Abstract
CFD is concerned with solution of Navier–Stokes (NS) equations in discretised space. It is important, therefore, to ensure that the discretised equations and their solutions obey the continuum condition embedded in Stokes’s stress–strain laws for an isotropic continuum fluid. In this paper, it is shown that adherence to this condition leads to three important conceptual/algorithmic outcomes: 1. Prevention of zig-zag pressure distribution when NS equations are solved for incompressible flow of a single fluid on colocated grids. 2. Prevention of loss of volume/mass at large times when NS equations are solved for interfacial incompressible flows of multi-fluids within single-fluid formalism. 3. Evaluation of surface tension force in interfacial flows without using phenomenology embedded in the definition of the surface tension coefficient. All the above benefits are justified on the basis of a thermodynamic principle rarely invoked in discretised CFD. A few problems are solved by way of case studies.
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Notes
- 1.
This can be appreciated from the Avogadro’s number which specifies that at normal temperature and pressure, a gas will contain \(6.022 \times 10^{26}\) molecules per kmol. Thus in air, for example, there will be \(10^{16}\) molecules per \(\text {mm}^{3}\).
- 2.
In [27], symbol \(\overline{\sigma } = (\sum _{i=1}^{3}\,\sigma _{xi})/3\) is used. Here, \(\overline{p} = - \overline{\sigma }\) is preferred. Both \(\overline{p}\) and q are newly introduced to serve a pedagogic purpose.
- 3.
It is important to recognise that in discretised CFD, the incompressible condition (\(\bigtriangledown \,.\,V_{f}\) = 0) is defined in terms of CV face velocities \(u_{fi}\) as shown in Fig. 1. In fact, when this definition is explicitly implemented, there results the SIMPLE staggered grid procedure of Patankar and Spalding [19]. Further, \(u_{fi}\) must satisfy momentum equations. In a continuum, \(u_{fi}\) and \(u_{i}\) fields coincide but in a discretised space, it is important to distinguish them. This will become apparent in the next section.
- 4.
In passing we note that in all three cases, it can be verified that the quantity q is invariant under rotation of the coordinate system or interchange of axes. This property ensures isotropy [27].
- 5.
Analysis of the discretised equations presented in the next section shows that \(\lambda = 0.5\).
- 6.
- 7.
- 8.
Incidentally, in the literature, several different types of interpolations have been proposed. Some of these are given below by way of example.
-
Rhie and Chow [24] (1D Pressure gradient interpolation)
$$\begin{aligned} u_{f1,e}= & {} \overline{u}_{1,e} - \frac{\Delta V}{AP^{u}}\,\left[ \frac{\partial p}{\partial x_{1}}\,|_{e} - \overline{\frac{\partial p}{\partial x_{1}}}\,|_{e} \, \right] \nonumber \\ \text{ where } \overline{\frac{\partial p}{\partial x_{1}}}\,|_{e}= & {} \frac{1}{2}\,\left[ \frac{\partial p}{\partial x_{1}}\,|_{P} + \frac{\partial p}{\partial x_{1}}\,|_{E} \,\right] \end{aligned}$$(33) -
Peric [8] (1D Mom-Outflow interpolation)
$$\begin{aligned} u_{f1,e} = \frac{1}{2}\,\left[ \frac{\sum A_{k}\,u_{1,k}}{AP^{u_{1}}}\,|_{P} + \frac{\sum A_{k}\,u_{1,k}}{AP^{u_{1}} }\,|_{E} \,\right] - \frac{\Delta V}{AP^{u}}\,\frac{\partial p}{\partial x_{1}}\,|_{e} \end{aligned}$$(34) -
Thiart [34] (Power Law Scheme [20])
$$\begin{aligned} u_{f1,e}= & {} \theta \,u_{1,P} + ( 1 - \theta )\,u_{1,E} \text{ where } \nonumber \\ \theta ( Pc_{e} )= & {} \left[ Pc_{e} - 1 + \text{ max }(0,-Pc_{e})\right] /Pc_{e} \nonumber \\+ & {} \text{ max }\left\{ 0, ( 1 - 0.1|Pc_{e}|)^{5}\right\} /Pc_{e} \end{aligned}$$(35)where cell-face Reynolds/Peclet number \(Pc_{e} = ({\rho _{m}\,u_{f1}\Delta x_{1}}/{\mu })_{e}\).
