Improving the accuracy of a 1D Gas Dynamics Model

This paper and its companion present the salient features of a 1D gas dynamics code which traces the propagation of finite amplitude waves along a quasi 1D duct. This model extends an originally first order method to second order, and re-works the way heat transfer is accounted for, including a method for maintaining full mass conservation. The model is fully nonhomentropic, accounting for the large variations in gas properties and temperature that routinely occur in internal combustion engine ducts. In this paper, the model is tested against three basic problems for constant-area frictionless flow – namely propagation of a small short wavelength pulse to test numerical smearing, the shock tube problem to test handling of steep gradients, and the Rayleigh flow problem to test the heat transfer implementation. The tests show the model performs very well in these simple cases which are a pre-requisite to more complex modeling tasks. This mode offers a useful alternative to finite difference based codes and addresses the criticisms usually made of wave action methods. It allows a uniform treatment of duct and cell boundaries to be used throughout. Introduction Unsteady gas dynamics is an important phenomenon in numerous fluid-mechanical devices – most well known of which is the reciprocating internal combustion engine, but also of importance in other unsteady devices such as pulse jet engines and in normally steady flow devices such as gas piping and turbomachinery. Gas dynamic modeling has a long history. The relationship between pressure and velocity for a wave travelling in one direction was derived by Samuel Earnshaw [7] in 1860. Bannister and Mucklow [1] experimentally validated the theory of finite waves using a shock tube experiment. Riemann [11] had earlier proposed the method of characteristics (MOC) and this was now developed into graphical solution for unsteady flow problems. Benson et al. [2] used digital computing and the MOC to evaluate unsteady flows in engine ducts. By the 1980’s gas dynamics in engine modelling using the MOC was well established [3]. Since then finite difference formulations have become popular for engine gas dynamics [12], though research using the MOC continue eg [13]. A vast plurality of numerical schemes and variations on schemes exists in the literature and in engine research laboratories around the world. A code should display good mass conservation, handle thermal and property discontinuities and handle rapid changes in pressure or even shocks. For engine modeling, the code must accuracy simulate flow in tapered ducts and sudden changes in area such as pipe entries, valves, junctions and the like. The accuracy of the discretisation method (be it first, second or higher order) is also important. Finally, the computational efficiency and complexity of the code should be considerd. 1D engine simulations require modeling of multiple boundary connections, tapers and area changes, and gas property changes. These requirements mean that the simple elegance of a finite difference solution is somewhat smothered. A critical comparison of the most popular schemes on all of the above issues is beyond the scope of this paper, and has indeed been attempted by others [5, 6, 10, 12]. This paper and its companion [9] merely present the salient features of another gas dynamics code. This is a wave action based code which addresses the main shortcomings of typical wave action methods. The 1D gas dynamics code introduced in here is based on the method developed at Queens University, Belfast eg. [4, 8] and subsequently adopted by the commercial simulation package Virtual Engines. That method has been slightly re-worked, with the original first order (linear) interpolation of pressure waves extended to second order. Also, the way heat transfer is incorporated is modified and includes a method for achieving full mass conservation.