50 Years of CFD in Engineering Sciences

This paper describes some contributions to Computational Fluid Dynamics (CFD) made by Professor Brian Spalding whilst working at Concentration Heat and Momentum Limited (CHAM) during the period 1993–2004. The discussions focus principally on those areas with which the author had been directly involved with Brian and colleagues at CHAM. Some of the material is now well known in the field, and some not, but familiar material is not submitted as a new or original contribution, but rather to provide examples of Brian’s unique approach to solving practical CFD problems and to explain their origin. The following areas of work are described together with their influence in the field, where this is appropriate: the differential-equation wall-distance calculator; the LVEL model of turbulence; the IMMERSOL model of thermal radiation; virtual mass modelling in Eulerian–Eulerian descriptions of two-phase flow; a space-marching method for hyperbolic and transonic flow; and an automatic convergence-promoting algorithm for SIMPLE-based CFD codes.

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.

The numerical simulation of fluid flow and heat/mass transfer phenomena requires the numerical solution of the Navier–Stokes and energy conservation equations coupled with the continuity equation. Numerical or false diffusion is the phenomenon of producing errors in the calculations that compromise the accuracy of the computational solution. According to Spalding, the Taylor series analysis that reveals the truncation/discretization errors of the differential equations terms should not be classified as false diffusion. Numerical diffusion, in the strict sense of definition, appears in multidimensional flows when the differencing scheme fails to account for the true direction of the flow. Numerical errors associated with false diffusion are investigated via two- and three-dimensional problems. A numerical scheme must satisfy some necessary criteria for the successful solution of the convection–diffusion formulations. The common practice of approximating the diffusion terms via the central-difference approximation is satisfactory. Attention is directed to the convection terms since their approximations induce false diffusion. The conservation equations of all the dependent variables in this study are discretized by the finite volume method. The performance of different numerical schemes (e.g. hybrid, van Leer, SUCCA and the novel SUPER version) is studied in this chapter by the numerical simulation of the transport of a scalar quantity in an inclined and tubular airflow, heat conduction in a cylindrical heat exchanger, and the water vapour condensation in an enclosed space. The numerical accuracy of the predictions obtained when using the various schemes was also studied in the classical cases of the backward-facing step and of an inclined inflow. The study focused on the transport of a contaminant concentration by means of an airflow, the diffusion of temperature in an water flow and at the solid surface of a triple tube, the mass transfer interaction between liquid droplets and humid air and on circular airflow predictions. An Eulerian one- and two-phase flow model is developed within the CFD general-purpose computer program PHOENICS, which considers the phases as interpenetrating continua. The phases may move at different velocities (slip velocity) in a manner that is dictated by the interphase friction. According to the numerical results obtained, it is concluded that the predictions improve when using SUPER in all cases of inclined flow and of humid air precipitation while they are similar to the predictions of the other schemes in the case of heat conduction in the tubular flow.

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.

Practical thermal processes and systems, in application areas such as energy, manufacturing, environmental control, heating/cooling, thermal management of electronics, and transportation, generally involve combined transport mechanisms and many different complex phenomena. The materials of interest are also frequently difficult to characterize, and their properties could involve large changes with temperature, concentration, and pressure. The boundary conditions are often unknown or not well defined. The configuration and the geometry are frequently quite complicated. However, in order to study, predict, design, and optimize most practical thermal processes, it is important to obtain accurate and realistic numerical results from the simulation. The mathematical and numerical models must be verified and validated to establish the accuracy and reliability of the simulation results if these are to be used for improving existing systems and developing new ones. This paper focuses on the main considerations that arise and approaches that may be adopted to obtain accurate numerical simulation results on practical thermal processes and systems. A wide range of systems is considered, including those involved in materials processing, energy, heat removal, and safety. Verification and validation, imposition of realistic boundary conditions, modeling of complex, multimode, transport phenomena, multiscale modeling, and time dependence of the processes are discussed. Additional aspects such as viscous dissipation, surface tension, buoyancy, and rarefaction that arise in several systems are also considered. Uncertainties that arise in material properties and in boundary conditions are also important in design and optimization. The methodology to treat these is outlined. Large variations in the geometry and coupled multiple regions are also of interest. The methods that may be used to address these issues are discussed, along with typical results for a range of important processes. Future needs in this interesting and important area are also presented. In many of these studies, the work done by Professor Spalding and his research group has been particularly valuable, since it has guided many of the simplifications and approaches that have been adopted. This paper is a brief tribute to the extraordinary contributions of Professor Spalding to the field of computational fluid dynamics and heat transfer.

