Application of modified Navier-Stokes equations to determine the unsteady force effects of a heterogeneous liquid

Abstract

The article is focused on calculating the force effects of a heterogeneous liquid on pipe walls. The solution is based on the concentration of solid particles. The base fluid is assumed to be incompressible. The solution will apply Euler-Lagrange's solution principle. Two tasks will be solved; with a rigid and a flexible tube wall. The solution will be carried out with non-stationary boundary conditions that were determined experimentally. Interaction of a heterogeneous fluid with a flexible wall assumes its deformation. The force effects will be solved by two methods; FSI simulation using ANSYS FEA solvers and CFD solvers ANSYS Fluent and using Navier-Stokes equations by direct integration through liquid volume. In this case, the unsteady term of the Navier-Stokes equations will be modified so that the Gauss Ostrogradsky theorem can be used to calculate the force. At the end, the force effects on the rigid and compliant wall will be compared with the unsteady turbulent flow of the heterogeneous liquid.

Methods for Simplification of the Mathematical Model for the Calculation of Flows in the Flow Path of Hydraulic Turbines

“…Differentiating continuity equation ( 5) with respect to time yields Application of a procedure similar to that used for ( 12)-( 14), gives (15) Integral equations ( 13) and ( 14) can also be written in vector form (16) The physical concept of equations ( 16) can be very useful for describing unsteady vortex structures inside the region V [18][19][20][21]. Obviously, local acceleration within the volume V depends only on the boundary conditions.…”

“…Substituting (11) and ( 15) into the Navier-Stokes equations (8), yields (20) Equation ( 20) is a new form of the Navier-Stokes equations for incompressible fluid. To analyze equation (20) further, a new tensor is introduced (21) Let us introduce a coordinate system meeting the conditions (22) We can then write (23) and, therefore, (…”

“…It follows from this equation that the force can be calculated without a knowledge of the pressure and velocity fields [20,21]. Therefore, equation (27) may be helpful not only for calculating the acting forces.…”

“…(34) Equations ( 27) and (34) make it possible to estimate the effect of boundary conditions on the resulting force. Moreover, equation (34) can be used for numerical calculations by the finite volume method [3,21].…”