# petebachant.me

• January 02, 2020

# Diesel Explorer: Sold

In the spring of 2019, after roughly 11 years and 80,000 miles of operation, I decided to sell the Diesel Explorer. It was hardly being driven since moving to Boston from Michigan, and I figured it would be better to find a new home where it could be put to good use. The eBay listing caught the attention of some Cummins diesel enthusiasts out west, and shortly thereafter, the Explorer’s new owner flew out to Boston and drove it all the way back to Montana, picking up a boat in Minnesota along the way.

## Conclusions and future work

A simple algorithm was developed to detect the governing PDE from a given analytical solution. The algorithm uses “reverse” finite differences to sample the solution at random points, assembles a linear system of equations, and solves these using singular value decomposition to find the constant, homogeneous coefficients associated with terms of interest. Using an analytical solution to the heat equation, the algorithm successfully identified that the system’s evolution was only affected by diffusion, whereas additional terms for convection and first order wave propagation were shown to be insignificant.

When subjected to Gaussian noise (to simulate experimental data), the algorithm failed for noise amplitudes roughly 6 orders of magnitude smaller than the amplitude of the initial condition. This shows an important weakness for looking at noisy data or higher derivatives, and may necessitate filtering or more robust linear solution techniques.

The example presented here is admittedly trivial. However, it could be used to develop new theories or models for more complex systems, e.g., turbulent flow. For example, we may take the Reynolds-averaged Navier–Stokes equations: $\frac{\partial U_i}{\partial t} + U_j \frac{\partial U_i}{\partial x_j} = -\frac{1}{\rho}\frac{\partial P}{\partial x_i} + \nu \frac{\partial^2 U_i}{\partial x_j^2} -\frac{\partial}{\partial x_j} \overline{u_i^\prime u_j^\prime},$ and attempt to solve for the Reynolds stresses (the last term on the RHS, a.k.a., the “closure problem”) in terms of many arbitrary combinations of the mean velocity and/or pressure, and their partial derivatives—where numerical values for all could be computed from a direct numerical simulation (DNS) of the exact Navier–Stokes equations.

Most likely, resulting equations would not be theoretically correct, and the dimensions of their coefficients may not be able to be formulated in terms of physical parameters, e.g., viscosity. However, the equations would still satisfy the exact equations within some tolerance, and could prove to be useful models that would not have been derived via conventional analytical or phenomenological means.