The purpose of this paper is to solve unsteady compressible Navier–Stokes equations without the commonly used dual-time loop. The authors would like to use an adaptive time-stepping (ATS)-based local error control instead of CFL-based time-stepping technique. Also, an all-speed flow algorithm is implemented with simple low dissipation AUSM convective scheme, which can be computed without preconditioning which in general destroys the time accuracy.
In transient flow computations, the time-step is generally determined from the CFL condition. In this paper, the authors demonstrate the usefulness of ATS based on local time-stepping previously used extensively in ordinary differential equations (ODE) integration. This method is implemented in an implicit framework to ensure the numerical domain of dependence always contains the physical domain of dependence.
In this paper, the authors limit their focus to capture the unsteady physics for three cases: Sod’s shock-tube problem, Stokes’ second problem and a circular cylinder. The use of ATS with local truncation error control enables the solver to use the maximum allowable time-step, for the prescribed tolerance of error. The algorithm is also capable of converging very rapidly to the steady state (if there is any) after the initial transient phase. The authors present here only the first-order time-stepping scheme. An algorithmic comparison is made between the proposed adaptive time-stepping method and the commonly used dual time-stepping approach that indicates the former will be more efficient.
The original method of ATS based on local error control is used extensively in ODE integration, whereas, this method is not so popular in the computational fluid dynamics (CFD) community. In this paper, the authors investigate its use in the unsteady CFD computations. The authors hope that it would provide CFD researchers with an algorithm based on an adaptive time-stepping approach for unsteady calculations.
The authors NK and AA would like to acknowledge the fellowship assistance provided by the Department of Higher Education, Ministry of Human Resource Development, Government of India. The authors would like to acknowledge Dr Amaresh Dalal, Department of Mechanical Engineering, IIT Guwahati, for providing the unstructured solver framework Anupravaha-2 over which the current compressible flow solver was developed.
Author contributions: The author NK has come with the idea of the work and did the initial work. The author AA along with NK then implemented it on the core solver and tested the algorithm further. The author VE supervised the entire work.
Financial disclosure: None reported.
Conflict of interest: The authors declare no potential conflict of interests.
Kalkote, N., Assam, A. and Eswaran, V. (2019), "Towards developing an adaptive time stepping for compressible unsteady flows", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29 No. 2, pp. 487-503. https://doi.org/10.1108/HFF-03-2018-0095Download as .RIS
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