Hayri Sezer

Error Estimation Using Hybrid Methods

New formalisms are developed for robust calculation of the discretization errors in CFD applications. The new methods are based on the premise that the true error (i.e. the difference between the exact solution and the numerical solution) on a given mesh is similar to the approximate error (i.e. the difference between the fine grid solution and the coarse grid solution). The proportionality constant can be calculated theoretically for a given scheme and it is, to a first order approximation, only a function of the grid refinement ratio. This method is called Approximate Error Scaling (AES) method.Although AES is a viable method for error estimation it does not directly take into account the transport of error nor the error sources at the boundaries. To remedy this, a new method which is referred to, here and after, as the residual source in transport (REST-IC) method is formulated. This paper presents the theoretical details and the results from several test cases that are used for validation of the newly proposed methods.Application to the 1D and 2D steady scalar transport equation and 2D Navier-Stokes (N-S) equations has revealed excellent performance with a pressure based segregated N-S finite volume solver. Implementation of these methods into CFD commercial codes such as ANSYS-FLUENT can be done with the help of expert developers of the code who can handle residuals for the selected equation and the variable through user defined subroutines.Copyright

Effect of local variations of the laminar flame speed on the global finger-flame acceleration scenario

Numerous formulations describing the dynamics and morphology of corrugated flames, including the scenarios of flame acceleration, are based on a “geometrical consideration”, where the wrinkled-to-planar flame velocities ratio, Sw /SL , is evaluated as the scaled flame surface area, while the entire combustion chemistry is immersed into the planar flame speed SL , which is assumed to be constant. However, SL may experience noticeable spatial/temporal variations in practice, in particular, due to pressure/temperature variations as well as non-uniform distribution of the equivalence ratio and/or that of combustible or inert dust impurities. The present work initiates the systematic study of the impact of the local SL -variations on the global flame evolution scenario. The variations are assumed to be imposed externally, in a manner being a free functional of the formulation. Specifically, the linear, parabolic and hyperbolic spatial SL -distributions are incorporated into the formulations of finger flame acceleration in pipes, and they are compared to the case of constant SL . Both two-dimensional channels and cylindrical tubes are considered. The conditions promoting or moderating flame acceleration are identified, and the revisited equations for the flame shape, velocity and acceleration rate are obtained for various SL -distributions. The theoretical findings are validated by the computational simulations of the reacting flow equations, with agreement between the theory and modelling demonstrated.

Flame acceleration due to wall friction: Accuracy and intrinsic limitations of the formulations

The analytical formulations on the premixed flame acceleration induced by wall friction in two-dimensional (2D) channels [Bychkov et al., Phys. Rev. E 72 (2005) 046307] and cylindrical tubes [Akkerman et al., Combust. Flame 145 (2006) 206] are revisited. Specifically, pipes with one end closed are considered, with a flame front propagating from the closed pipe end to the open one. The original studies provide the analytical formulas for the basic flame and fluid characteristics such as the flame acceleration rate, the flame shape and its propagation speed, as well as the flame-generated flow velocity profile. In the present work, the accuracy of these approaches is verified, computationally, and the intrinsic limitations and validity domains of the formulations are identified. Specifically, the error diagrams are presented to demonstrate how the accuracy of the formulations depends on the thermal expansion in the combustion process and the Reynolds number associated with the flame propagation. It is shown that the 2D theory is accurate enough for a wide range of parameters. In contrast, the zeroth-order approximation for the cylindrical configuration appeared to be quite inaccurate and had to be revisited. It is subsequently demonstrated that the first-order approximation for the cylindrical geometry is very accurate for realistically large thermal expansions and Reynolds numbers. Consequently, unlike the zeroth-order approach, the first-order formulation can constitute a backbone for the comprehensive theory of the flame acceleration and detonation initiation in cylindrical tubes. Cumulatively, the accuracy of the formulations deteriorates with the reduction of the Reynolds number and thermal expansion.

Implications of Using Dual Number Derivatives With a Numerical Solution

Dual number arithmetic is a well-known strategy for automatic differentiation of computer codes which gives exact derivatives, to the machine accuracy, of the computed quantities with respect to any of the involved variables. A common application of this concept in Computational Fluid Dynamics, or numerical modeling in general, is to assess the sensitivity of mathematical models to the model parameters. However, dual number arithmetic, in theory, finds the derivatives of the actual mathematical expressions evaluated by the computer code. Thus the sensitivity to a model parameter found by dual number automatic differentiation is essentially that of the combination of the actual mathematical equations, the numerical scheme and the grid used to solve the equations not just that of the model equations alone as implied by some studies. This aspect of the sensitivity analysis of numerical simulations using dual number auto derivation is explored in the current study. A simple one-dimensional advection diffusion equation is discretized using different schemes of finite volume method and the resulting systems of equations are solved numerically. Derivatives of the numerical solutions with respect to parameters are evaluated automatically using dual number automatic differentiation. In addition the derivatives are also estimated using finite differencing for comparison. The analytical solution was also found for the original PDE and derivatives of this solution are also computed analytically. It is shown that a mathematical model could potentially show different sensitivity to a model parameter depending on the numerical method employed to solve the equations and the grid resolution used. This distinction is important since such inter-dependence needs to be carefully addressed to avoid confusion when reporting the sensitivity of predictions to a model parameter using a computer code. A systematic assessment of numerical uncertainty in the sensitivities computed using automatic differentiation is presented.Copyright