The grid generation is based on the equidistribution principle , which states that at every timestep in the simulations the mesh points are arranged such that this product remains constant. Here, is the cell size and W is a cell-averaged weight function. This principle leads to an explicit grid generation technique that has some advantages over an implicit one. It is easy to apply and easy to control such a mesh distribution. Mesh smoothness is simple and robust. In the present work, the smoothing is done on the weight function W, which indeed results in a smooth grid distribution.
Governing partial differential equations for fluid systems are generally described using a laboratory frame of reference. They can be transformed to a more general reference frame that reduces to the eulerian and lagrangian frames when the reference velocity is zero or equal to the local fluid velocity. In order to transfer the governing equations from to an adaptive system (,), we use a variable transformation as and
We choose and . That leaves us with and or in another representation; and Here, and are the mesh metric (jacobian) and speed that will appear with the unknowns in the transformed equations. In order to solve the set of transformed equations, which are given in section 4, we must estimate these mesh quantities in advance.
The information required about these quantities could be sought for the time as well as for the time . However, before any attempt to solve physical equations at , we let the mesh points move from to depending on the values and gradients of some chosen physical variables at . This procedure of determining the mesh distribution at in terms of information given at the is an explicit procedure. The equidistribution principle and the variable transformation described before leads to the following expression that can be used in an explicit manner for mesh generation
Here is simply a function whose value is chosen to be equal to the mesh number. The second term on the right hand side represents the average for each mesh cell. We call this procedure ``explicit'' because all the quantities on the right hand side are given at and the weight function W on the left hand side is also defined at with the exception of the integral's upper limit x being given at . After is calculated, the mesh edge velocity could be found by differencing the old and new values of x with respect to time. The formula for this, , will be given later in another section. The mesh metric is simply the difference in the space locations, equal to if is taken to be 1.
Therefore, knowing and beforehand, the physical equations can be evaluated without any difficulty. However, one has to determine what the weight function should be. Many forms of weight function can be postulated. One that proves to be both simple and robust is the following
where A and B are some normalized physical quantity such as velocity, pressure, mass, density, momentum density or temperature. Also , and are the first and second derivatives of A and B with respect to the spatial coordinate x. Dwyer [1,2] has developed a strategy to determine and through a specified fraction of points to be assigned to each function variation. That is, if is defined as the fraction of grid points to be assigned to the first derivative variation, , then
and also for the second degree of variation
If and are held constant for the problem, then and will be determined at each time step while the solution develops. Determining 's and 's from these equations follows an explicit procedure that uses the old values at time on the right-hand-side to come up with new values at time on the left-hand-side. In fact, one could time-average the old and new values as we have done. Also, one does need an initial guess, such as the following, to start the procedure: and thus
Thus, R is the fraction of mesh points that are reserved for chosen gradients of A or B. Notice that Eq.(4) enables one to construct W out of two variables, A and B, which means one can adapt using more than one function. This obviously enhances the power of solving multigradient problems accurately.
Finally, it is important to recognize that for curvilinear geometries the identity
must be preserved in the difference equations in order for them to preserve their conservative forms. That is, when differenced equations are solved on the discretized coordinates, care should be taken to provide this identity relation which also introduces a formula to calculate the grid speed while the grid points move from to .