-
- 9.
- 10.
This is unlike the staggered grid practice in which the mass error is estimated from discretised version of Eq. 1.
- 11.
Incidentally, the superficial viscosity is now evaluated as \(\mu _{m} = F\,\mu _{a} + ( 1 - F )\,\mu _{b}\).
- 12.
- 13.
Volume error is defined as
$$\begin{aligned} \text{ Error }\,(t) = \left( \sum \,F_{i,j}\,\Delta V_{i,j}\right) / \left( \sum \,F^{0}_{i,j}\,\Delta V_{i,j}\right) \end{aligned}$$(72)where \(F^{0}\) is the initial F-distribution at \(t = 0\) and \(\Delta V_{i,j}\) is the volume of the cell surrounding node (i, j).
- 14.
This ignores the fact that \(\sigma \) is essentially a property of a specified fluid pair (a, b).
References
Andrillon, Y., & Alessandrini, B. (2004). A 2D+T VOF fully coupled formulation for the calculation of breaking free-surface flow. Journal of Marine Science and Technology, 8, 159–168.
Daly, B. J. (1969). Numerical study of the effect of surface tension on interface instability. Physics of Fluids, 17(7), 1340–1354.
Date, A. W. (1998). Solution of Navier-Stokes equations on non-staggered grids at all speeds. Numerical Heat Transfer, Part B, 33, 451.
Date, A. W. (2002). SIMPLE procedure on structured and unstructured meshes with collocated variables. In Proceedings of the 12th International Heat Transfer Conference, Grenoble, France.
Date, A. W. (2004). Fluid dynamic view of pressure checker-boarding problem and smoothing pressure correction on meshes with collocated variables. International Journal of Heat and Mass Transfer, 48, 4885–4898.
Date, A. (2005). Introduction to computational fluid dynamics. New York: Cambridge University Press.
Date, A. W. (2008). Computational fluid dynamics. In Kirk-Othmer encyclopedia of chemical technology, NY, USA.
Ferziger, J. H., & Peric, M. (1999). Computational methods for fluid dynamics (2nd ed.). Springer.
Gerlach, D., Tomar, G., Biswas, G., & Durst, F. (2005). Comparison of volume-of-fluid methods for surface tension dominant two phase flows. International Journal of Heat and Mass Transfer, 49, 740–754.
Holman, J. P. (1980). Thermodynamics (3rd ed.). Tokyo: McGraw-Hill Kogakusha.
Jagad, P. I., Puranik, B. P., & Date, A. W. (2015). A novel concept of measuring mass flow rates using flow induced stresses. SADHANA, 40(Part 5), 1555–1566.
Jun, L., & Spalding, D. B. (1988). Numerical simulation of flows with moving interfaces. Physico-Chemical Hydrodynamics, 10, 625–637.
Karki, K. C., & Patankar, S. V. (1988). A pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations. In AIAA 26th Aerospace Science Meeting, Paper No AIAA-88-0058, Nevada, USA.
Lin, C. H., & Lin, C. A. (1997). Simple high-order bounded convection scheme to model discontinuities. AIAA Journal, 35, 563–565.
Martin, J. C., & Moyce, W. J. (1952). An experimental study of collapse of liquid columns on a rigid horizontal plane. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 244, 312–324.
Mason, M. L., Putnam, L. E., & Re, R. J. (1980). The effect of throat contouring on two-dimensional converging-diverging nozzles at static conditions. NASA Technical Paper, 1704.
Nandi, K., & Date, A. W. (2009). Formulation of fully implicit method for simulation of flows with interfaces using primitive variables. International Journal of Heat and Mass Transfer, 52, 3217–3224.
Nandi, K., & Date, A. W. (2009). Validation of fully implicit method for simulation of flows with interfaces using primitive variables. International Journal of Heat and Mass Transfer, 52, 3225–3234.
