Measurement of the unsteady thermal Green's function in a boundary layer flow: A preliminary theory

Compared to the heat transfer coefficient, the thermal Green's function concept is a more fundamental method of describing the relationship between local wall heat transfer and wall temperature. It is far more amenable to situations involving strong spatial, temporal, and boundary condition variability. The utility of this methodology has been established, in particular for the analysis of the conjugate heat transfer problem. A necessary element in this technique is an inverse theoretical model to infer the Green's function from laboratory thermal response data. This paper presents preliminary results from a first attempt to develop such a measurement theory to extract local approximations to the unsteady thermal Green's function (UTGF) in 3-D boundary layer flows. The flow model used is a linear shear flow, which is a solution valid in the near wall region of laminar flows, as well as the viscous sublayer region of a turbulent boundary layer. This model is governed by the shear velocity, which is a measure of the local wall shear stress. The solution methodology employs the mathematical theory of Green's function solutions to the energy equation for a general 3-D boundary layer flow, where the UTGF of interest is the thermal response to an impulsive heat load. Analytic methods are used to condense the equation from a 3-D to a 2-D transient PDE, and the reduced equation is solved using a Petrov-Galerkin Finite Element Method. These data are used to construct a numerical UTGF uniquely determined by the shear velocity, flow angle, and the thermodynamic properties of the fluid. An error minimization scheme is proposed to find the appropriate value of the shear velocity, thus providing local UTGF, and shear velocity measurements.