Parameter Sensitivity of an Eddy Viscosity Model: Analysis, Computation and Its Application to Quantifying Model Reliability

Abstract

An eddy viscosity model can be used as a computationally tractable alternative to that of the Navier-Stokes equations. Model errors immediately become a concern when considering such an approach, and quantifying this error is essential to understanding and using model predictions within an engineering design process. In this paper, sensitivity analysis is presented for a subgrid eddy viscosity model with respect to variations of the eddy viscosity parameter. We demonstrate the analysis utilizing the sensitivity equation method. Approximating the sensitivity requires the solution of the eddy viscosity model. Therefore, the eddy viscosity model and sensitivity equation are coupled in our analysis and computations. An implicit-explicit time-stepping method is developed and analyzed for this set of equations. Our numerical assessments present the role of the sensitivity in quantifying the modeling error arising from the choice of various values of the eddy viscosity parameter. The sensitivity computation allows one to identify an interval of reliability for the eddy viscosity parameter. This gives the user a range of parameter values for which the eddy viscosity model can be considered to be a reliable approximation to the Navier-Stokes equations. A two-dimensional cavity problem is used to illustrate the ideas. In addition, for the standard model problem of two-dimensional flow around a cylinder, the sensitivity computations are shown to be very useful in improving the flow functional approximations that may be used within an optimal design algorithm.

A Review of Time Relaxation Methods

“…Sensitivity studies can be used with great effect to help quantify the reliability of such calculations. While [22] did not explore particular methods to directly quantify model reliability, examples can be found in [23,24]. One such technique is to quantify reliability by using the linear estimate, ||v h (0) − v h (χ)|| ≤ χ||s h || + O(χ 2 ).…”

A sensitivity study of relaxation parameter in Uzawa algorithm for the steady natural convection model

“…Further, these authors have showed a numerical study of the sensitivity of a fluid model known as time relaxation model with respect to variations of the time relaxation coefficient (Neda et al , 1999). (Pahlevani and Davis, 2013) have developed and analyzed a numerical method for simulating an eddy viscosity model and its corresponding sensitivity with respect to the eddy viscosity parameter.…”

Sensitivity Analysis and Computations of the Time Relaxation Model

“…Over the years, there have been studies on the sensitivity of fluid flows in different aspects, [3,9,23,39,41,42]. Assessing model error that leads to uncertainty quantification is an important application of sensitivity analysis, see [35,40] for recent works in this direction. Defining the sensitivity of state variables in a physical system as derivatives of state varaibles with respect to the selected parameter, there are basically two methods of numerically calculating the sensitivities: Forward Finite Difference method (FFD) and Sensitivity Equation Method (SEM).…”