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AHPCRC Bulletin: Spring 1994 - Volume 4 Number 2

Application of Finite Element Strategies to Nearly Incompressible Liquid Flow

Gloria Wren, U.S. Army Research Laboratory

A problem of long-term significance to the U.S. Army is the understanding of flow properties of nearly incompressible fluids over moving, complicated geometries at pressures up to 700 MPa. Such problems pose a unique challenge, both for the diagnostic and modeling communities, due to the difficulty in obtaining direct diagnostic data, e.g. flow velocity at these extreme conditions and the numerical challenges imposed by realistic conditions. Fortunately, recent advances in finite element (FE) strategies to solve compressible flow problems have allowed the solution of equations describing challenging applied problems. In particular, a new class of numerical methods has been developed at the University of Minnesota (UM) under the sponsorship of the Army High Performance Computing Research Center (AHPCRC) and the Army Research Office (ARO), to simulate nearly incompressible, barotropic flows, i.e., flows in which small changes in density lead to large changes in pressure. In some cases, limited experimental data for related parameters are available which can be utilized to validate the model. These methods are being applied in a joint effort between the Weapons Technology Directorate, U.S. Army Research Laboratory (WTD/ARL), and the AHPCRC.

For example, in the 155 mm, regenerative liquid propellant gun under development by the Project Manager - Advanced Field Artillery System/Future Armored Resupply Vehicle, Picatinny Arsenal, New Jersey, a liquid monopropellant is injected at high pressures from a reservoir through the annulus formed by two moving, cylindrical pistons into the combustion chamber of the gun where it burns, creating combustion gases to accelerate the projectile. Current Army injector designs have evolved empirically based on experimentally measured piston motion and mass flow rate requirements. However, modifications to the injector have been observed to yield unexpected results in some tests, indicating a lack of fundamental understanding of fluid flow in the injector and conditions leading to flow separation and cavitation. In extreme cases, the pistons have been observed to reverse their motion, leading to uncontrolled bulk-burning of the propellant. The modeling effort currently underway seeks to develop a fundamental understanding of the interaction between the piston motion and the liquid propellant flow. Experimentally measured pressures and piston motions can be utilized to validate the model, while the axisymmetric solution provides data on flow field such as velocity and pressure field which cannot be experimentally obtained.

Numerical simulation of this problem has been complicated in the past due to a number of serious challenges. The mathematical modeling of the problem is based on the Navier-Stokes equations of compressible flows for the fluid, the equations governing the dynamics of moving components, and the interface conditions governing the interaction between the fluid and the moving components. This results in a time- dependent, nonlinear partial differential equation system that needs to be solved over an intricate computational domain that changes its shape with time. The change in the shape of the computational domain is one of the unknowns of the problem, and needs to be determined as part of the overall solution. In addition, the computational domain in this problem contracts dramatically during the injection process. Furthermore, the fluid dynamics of the problem involve high Reynolds numbers and very low Mach numbers (at the nearly incompressible limit, but with large pressure variations), and each of these two extreme conditions poses a different kind of numerical stability challenge.

The geometric complexities involved in the problem require a formulation that can handle moving boundaries and interfaces. The Deformable-Spatial-Domain/Stabilized-Space-Time (DSD/SST) formulation introduced in 1990 by the AHPCRC team, headed by T. Tezduyar, has this capability. This is a new, accurate, general-purpose stabilized finite element formulation for the computation of unsteady flows involving free surfaces, two-liquid interfaces, moving mechanical components, and fluid- structure and fluid-particle interactions. In the DSD/SST method, the stabilized finite element formulation of the governing equations is written over the space-time domain of the problem and therefore, the deformation of the spatial domain with respect to time is automatically taken into account. With the advanced stabilization techniques used in the DSD/SST formulation, the numerical stability challenges involved in the problem are overcome with minimal numerical dissipation and therefore, with minimal loss of accuracy.

During the computation, the outer and inner pistons move rearward, with the inner piston motion provided from experimental data. The downstream pressure on the right-hand boundary, representing a location at which the fluid expansion and gas properties will induce atomization, is also provided from experimental data. The mesh updating is performed only when it becomes necessary to do so to prevent unacceptable degrees of mesh distortion. The computation then provides the flow field and the motion of the outer piston. Since both pistons move independently, coupled via the fluid pressure history, the computational domain may change in shape as well as size. A comparison between the measured and simulated outer piston motion and liquid pressure at the rear wall, although not shown, serves to validate the model.

Figure 1. Sample output is shown as Mach number at three different instants: 7.2, 8.9, and 10.6 ms Figure 2. The computational domain and finite element mesh at 7.2, 8.9, and 10.6 ms

A detailed examination of the Mach number contours in the sample calculations reveals that the flow shows no tendency to separate from the injector on the liquid reservoir side (i.e., on the left-hand side of the injection orifice) at any time step and thus, from an engineering point of view, the injector appears to be well designed.

The flow on the combustion chamber side of the injector (i.e., on the right-hand side of the injection orifice) displays more structure. At early times, the flow follows the contour of the outer piston into the combustion chamber. At later times, with higher pressures and velocities, a recirculation zone is established in the combustion chamber, and the fluid appears to have a larger axial component. Interestingly, the fluid appears to lift from the inner piston just downstream of the injection orifice. Thus, the inner piston nose may be exposed to hot gases during the combustion process, with implications for erosion processes. However, we acknowledge that the computation is completed without jet breakup or combustion, and that the observations about flow in the combustion chamber may be modified in the actual gun environment.

The methods discussed above in two- and three-dimensional form are applicable to a wide range of fluid flow problems. They have been applied to many problems, including flow around a Los Angeles-class submarine, supersonic flow past a delta-wing, and flow created by the passing of high speed trains, in addition to the liquid propellant gun. Current efforts involve computation on the AHPCRC's CM-5, which has offered a new level of computational capability in solving this class of problems. Thus, it appears that application of the methods discussed can aid our understanding of the fundamental behavior of nearly incompressible flows in reasonable computational time. This work has the capability to impact the design of future U.S. Army hardware.