# Some Mathematical Aspects of Thermo and Fluid Dynamics Doktorsavhandling, 2000

This thesis contains mathematical treatments of three issues in thermo and fluid dynamics: swirling jets and vortex breakdown, Taylor's hypothesis and conjugate heat transfer. The work on swirling jets is focused on the study of the well-posedness of the conically self-similar free-vortex solutions to the Navier--Stokes equations. Sufficient conditions for the existence and non-existence of such solutions are established. In addition, it is proven that these solutions are all real analytic except at the origin. Furthermore, these results are extended to establish that a conically self-similar free-vortex solution to the Navier--Stokes equations is uniquely determined by the opening angle of the bounding conical streamsurface and by two flow properties thereon. Finally, in the case of high circulation over viscosity ratio, asymptotic formulae are derived for the dependence of the opening angle of the separating conical streamsurface for two-cell flows, in terms of quantities defined on the bounding conical streamsurface. The remaining work on swirling jets includes a numerical study of the relevance of some commonly used swirl parameters for the onset of vortex breakdown, as well as a brief study into the existence properties for Long's jet as well as for general self-similar axisymmetric vortex cores. In the work on Taylor's hypothesis, it is argued that in flow cases when the acceleration terms of the Navier--Stokes equations are negligible, a drastic increase in the complexity of the solutions to the Navier--Stokes equations for the fluctuating velocity occurs when the quantity Uj .DELTA.Ui /.DELTA.xj changes from being a linear function in space to being a non-linear one. In addition, for uniform mean flow with negligible acceleration terms, sufficient conditions for the coincidence of the advection and local mean velocity are proven. Moreover, a method to take into account the effect of mean velocity gradients when using Taylor's hypothesis is proposed, which should allow a substantial relaxation of Lin's criterion in many cases. Finally, an asymptotic solution is presented to the conjugate heat transfer problem for a flush mounted heat source on the fluid-solid interface, in the case that the bottom of the solid is perfectly insulated and the velocity profile in the fluid is linear in the wall-normal direction, and constant in the other space dimensions as well as in time.

swirling jets

asymptotic analysis

Long's jet

Taylor's hypothesis

conjugate heat transfer

conical self-similarity

vortex breakdown

frozen turbulence hypothesis

## Författare

### Carl Fredrik Stein

Institutionen för matematik

Göteborgs universitet

Matematik

### ISBN

91-7197-925-5

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 1610

2017-10-07