Pressure from Velocity

This chapter describes extraction of static pressure from velocity as an inverse problem. The non-dimensional relation between the total pressure and the velocity-related source term is given as an adapted form of the Navier-Stokes (NS) equations. Projection of the gradient of the total pressure on a given test vector field leads to a first-order partial differential equation. For a constant test vector field, the method of characteristics is described to calculate a pressure field by using a line integral along a given characteristic line, and then an error analysis for this line integral method is given. Further, a global variational formulation is proposed and the Euler-Lagrange equation is derived for the total pressure. In particular, for a constant test vector field, a second-order elliptical-type particle differential equation with constant coefficients is obtained, where the Neumann condition is imposed on the boundary of a domain. To solve this Euler-Lagrange equation with suitable boundary conditions for a field of the total pressure, the Direct-QR method and the iterative method are described. The accuracy of these methods is evaluated through simulations in the oblique Hiemenz flow. As an example, the pressure fields near a freely flying hawkmoth are obtained from high-resolution velocity fields extracted from Schileren visualization images by the optical flow method.