Parameter‐uniform numerical methods for a laminar jet problem

Abstract

SUMMARYWe consider the classical problem of a two-dimensional laminar jet of incompressible uid owing into a stationary medium of the same uid. The equations of motion are the same as the boundary layer equations for ow past an inÿnite at plate, but with di erent boundary conditions. Numerical experiments show that, using appropriate piecewise-uniform meshes, numerical solutions together with their scaled discrete derivatives are obtained which are parameter (i.e., viscosity ) robust with respect to both the number of mesh nodes and the number of iterations required for convergence. While the method employed is non-conservative, we show with the aid of numerical experiments that the loss in conservation of momentum is minimal.

A robust numerical method for flow through a pipe driven by an oscillating pressure gradient

“…in Reference [2], and more recently for nonlinear problems in Reference [5], where it is also shown that this difficulty is not resolved unless the meshes are appropriately fitted to the boundary layer. The numerical method constructed here comprises appropriately fitted piecewise-uniform meshes [4,5,11,12] in conjunction with an upwind finite difference operator [4,12]. This is a parameter-robust numerical method for problems (1)- (6), as is shown experimentally in the following sections by extensive numerical computations.…”

“…Linear problems of this type have been solved with these techniques yielding numerical solutions that exhibit convergence uniformly with respect to the singular perturbation parameter, while the computational work required to obtain these solutions is also independent of this parameter [2][3][4]; we refer to numerical methods with this property as parameter-robust methods. These fitted mesh techniques have also been applied successfully to the computation of parameter-robust solutions of non-linear problems [4][5][6][7]. Ultimately we are interested in the development of parameter-robust numerical methods for solving the Navier-Stokes equations.…”

“…Note that the boundary condition (5) ensures that the solution is periodic in time, while the boundary condition (6) reflects the symmetry of the problem.…”

On the singularly perturbed character of the turbulent free jet and its robust numerical solution

“…Our objective was to solve numerically problems , in the quarter plane in a similar way as to the laminar jet problem . One of the key questions to address is what parameter now affects the errors in the problem, as the viscosity, used previously for the laminar jet , is no longer a parameter as it has now become variable with x . Now, this parameter role for the turbulent jet is taken up by α ∈ (0,1].…”

“…Now, this parameter role for the turbulent jet is taken up by α ∈ (0,1]. Using piecewise‐uniform meshes , in conjunction with an upwind finite difference method, we will show by experimental results that the numerical solutions are parameter robust , that is, numerical solutions where the maximum pointwise error tends to zero independently of the perturbation parameter α , whereas the work required to obtain the solutions is also independent of α . As the analytical solution of this particular problem is available, we will use it to compute the discretisation errors, with respect to the discrete L ∞ ‐norm .…”