\partial y}{\partial q_i}\right)^2 + \left(\frac{\partial z}{\partial q_i}\right)^2. \end{equation} With each family of surface $ q_i $ = constant, we can associate a unit vector $ \textbf{a}_i $ normal to the surface and in the direction of increasing $ q_i $.

Now it is possible to develop the operators like gradient, divergence, curl, and Laplacian by means of $ \textbf{a}_i $ and $ h_i $ for every coordinate system:
\begin{equation} \boldsymbol{\nabla} \psi = \mathbf{a}_1 \frac{\partial \psi}{h_1 \partial q_1} + \mathbf{a}_2 \frac{\partial \psi}{h_2 \partial q_2} + \mathbf{a}_3 \frac{\partial \psi}{h_3 \partial q_3}, \end{equation} \begin{equation} \boldsymbol{\nabla} \cdot \mathbf{V}(q_1, q_2, q_3) = \frac{1}{h_1 h_2 h_3} \left[\frac{\partial}{\partial q_1}(V_1 h_2 h_3) + \frac{\partial}{\partial q_2}(V_2 h_3 h_1) + \frac{\partial}{\partial q_3}(V_3 h_1 h_2) \right], \end{equation} \begin{equation} \boldsymbol{\Delta} \psi = \frac{1}{h_1 h_2 h_3} \left[\frac{\partial}{\partial q_1}\left(\frac{h_2 h_3}{h_1} \frac{\partial \psi}{\partial q_1} \right) + \frac{\partial}{\partial q_2}\left(\frac{h_3 h_1}{h_2} \frac{\partial \psi}{\partial q_2} \right) + \frac{\partial}{\partial q_3}\left(\frac{h_1 h_2}{h_3} \frac{\partial \psi}{\partial q_3} \right) \right], \end{equation} \begin{equation} \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{V} = \frac{1}{h_1 h_2 h_3} \begin{vmatrix} \mathbf{a}_1 h_1 & \mathbf{a}_2 h_2 & \mathbf{a}_3 h_3 \\ \partial / \partial q_1 & \partial / \partial q_2 & \partial / \partial q_3 \\ h_1 V_1 & h_2 V_2 & h_3 V_3 \end{vmatrix} . \end{equation} Using these general formulations, it is possible to derive the differential operators for every curvilinear coordinate system.