E. Turgeon

A CONTINUOUS SENSITIVITY EQUATION APPROACH TO OPTIMAL DESIGN IN MIXED CONVECTION

In this paper, we consider the design of a mixed convection system to achieve optimum heat transfer properties. Our design methodology consists of using an adaptive finite element method to obtain approximations for both flow and sensitivity variables. These quantities allow us to calculate values of a design objective function and its gradient. A BFGS/trust-region optimization algorithm uses this information to find optimum parameter values. The adaptive remeshing strategy is constructed to ensure accurate resolution of both flow and sensitivity variables throughout the design iterations. We present this methodology along with numerical results that include a validation of the effectiveness of our remeshing strategy along with the solution to an optimal design problem.In this paper, we consider the design of a mixed convection system to achieve optimum heat transfer properties. Our design methodology consists of using an adaptive finite element method to obtain approximations for both flow and sensitivity variables. These quantities allow us to calculate values of a design objective function and its gradient. A BFGS/trust-region optimization algorithm uses this information to find optimum parameter values. The adaptive remeshing strategy is constructed to ensure accurate resolution of both flow and sensitivity variables throughout the design iterations. We present this methodology along with numerical results that include a validation of the effectiveness of our remeshing strategy along with the solution to an optimal design problem.

A GENERAL CONTINUOUS SENSITIVITY EQUATION FORMULATION FOR COMPLEX FLOWS

In this article, we develop a general formulation of the continuous sensitivity equation method that accounts for a complex parameter dependence in both flow variables and physical fluid properties (such as viscosity, thermal conductivity, etc.). This formulation unifies the treatment of shape sensitivities and value sensitivities. The result leads to the development of software that is suitable for a wide range of problems. In addition to details of an implementation within an existing adaptive finite-element program, we perform a careful verification study as well as demonstrate the flexibility of the software by computing cost function gradients for an optimization algorithm.

A General Continuous Sensitivity Equation Formulation for the k-ε Model of Turbulence

In this paper, we develop a general formulation of the continuous sensitivity equations (CSEs) for the standard model of turbulence with wall functions. The development is performed for value parameters that do not affect the geometry of the computational domain. The formulation accounts for complex parameter dependencies and results in the development of software that is suitable for a wide range of problems. In addition to details of an implementation within an existing adaptive finite element program, we perform a careful verification study and present an application of sensitivity analysis to turbulent flow over a flat plate.

A CONTINUOUS SENSITIVITY EQUATION METHOD FOR FLOWS WITH TEMPERATURE DEPENDENT PROPERTIES

Sensitivity analysis for free convection of corn syrup (modeled with temperature dependent viscosity, conductivity and specific heat) is carried out by solving the continuous sensitivity equation with an adaptive finite element solver. The sensitivity of the flow to changes in boundary conditions, geometric variation and coefficients in material property relationships are presented. Additionally, we emphasize three applications of sensitivity analysis: its role in identifying the significance of design parameters on the flow, its use in calculating nearby flows, and its utility in uncertainty analysis.

Adaptivity, Sensitivity, and Uncertainty: Toward Standards of Good Practice in Computational Fluid Dynamics

Three issues related to good computational e uid dynamics (CFD) practice are discussed. First, adaptive meth- ods are shown to be a simple tool to perform systematic grid ree nement studies needed to achieve solutions with controlled accuracy (verie cation of simulations). Second, it is shown that the sensitivity equation method pro- vides insights about which parameters critically affect the e ow response. Finally, e ow sensitivities are used to propagate model parameter uncertainties through the CFD code to yield uncertainty estimates of the CFD predic- tions. This provides a rigorous framework for comparing predictions to measurements (validation of predictions). These combined approaches help to build cone dence in CFD predictions and develop good CFD practice. The resulting uncertainty bars put CFD on par with experimental techniques. The approaches are demonstrated on two-dimensional problems: a k-≤ model of the e ow in an annular turn-around duct and conjugate free convection with variable e uid properties. Taken together, these approaches offer a good prospect for developing families of computing methods that can be viewed as standards of good practice in CFD, ensuring that verie cation and validation studies are performed on solid grounds.

APPLICATIONS OF CONTINUOUS SENSITIVITY EQUATIONS TO FLOWS WITH TEMPERATURE-DEPENDENT PROPERTIES

An adaptive finite-element solver is used to perform sensitivity analysis for free convection in corn syrup, a fluid with strong temperature dependence of viscosity, conductivity, and specific heat. Results are presented for sensitivity of the flow with respect to parameters describing boundary conditions, the geometry of the domain, and constitutive equations for the physical properties. This model is used to emphasize three applications of sensitivity analysis: its role in identifying key parameters controlling the flow, its use in fast calculation of nearby flows, and its utility in uncertainty analysis.

AN ADAPTIVE FINITE ELEMENT METHOD FOR FORCED CONVECTION

SUMMARY This paper presents an adaptive finite element method to solve forced convective heat transfer. Solutions are obtained in primitive variables using a high-order finite element approximation on unstructured grids. Two general-purpose error estimators are developed to analyse finite element solutions and to determine the characteristics of an improved mesh which is adaptively regenerated by the advancing front method. The adaptive methodology is validated on a problem with a known analytical solution. The methodology is then applied to heat transfer predictions for two cases of practical interest. Predictions of the Nusselt number compare well with measurements and constitute an improvement over previous results. # 1997 John Wiley & Sons, Ltd.

COMPUTATION OF JET IMPINGEMENT HEAT TRANSFER BY AN ADAPTIVE FINITE ELEMENT ALGORITHM

This paper presents adaptive finite element computations of laminar and turbulent jet impingement heat transfer. Turbulence is modeled using the standard k — c model for high Reynolds number, coupled with wall functions. The turbulence model is solved in logarithmic form. The error estimator uses a local least squares projection of derivatives and accounts for errors in all dependent variables, including the eddy viscosity. The performance of the methodology is verified by solving a problem possessing a closed form solution. Two applications are then considered: laminar and turbulent impinging jets. In both cases, heat transfer is a key element of the study. Results indicate that the methodology can produce grid independent solutions even for derived quantities and in thin boundary layers. Numerical predictions are compared to experimental data. Nomenclature

Prediction of Turbulent Separated Flow in a Turnaround Duct Using Wall Functions and Adaptivity

Abstract This paper presents an application of adaptive remeshing to the prediction of turbulent separated flows. The paper shows that the κ - ϵ model with wall functions can predict separated flows along smooth curved surfaces. Success is achieved if the wall functions exhibit values of y+ close to 30, and if meshes are fine enough to guarantee that wall function boundary conditions are grid converged. Adaptive remeshing proves to be a very cost effective tool in this context. The methodology is demonstrated on a problem possessing a closed form solution to establish the performance and reliability of the proposed approach. The method is then applied to prediction of turbulent flow in an annular, axisymmetric turnaround duct (TAD). Predictions from two computational models of the TAD are compared with experimental measurements. The importance of appropriate meshes to achieve grid independent solutions is demonstrated in both cases. Better agreement with measurements is obtained when partially developed profiles of u, κ, and ϵ are specified at the TAD inlet.