Analysis of Real Fluid Flows in Converging Diverging Nozzles

A computational method for the simulation of general fluids with arbitrary equation of state and multiple component and multiple phases is outlined along with corresponding boundary condition methods. The capabilities of the approach are demonstrated by nozzle flow calculations for four fluids with very different properties: air at normal pressures and temperatures where perfect gas assumptions apply, air at ultra-high pressures where real-gas effects dominate the solutions, argon expanded to low pressures where condensation and two-phase flow take place, and a purely incompressible fluid to demonstrate the method in this ‘singular’ limit. The results show a variety of interesting physical phenomena that are introduced by the unique fluid physics. INTRODUCTION In the present paper we use computational methods to compare the flowfield characteristics of four fluids whose physical properties are very different. Our goals are twofold. First, we wish to highlight the interesting effects that fluid properties can have on the ensuing flowfields, and second, we wish to demonstrate that computational algorithms can be unified such that they remain accurate and efficient for essentially all classes of fluids over wide ranges of flow conditions. As specific examples of this capability, we present results of calculations for three compressible fluids with distinct equations of state as well as for an incompressible fluid. Research Associate Professor; Department of Mechanical Engineering, Computational Fluids Dynamics Research roup; Member AIAA. G PhD Student; Department of Mechanical Engineering, tudent Member AIAA. S Professor, Department of Mechanical Engineering; H. H. Arnold Chair of Excellence in Computational Mechanics; Senior Member AIAA. As a representative geometry, we consider the internal flow through converging-diverging nozzles. This geometry allows us to investigate flow conditions from subsonic through hypersonic speeds in a single example for the compressible fluids while also providing an effective geometry for which the characteristics of incompressible fluids can be contrasted. Because of the very different properties of the chosen fluids, as well as particular applications of interest to us, we use different nozzle shapes for each fluid to ensure that the flow conditions remain interesting. For the compressible fluids, we have selected nozzle shapes that are characterized by strong convergence in the subsonic portion leading to very low inlet speeds followed by expansion sufficient to produce hypersonic speeds in the divergent section. Typical Mach number ranges go from M = 0.01 to 8. The incompressible calculations are based on the nozzle shape from one of the compressible flow cases and serve to demonstrate the computation of incompressible flow as the limit of a compressible flow system. The geometric scales and thermodynamic conditions are chosen to give Reynolds numbers ranging from less than 1000 to greater than ten million based on diameter. This combination of factors (Mach number range, Reynolds number range, diverse equation of state) provides a severe test of any CFD algorithm. The equations of state for the three compressible fluids are chosen as follows. As a first case, we consider the familiar perfect gas relations and express the equation of state in algebraic fashion. In this simplest example, we introduce isolated streams of air and carbon dioxide into the upstream end of the flowfield and observe the effect of two distinct ratios of specific heats on the downstream flow. For the second case, we choose the ultra-high pressure limit of recent equation of state information for air [5] where the density of air exceeds that of water and realgas effects dominate. Here we use a table look-up procedure for the equation of state and choose a 1 American Institute of Aeronautics and Astronautics 33rd AIAA Fluid Dynamics Conference and Exhibit 23-26 June 2003, Orlando, Florida AIAA 2003-4132 Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. geometry and conditions similar to those recently investigated by Brown and co-workers for a proposed hypersonic test facility [10,11]. For the third compressible fluid, we expand argon through a nozzle to temperatures substantially below the critical point where equilibrium thermodynamics dictates that two-phase conditions will appear. Argon represents an interesting contrast with air in that its equation of state remains identically that of a perfect gas until it crosses the condensation or sublimation line and the only ‘real’-gas effects are those do to condensation. We represent the two-phase region by an approximate non-equilibrium process that allows us to retain pressure-temperature independent variables even in the dome region. Here we again use table look-up procedures obtained from the REFPROP [6] database. Finally, as the fourth case, we consider an incompressible fluid to demonstrate capability for using similar algorithms and solution methods for compressible and incompressible fluids. Our goal here is to emphasize that incompressible fluids can be distinguished from compressible fluids by only changing the equation of state information, not the describing partial differential equations or the solution algorithm. The trivial equation of state in this case is specified ‘algebraically’, and is based on properties corresponding to those of room temperature water. FORMULATION OF THE EQUATIONS It is straightforward to write the conservation equations in a form that applies to all types of fluids. The primary issues are to retain density inside the differentials even for the incompressible fluid case and to retain either enthalpy or internal energy (as opposed to the specific heats or the specific heat ratio) in the energy equation. Here we choose enthalpy and write the equations of motion in the vector form, ( ) S F F v = − ⋅ ∇ v v (1) The inviscid flux vectors in Eq. 1 are given in their standard conservative form,