Sensitivity analysis of low Reynolds number channel flow using the finite volume method

Abstract

SUMMARYAn unsteady finite volume-based fractional step algorithm solved on a staggered grid has been developed for computing design sensitivity parameters in two-dimensional flows. Verification of the numerical code is performed for the case of low Reynolds number, pressure-driven flow through a straight channel, which has an exact steady-state solution to the Navier-Stokes equations. Sensitivity of the flow to the channel height, fluid viscosity, and imposed pressure gradient is considered. Three different numerical techniques for computing the design sensitivity parameters: finite difference, complex-step differentiation, and sensitivity equation method (SEM), are compared in terms of numerical error (relative to the exact solution), computational expense, and ease of implementation. Results indicate that, of all the three methods, complex step is the most accurate and requires the least computational time. In addition, treatment of the boundary conditions in SEM is addressed, within the framework of the present finite volume approach, with special attention given to parameter dependence in the boundary conditions. Error estimation based on the Grid Convergence Index provides a good indication of the exact error in the SEM solutions. An example of application of the use of sensitivity parameters to estimate the propagation of input uncertainty through the numerical simulation is also provided.

Coherent features in the sensitivity field of a planar mixing layer

“…As such, the complex-step derivative remains essentially insensitive to the parameter step size, 9,15 in contrast to the finite difference method.…”

“…Importantly, because of this, SEM provides a quantitative measure for interrogating how inherently unsteady coherent features of the turbulence are instantaneously affected by changes in the parameter͑s͒ of interest. 15,24 Note that the complex-step differentiation method also has this capability. 25…”