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Examples of Decompositions for Time and Space Domains and Discretization of Equations for General Purpose Computational Fluid Dynamics Programs and Historical Perspective of Some Key Developments

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Abstract

This paper presents two examples from mid-eighties related to decomposition for time and space domains and discretization of equations for the general purpose Computational Fluid Dynamics (CFD) programs. The first example is related to the implementation of rectangular coordinates to simulate flow and heat transfer in the arbitrarily shaped domains with various heat transfer boundary conditions. The second example demonstrates capabilities to introduce and test implicit and explicit higher order numerical schemes. In both cases the implementation of linearized source terms for various equations is used to allow regrouping and adding new terms in equations without the need for major changes to the general purpose CFD programs. Presented examples provide a historical perspective of some key developments based on the well-planned code architecture. These developments are contrasted with the other selected historical developments and current practices.

Presented work had been performed at CFD Unit, ME Building, Imperial College, Exhibition Road, London, U.K.

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Abbreviations

a :

Coefficient in the FDE

\(a^{C}\) :

Convection coefficient for FDE

\(a^{d}\) :

Diffusion coefficient for FDE

\(a^{{\prime }}\) :

Coefficients for \(\Phi ^{{\prime }}\) in FDE

\(a^{*}\) :

Coefficients for \(\Phi ^{*}\) in FDE

A :

Surface area of the relevant cell area

c :

Concentration

\(\left\langle c \right\rangle\) :

Concentration in the polluted cell at the beginning

C :

Courant number

\(C^{{\prime }} = c/\left\langle c \right\rangle\) :

Dimensionless concentration

d :

Distance between the two neighbor grid nodes

D :

Tube diameter or enclosure width

D :

Diffusion coefficient

f :

Friction factor

g :

Acceleration of gravity

\(Gr = g\beta \left( {\overline{T}_{w} - T_{b} } \right)D^{3} /\nu^{2}\) :

Grashof number

\(Gr^{ + } = \frac{{g\beta Q^{{\prime }} D^{3} }}{{\left( {k\nu^{2} 8} \right)}} = \frac{{Gr\overline{Nu} \pi }}{16}\) :

Modified Grashof number

\(h = q/\left( {T_{w} - T_{b} } \right)\) :

Local heat transfer coefficient

\(\bar{h} = q/\left( {\overline{T}_{w} - T_{b} } \right)\) :

Average heat transfer coefficient

k :

Thermal conductivity

L :

Dimension of the cell, see Fig. 9

\(Nu = hD/k\) :

Local Nusselt number

\(\overline{Nu} = \bar{h}D/k\) :

Average Nusselt number

\(Nu_{0}\) :

Forced convection value of \(\overline{Nu}\)

p :

Pressure

\(p^{*}\) :

Reduced pressure

P :

Dimensionless pressure

\(Pe = \rho u\left( {\delta x} \right)/\Gamma\) :

Local (grid) Peclet number

\(Pe = Re\,Pr = u\,D/\Gamma\) :

Enclosure Peclet number

\(Pr = \nu /\Gamma\) :

Prandtl number

\(q = \bar{h}\left( {\overline{T}_{w} - T_{b} } \right) = 2Q^{{\prime }} /\left( {\pi D} \right)\) :

Rate of heat transfer per unit area

\(Q^{{\prime }} = \bar{h}\left( {\overline{T}_{w} - T_{b} } \right)\pi D/2\) :

Rate of heat transfer per unit length

r :

Radial coordinates

R :

Tube radius

\(Re = \bar{w}D/\nu\) :

Reynolds number

\(S_{T}\) :

Source term for the cells near the heated boundary

t :

Time

\(T = t/\Delta t\) :

Dimensionless time

T :

Temperature

\(T_{b}\) :

Bulk temperature

\(T_{w}\) :

Local wall temperature

u, v, w :

Velocity components in x, y, z directions

U, V, W :

Dimensionless velocities

\(\bar{w}\) :

Mean axial velocity

\(\overline{W}\) :

Mean value of W

x, y, z, X, Y, Z :

Cartesian coordinates

\(Z = \frac{z}{L}\) :

Dimensionless distance

α :

Angle of gravitation action, see Fig. 1

β :

Thermal expansion coefficient

\(\Delta r\) :

Distance between the boundary and the boundary grid node, see Fig. 4

\(\Delta t\) :

Time increment

\(\Delta \theta\) :

Corresponding angle for the boundary cell, see Fig. 4

θ :

Angular coordinate

\(\nu\) :

Kinematic viscosity

ρ :

Density

\(\Gamma\) :

Diffusion coefficient

\(\Phi = (T - T_{b} )/\left( {Q^{{\prime }} /k} \right)\) :

Dimensionless temperature

\(\Phi\) :

Value at grid node

\(\Phi ^{{\prime }}\) :

Correction to \(\Phi\)

\(\Phi ^{*}\) :

Stored values of the \(\Phi\)

\(\Phi _{h}\), \(\Phi _{l}\):

Values at higher and lower cell faces, see Fig. 10

\(\Phi _{T}\) :

Value at grid point P at previous time step

\(\psi\) :

Stream function

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Acknowledgements

The author would like to thank Professor Brian D. Spalding for guidance throughout the specialization at Imperial College, and for the privilege to experience a unique educational approach resulting in step-by-step gradual increase of the complexity and challenge of the problems to be solved. At the same time, giving me the freedom and time to follow intuition, search literature, and explore alternative ways instead of just recommending certain courses. The author would like to thank Dr. W. M. Pun for his encouragement and guidance throughout the work on the second example which was performed in connection with U.K. Department of Environment contract (PECD 7/9/322). The author is thankful to the British Council for providing the fellowship for study at the Imperial College in London, England, U.K., and to the Community of Science and Education of the Socialistic Republic of Serbia for the financial support of my family in Belgrade, Serbia, Yugoslavia, during the 1985/86 school year.

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  1. Westinghouse Electric Company, Cranberry, USA

    Milorad B. Dzodzo

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    Akshai Runchal

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Dzodzo, M.B. (2020). Examples of Decompositions for Time and Space Domains and Discretization of Equations for General Purpose Computational Fluid Dynamics Programs and Historical Perspective of Some Key Developments. In: Runchal, A. (eds) 50 Years of CFD in Engineering Sciences. Springer, Singapore. https://doi.org/10.1007/978-981-15-2670-1_4

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