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Numerical studies of laminar flow over two tandem elliptical cylinders using Ramanujan approximation

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Abstract

This paper studies a laminar flow over tandem elliptical cylinders at Re = 200. While the aspect ratio (AR) of the upstream cylinder varies from AR = \(a_{1}\)/\(b_{1}\) = 0.25 to 2.00, the aspect ratio of the downstream cylinder is kept constant at AR = 1.00 (e.g., \(a_{2}\)/\(b_{2}\) = 1). This range of AR covers the most important practical cross sections of elliptical cylinders, including normal elliptic cylinder (with 90° incidence) and parallel elliptic cylinders (with 0° incidence). Although the spacing ratio between the centers of the cylinders is kept constant at \(L^{*}\) = 4\(D_{2}\), the gap ratio (G* = G/\(D_{2}\)) between the surfaces of the cylinders is varied due to the alteration of AR of the upstream cylinder. Unlike the previously published studies, which estimated the hydraulic diameter of the elliptical cylinder, in this paper, the precise hydraulic diameter is evaluated and used to analyze the wake instabilities and the variation of the imposed pressure as well as forces coefficients on the cylinders. The results reveal that with the estimation of the hydraulic diameter of the elliptic cylinder, the maximum error of 178% has arisen, which significantly affects forces (lift and drag) coefficients. It was found that the phase lag between the sinusoidal lift coefficients of the cylinders varies and it reaches a minimum at AR = 1.5 and it slightly increases once AR = 2.0. Besides, the amplitude of the fluctuating drag coefficient is larger for the upstream cylinder as compared with that of the downstream cylinder. Besides, it is found that the minimum St = 0.065 occurs at AR = 0.25, and it is regularly increased to the maximum value of St = 0.211 at AR = 1.75. A parabolic equation is deduced with high accuracy and a reasonable error of less than 1.8% to show the relationship between St and AR.

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Abbreviations

\(a_{{\text{n}}}\) :

Horizontal radius of the ellipse

A :

Area of the cross section

\(b_{{\text{n}}}\) :

Vertical radius of the ellipse

\(C_{{\text{D}}}\) :

Drag coefficient

\(C_{{{\text{lf}}}}\) :

Fluctuating lift coefficient

\(C_{{\text{P}}}\) :

Pressure coefficient

\(d_{{\text{h}}}\) :

Hydraulic dimension

D :

Diameter of the cylinder

e :

Error

f :

Shedding frequency

G :

Gap between tandem cylinders

H :

Vertical length of CFD domain

L :

Length between the center of the cylinders

n :

Stands for integer value (1 and 2)

P :

Dynamic pressure

\(P_{{\text{w}}}\) :

Wetted perimeter of the upstream cylinder

Re:

Reynolds number

St:

Strouhal number

t :

Time

\(\Delta t\) :

Time step

U :

Freestream velocity

v :

Local cross-stream velocity

\(\mu\) :

Dynamic viscosity of the fluid

\(\rho\) :

Density of the fluid

AR:

Aspect ratio

CFD:

Computational fluid dynamics

CFL:

Courant–Friedrichs–Lewy number

FVM:

Finite volume method

FFT:

Fast Fourier transform

MR:

Mesh resolution

NS:

Navier Stokes

NEC:

Normal elliptic cylinder

TS:

Transverse spacing

PEC:

Parallel elliptic cylinder

VKS:

Von Karman Street

VIV:

Vortex-induced vibration

Superscript (*):

Stands for dimensionless parameters

Superscript (–):

Stands for mean parameters

Subscript (min):

Stands for minimum

(x, y):

Cartesian coordinate system

(i, j):

Directions in “x” and “y

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Authors and Affiliations

  1. College of Engineering and Technology, American University of the Middle East, Kuwait City, Kuwait

    Javad Farrokhi Derakhshandeh & Nima Gharib

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  1. Javad Farrokhi Derakhshandeh
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  2. Nima Gharib
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Correspondence to Javad Farrokhi Derakhshandeh.

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Derakhshandeh, J.F., Gharib, N. Numerical studies of laminar flow over two tandem elliptical cylinders using Ramanujan approximation. J Braz. Soc. Mech. Sci. Eng. 43, 169 (2021). https://doi.org/10.1007/s40430-021-02890-0

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