Hyperbolic tangent function to create stretched grid

Hyperbolic tangent function \({\rm tanh}\) is often used to generate the stretched structured grid.

In this blog post, I will introduce some examples I have found in the references.

Example #1 [1]

y_j = \frac{1}{\alpha}{\rm tanh} \left[\xi_j {\rm tanh}^{-1}\left(\alpha\right)\right] + 1\;\;\;\left( j = 0, …, N_2 \right), \tag{1}
\xi_j = -1 + 2\frac{j}{N_2}, \tag{2}
where \(\alpha\) is an adjustable parameter of the transformation \((0<\alpha<1)\) and \(N_2\) is the grid number of the direction. As shown in the following figure, the grids are more clustered towards the both ends as the parameter \(\alpha\) approaches 1.

Example #2 [2]

y_j = 1 -\frac{{\rm tanh}\left[ \gamma \left( 1 – \frac{2j}{N_2} \right) \right]}{{\rm tanh} \left( \gamma \right)}\;\;\;\left( j = 0, …, N_2 \right), \tag{3}
where \(\gamma\) is the stretching parameter and \(N_2\) is the number of grid points of the direction.

Grid Images

Coming soon.


[1] H. Abe, H. Kawamura and Y. Matsuo, Direct Numerical Simulation of a Fully Developed Turbulent Channel Flow With Respect to the Reynolds Number Dependence. J. Fluids Eng 123(2), 382-393, 2001.
[2] J. Gullbrand, Grid-independent large-eddy simulation in turbulent channel flow using three-dimensional explicit filtering. Center for Turbulence Research Annual Research Briefs, 2003.

Japanese Technical Report


CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences

I found very interesting technical report titled “CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences” by NASA.


This report documents the results of a study to address the long range, strategic planning required by NASA’s Revolutionary Computational Aerosciences (RCA) program in the area of computational fluid dynamics (CFD), including future software and hardware requirements for High Performance Computing (HPC). Specifically, the “Vision 2030” CFD study is to provide a knowledge-based forecast of the future computational capabilities required for turbulent, transitional, and reacting flow simulations across a broad Mach number regime, and to lay the foundation for the development of a future framework and/or environment where physics-based, accurate predictions of complex turbulent flows, including flow separation, can be accomplished routinely and efficiently in cooperation with other physics-based simulations to enable multi-physics analysis and design. Specific technical requirements from the aerospace industrial and scientific communities were obtained to determine critical capability gaps, anticipated technical challenges, and impediments to achieving the target CFD capability in 2030. A preliminary development plan and roadmap were created to help focus investments in technology development to help achieve the CFD vision in 2030.

I want to read it thoroughly 🙂

Impinging Jet part1

Jet flows can be classified in terms of

  • the flow conditions (laminar and turbulent)
  • the existence of objects: free jet and impinging jet
  • the differences of physical properties between a projected fluid and an ambient fluid: submerged jet and unsubmerged jet
  • the geometry of a nozzle: round jet and slot jet

and so on.

Free Jet

The following video visualizes the flow pattern of a submerged free jet (created by Bjarke Ove Andersen and Mathies Hjorth Jensen of Technical University of Denmark):

Flow Regions of Impinging Jet [1, 2]


  • Region Ⅰ is the region of flow establishment. It extends from the nozzle exit to the apex of the potential core. The so-called potential core is the central portion of the flow in which the velocity remains constant and equal to the velocity at the nozzle exit.
  • Region Ⅱ is a region of established flow in the direction of the jet beyond the apex of the potential core; it is characterized by a dissipation of the centerline jet velocity and by a spreading of the jet in the transverse direction.
  • Region Ⅲ is that region in which the jet is deflected from the axial direction.
  • Region Ⅳ is known as the wall jet region, where the directed flow increases in thickness as the boundary layer builds up along the solid surface.

– Gauntner et al. [1]

  2. T. Dairay, DNS of a turbulent jet impinging on a heated wall (accessed 2016-09-04)
  3. N. ZUCKERMAN and N. LIOR, Jet Impingement Heat Transfer: Physics, Correlations, and Numerical Modeling (accessed 2016-09-04)
  4. Y. M. Chung and K. H. Luo, Unsteady Heat Transfer Analysis of an Impinging Jet (accessed 2016-09-04)