Hierarchical cascade model leading to 7-th order initial value problem. (English) Zbl 1308.76195
Summary: In turbulent flows, local velocity differences often obey a cascade-like hierarchical dynamics, in the sense that local velocity differences at a given scale \(k\) are driven by deterministic and random forces from the next-higher scale \(k-1\). Here we consider such a hierarchically coupled model with periodic boundary conditions, and show that it leads to an \(N\)-th order initial value problem, where \(N\) is the number of cascade steps. We deal in detail with the case \(N = 7\) and introduce a non-polynomial spline method that solves the problem for arbitrary driving forces. Several examples of driving forces are considered, and estimates of the numerical precision of our method are given. We show how to optimize the numerical method to obtain a truncation error of order \(O(h^5)\) rather than \(O(h^2)\), where \(h\) is the discretization step.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76F02 | Fundamentals of turbulence |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 65D07 | Numerical computation using splines |
Keywords:
driving force; initial value problem; cascade model; seventh order differential equation; turbulent flowReferences:
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