**Unsteady Euler Solutions of Moving Boundary Problems Using Unstructured Deforming Mesh and ALE Formulation**

**Summary**

A parallel algorithm for the solution of unsteady Euler equations on unstructured and moving meshes is developed. A cell-centered finite-volume scheme is used. The temporal discretization involves an implicit time-integration scheme based on backward-Euler time differencing. The movement of the computational mesh is accomplished by means of a dynamically deforming mesh algorithm. The parallelization is based on decomposition of the domain into a series of sub domains with overlapped interfaces. The scheme is computationally efficient, time accurate, and stable for large time increments.

**Dynamically Deforming Mesh Algorithm**

This algorithm moves the computational mesh to conform to the instantaneous position of the moving body at each time step.

The algorithm treats the computational mesh as a system of interconnected springs. This system is constructed by representing each edge of each triangle by a tension spring. The spring stiffness for a given edge i-j is taken to be inversely proportional to the length of the edge and it is defined as follows:

_{}

Grid points on the outer boundary of the mesh are held fixed and the instantaneous location of the points on the inner boundary (i.e., moving body) is given by the body motion. At each time step, the static equilibrium equations in the x, y, and z directions that result from a summation of forces are solved iteratively at each interior node i of the mesh for the displacements.

**The Ale Formulation Of The Three-Dimensional Time-Dependent Inviscid Fluid-Flow Equations**

_{}

Here *W* represents the physical domain with a boundary * The vectors **Q* and *F* are given by

_{}

Here *n*_{x }*, n*_{y }*, n*_{z }*, *are the Cartesian components of the exterior surface unit outward normal vector *n *on the boundary_{ }** V = ***u ***i + ***v ***j + ***w ***k**** **is the fluid velocity,_{ }**W = ***x*_{t }**i + ***y*_{t }**j + ***z*_{t }* k *is the mesh velocity and

*W =*

**W . n***=*

*x*

_{t }

*n*

_{x }

*+ y*

_{t }

*n*

_{y }

*+ z*

_{t}

*n*

_{z }is the face speed in the normal direction. Pressure can be expressed as

_{}** Solution Method**

The spatial integration employed in the flow solver is the cell-centered finite volume formulation. The volume-averaged values are adopted to represent the flow variables. For time integration Implicit Backward-Euler Time-Differencing is used.

**Geometric Conservation **

Due to mesh movements, a geometric conservation law has to be solved in addition to the mass, momentum, and energy conservation laws. This law is expressed in integral form as :

_{}

This geometric conservation law provides a self-consistent solution for the local cell volumes and is solved using the same scheme used for all other flow conservation equations.

**Boundary Conditions**

Characteristic boundary conditions are applied on the far field using Riemann invariants. For moving boundaries, the wall boundary conditions are modified by taking the mesh movements into account. Thus, the flow tangency condition is imposed by calculating the flow velocity on solid walls from:

_{}

The wall pressure is calculated from the normal momentum equation as:

_{}

where, **a**_{w} is the acceleration of the body, subscripts *c* and *w* denote cell-centered and wall values, respectively. The above boundary conditions reduce to steady flow conditions by setting the mesh velocity * W *to zero

*Figure 1. Pitching motion of missile and deforming grid around it*

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*Figure 2. Rolling motion of a missile and deforming grid around it *

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*Figure 3. Mach number variation around a missile making pitching motion *