Patankar, S. V., & Spalding, D. B. (1971). A calculation procedure for heat mass and momentum transfer in three-dimensional parabolic flows. International Journal of Heat and Mass Transfer, 15, 1787.
Patankar, S. V. (1981). Numerical fluid flow and heat transfer. NewYork: Hemisphere Publ Co.
Perry, R. H., & Chilton, C. H. Chemical engineers handbook (5th ed.). Tokyo: McGraw-Hill Kogakusha.
Pimpalnerkar, S., Kulkarni, M., & Date, A. W. (2005). Solution of transport equations on unstructured meshes with cell-centered collocated variables. Part II: Applications. International Journal of Heat and Mass Transfer, 48, 1128–1136.
Ray, S., & Date, A. W. (2003). Friction and heat transfer characteristics of flow through square duct with twisted tape insert. International Journal of Heat and Mass Transfer, 46, 889–902.
Rhie, C. M., & Chow, W. L. (1983). A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation. AIAA Journal, 21, 1525.
Rudman, M. (1997). Volume-tracking methods for interfacial flow calculations. International Journal for Numerical Methods in Fluids, 24, 671–691.
Salih, A., & Ghosh Moulic, S. (2006). Simulation of Rayleigh-Taylor instability using level-set method. In 33rd National and 3rd International Conference on Fluid Mechanics and Fluid Power (p. 2006), Paper no 1303. India: IIT Bombay.
Sclichting. (1968). Boundary layer theory (English trans. Kestin J.). McGraw-Hill.
Shih, T. M., & Ren, A. L. (1984). Primitive variable formulations using non-staggered grid. Numerical Heat Transfer, 7, 413–428.
Shyy, W. (1994). Computational modeling for fluid flow and interfacial transport. Amsterdam: Elsevier.
Soni, B., & Date, A. W. (2011). Prediction of turbulent heat transfer in radially outward flow in twisted-tape inserted tube rotating in orthogonal mode. Computational Thermal Science, 3, 49–61.
Sussman, M., Smereka, P., & Osher, S. (1994). A level set approach for capturing solutions to incompressible two-phase flow. Journal of Computational Physics,114, 146–159.
Sussman, M., Smith, K. M., Hussaini, M. Y., Ohta, M., & Zhi-Wei, R. (2007). A sharp interface method for incompressible two-phase flows. Journal of Computational Physics, 221, 469–505.
Takahira, H., Horiuchi, T., & Banerjee, S. (2004). An improved three dimensional level set method for gas-liquid two-phase flows. Transaction of the ASME Journal of Fluids Engineering, 126, 578–585.
Thiart, G. D. (1990). Improved finite-difference formulation for convective-diffusive problems with SIMPLEN algorithm. Numerical Heat Transfer, Part B, 18, 81–95.
van Leer, B. (1977). Towards the ultimate conservative difference scheme IV, a new approach to numerical convection. Journal of Computational Physics, 23, 276–283.
Warsi, Z. U. A. (1993). Fluid dynamics—Theoretical and computational approaches. London: CRC Press.
Acknowledgements
It is a privilege for me to contribute to this volume in honour of Prof. D. B. Spalding to whom, in 2004, I sent the initial draft of my book [6] to seek his approval of the contents of the book. He not only read the draft but also gave me a task to solve a 1D problem of highly resisted flow through a porous medium. He solved the problem himself by using what he called Date-Colocated procedure using the PHOENICS code. Over exchange of four emails, he was satisfied that my solutions and his were in agreement. I have included with pride that 1D problem in my book. Material of section 3 was developed during sponsorship of Project No: 2005/36/47/BRNS by the Board of Research in Nuclear Science, Department of Atomic Energy, Govt of India. Contribution of Dr. Kausik Nandi, Scientist F, BARC is gratefully acknowledged.
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Date, A.W. (2020). Some Observations on Thermodynamic Basis of Pressure Continuum Condition and Consequences of Its Violation in Discretised CFD. In: Runchal, A. (eds) 50 Years of CFD in Engineering Sciences. Springer, Singapore. https://doi.org/10.1007/978-981-15-2670-1_2
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