In Boundary-Layer Theory by Schlichting he gives the boundary-layer equations for a body of revolution according to the paper by Boltze$^1$. Unfortunately, this paper is in German. He apparently uses the following local curvilinear coordinate system to derive the boundary-layer equations:
I'm not specifically interested in deriving the boundary-layer equations for a body of revolution, however, I am working on a problem with a very similar geometry and am having trouble wrapping my head around the coordinate system.
Question:
For the locally orthogonal basis vectors $\mathbf{e}_x$, $\mathbf{e}_y$, and $\mathbf{e}_\theta$, what are the gradient, divergence, and Laplacian operators?
This is what I have tried:
$$x = \hat{x}$$ $$y = \hat{y}$$ $$r(x)\cos\theta = \hat{z}$$
where $\hat{x}$, $\hat{y}$, and $\hat{z}$ are coordinates in a Cartesian system with the same origin. Lamé's coefficients are then:
$$h_x = \left\vert\left(\frac{\partial \hat{x}}{\partial x}, \frac{\partial \hat{y}}{\partial x}, \frac{\partial \hat{z}}{\partial x}\right)\right\vert = 1$$ $$h_y = \left\vert\left(\frac{\partial \hat{x}}{\partial y}, \frac{\partial \hat{y}}{\partial y}, \frac{\partial \hat{z}}{\partial y}\right)\right\vert = 1$$ $$h_\theta = \left\vert\left(\frac{\partial \hat{x}}{\partial \theta}, \frac{\partial \hat{y}}{\partial \theta}, \frac{\partial \hat{z}}{\partial \theta}\right)\right\vert = \left\vert r(x)\sin\theta\right\vert$$
With these coefficients it is straightforward to define the operators since this system is orthogonal (for example here are the formulas).
I do not think that the results above are correct though since it compares to a local Cartesian system at each point. What are the correct transformations?
- Boltze, Ernst. Grenzschichten an Rotationskorpern in Flussigkeiten mit kleiner Reibung. Georg-August-Universitat zu Gottingen., 1908.
UPDATE:
For any future Googlers, the correct transformation is: (based on Fig. 4 of @Floris's answer, $\xi$ is aligned with $U_\infty$ in the picture above)
$$\xi = x\cos\phi - y\sin\phi$$ $$\eta = -(r(\xi) + y\cos\phi + x\sin\phi)\sin\theta$$ $$\zeta = (r(\xi) + y\cos\phi + x\sin\phi)\cos\theta$$
where $\phi$ is the local angle of the body with respect to the axis of rotation, i.e. $\phi = arctan(\frac{dr}{dx})$. The paper by Boltze assumes this angle is small. The Jacobian is then:
$$J = r(\xi) + y\cos\phi + x\sin\phi$$
which is equivalent to Boltze's value for small $\phi$ and $y \ll r(\xi)$. And the Lamé coefficients are:
$$h_x = 1$$ $$h_y = 1$$ $$h_\theta = r(\xi) + y\cos\phi + x\sin\phi$$