Finding a weak solution of the heat diffusion differential equation for turbulent flow by Galerkin's variation method using p-version finite elements

The stochastic turbulence model developed by Professor Czibere provides a means of clarifying the flow conditions in pipes and of describing the heat evolution caused by shear stresses in the fluid. An important part of the theory is a consideration of the heat transfer-diffusion caused by heat generation. Most of the heat is generated around the pipe wall. One part of the heat enters its environment through the wall of the tube (heat transfer), the other part spreads in the form of diffusion in the liquid, increasing its temperature. The heat conduction differential equation related to the model contains the characteristics describing the turbulent flow, which decisively influence the resulting temperature field, appear. A weak solution of the boundary value problem is provided by Bubnov-Galerkin’s variational principle. The axially symmetric domain analyzed is discretized by a geometrically graded mesh of a high degree of p-version finite elements, this method is capable of describing substantial changes in the temperature gradient in the boundary layer. The novelty of this paper is the application of the p-version finite element method to the heat diffusion problem using Czibere’s turbulence model. Since the material properties depend on temperature, the problem is nonlinear, therefore its solution can be obtained by iteration. The temperature states of the pipes are analyzed for a variety of technical parameters, and useful suggestions are proposed for engineering designs.