The initial condition is important since the method used to solve the Navier-Stokes Equations (NSE) is iterative. Hence, a good initial condition is important, because as close it is from the final solution, faster will be the solution computation. In addition, the computation effort will be lower. The choice of the initial condition is critical, it usually requires some experience from the analyst.

Strategies for the initial condition

The common strategies to define an initial flow field are the uniform and the Laplacian flow fields.

Uniform flow field

The common procedure is to set a fixed initial condition everywhere. For instance, to impose free stream velocity condition, turbulence, quick de-pressure, starting from the outlet to everywhere. This is not the ideal condition, because it is far from the solution in terms of boundary layer evolution of the wake. This solution is called, the uniform flow field (Figure 1). The initial condition of U_∞ is far from the final one, because this velocity is fixed everywhere in the free stream and wake are not generated, so as the stagnation points and boundary layers. The uniform flow field results in a very small boundary layer, it is the worst estimation.

Laplacian flow field

The Laplacian flow field is a better solution than the uniform one, it assumes an incompressible and non viscous flow. The Laplacian equation is set equal to zero and then the velocity is set equal to the gradient of the potential flow, as it is described below.

∇²φ = 0 → V = ∇φ

∇²P = 0

This establishes a stagnation point and exhibits some evolutions on the velocity. However, this still is pretty far from the solution, because the non-viscosity assumption is too strong for a flow that will generate a boundary layer and this will not be present if this assumption is adopted. This assumption is slightly better, because there is some stagnation, different velocities and some accelerations. However, the boundary layer still is not correct. Its solving procedure, boundary layer, wake and stagnation point are very different from the assumptions. Therefore, this is a better choice. An interesting point is that the Laplacian uses a potential flow to represent the velocity. The potential flow solution is converted to the viscous one by the iterative solver using the same volume mesh that it will be used as the final mesh.

From the continuum to the discrete environment

Therefore, the CFD process is a set of processes for a given geometry mesh, then this is used to simulate the incompressible Navier-Stokes Equations (NSE) at the steady-state. It is assigned the boundary conditions and it is defined as the initial conditions. From this point it is started the real solver, that will get the process from the initial to the final conditions. Hence, it is necessary to start from the continuous equation and discretize it to a finite volume equation. Even though in the final form of NSE there are still some parts which are continuous, these are the operators that multiply the variables which are being solved for. For instance, the velocity, the pressure and the turbulence, since it is the turbulence model which is being used. Hence, it is necessary to discretize both the partial time and the partial operators if it is considered a steady-state simulator. Usually the solver allows to choose different options in terms of the compromise between stability and accuracy. The reason is to get a better solution, closer to the reality and accurate simulations are advisable. The problem is that these are expensive and unstable. In other words, these tend to crash during the solver process, because the initial conditions are not so close to the final condition and the high orders used to make the solver diverge. The solution for this problem is to use low order schemes at the first runs, because their outputs are still far from the last convergence. It is less accurate, but more stable, thus useful for the first stages of the solving. Hence, after a certain number of interactions it is possible to change for a higher order scheme, because the solution is better converged and near to the final solution.

References

• This is article is based on the lecture notes taken by the author during the Industrial Aerodynamics lectures hold by Muner at Dallara Academy.