Some observations on numerical predictions of temporally periodic fluid flow and heat transfer in spatially periodic geometries, in both spatially developing and fully developed regions, are presented and discussed in this chapter. Special attention is given to several issues that have not been fully resolved in earlier publications. The key points and ideas are demonstrated in the context of computationally convenient finite volume solutions of the mathematical models of two-dimensional, laminar, constant-property Newtonian fluid flow and forced convection heat transfer in uniform arrays of staggered rectangular plates. A dimensionless plate length of 1, dimensionless plate thicknesses of 1/4, 1/8, 1/12, and 1/16, time-mean Reynolds number values ranging from 100 to 1000, and a Prandtl number of 0.7 were considered. The simulations of developing fluid flow and heat transfer were conducted with calculation domains consisting of one row of ten consecutive geometric modules, followed by a plate-free exit zone of suitable length. Calculation domains consisting of single and multiple geometric modules were considered in simulations of fluid flow and heat transfer in the temporally and spatially periodic region. Findings of particular interest include the following: (1) multiple-module simulations of temporally and spatially periodic fluid flow and heat transfer yielded multiple solutions, but the absolute percentage differences in the corresponding values of time-mean modular friction and Colburn factors were all less than 6.4% and 5.1%, respectively; (2) in simulations of unsteady temporally periodic flows, the values of fully developed time-mean modular friction factor obtained from the predictions of the developing flow and the flow in a single module in the spatially periodic region differed by up to 22%; and (3) the instantaneous spatial periodicity conditions imposed in simulations of temporally periodic flows in single or multiple modules in the spatially periodic region are less restrictive than the boundary conditions employed in the corresponding simulations of developing flows, so the former yielded unsteady fluid flow over a wider range of Reynolds number.

This paper describes a finite volume procedure for network flow analysis in a thermofluid system. A flow network is defined as a group of interconnected control volumes called ‘nodes’ that are connected by ‘branches.’ The mass and energy conservation equations are solved at the nodes and momentum conservation equations are solved at the branches. The flow network also includes solid nodes to account for fluid to solid heat transfer. The heat conduction equation is solved at the solid nodes in conjunction with the flow equations. The properties of a real fluid are calculated using a thermodynamic property program and used in the conservation equations. The system of equations describing the fluid–solid network is solved by a hybrid numerical method that is a combination of the Newton-Raphson and successive substitution method. This procedure has been incorporated into a general-purpose computer program, the Generalized Fluid System Simulation Program (GFSSP). This paper also presents the application and verification of the method by comparison with test data for several applications that include (1) internal flow in a rocket engine turbopump, (2) pressurization and loading of a cryogenic propellant tank, (3) fluid transient during a sudden opening of the valve for priming of an evacuated feed line, and (4) chilldown of a cryogenic transfer line with phase change and two-phase flows. This paper also presents the extension of this finite volume-based network flow method to perform multidimensional flow calculation.

The turbulent round jet is considered including the effects of introducing swirl, tabs and rings so as to modify its behaviour. The paper traces the influence of Ricou and Spalding [J Fluid Mech 11(1):21–32, 1961] through almost 60 years of research into this proto-typical engineering and canonically fundamental flow, albeit from a personal perspective. It culminates on current efforts to unravel the turbulent, non-turbulent interface and a brief examination of Spalding’s population modelling ideas, which could assist exploration of this fascinating multi-scale flow.

This contribution presents the authors’ view of the historical evolution of modelling turbulence by way of the simple (though, some would say, outrageously simplistic) notion that the local turbulent stress–strain connection should be the same as in a laminar Newtonian flow. The principal emphasis is on modelling at a level where two transport equations are solved for scalar properties of turbulence, the level of approximation popularized (though not invented) by D. B. Spalding at Imperial College in the early 1970s. The successes and failures of the approach are examined. The chapter concludes by showing examples of closure at eddy viscosity level of what would be regarded as steady flows though treated by way of a time-dependent solution of the transport equations. These lead, in appropriate circumstances, to time-dependent structures which contribute additional momentum and heat transport thereby enhancing agreement with experiment. 

Studies on mixed convective fluid flow and heat transfer are much more scarce compared to the large volume of literature available on either forced or natural convection. This is primarily because it was thought that applications of comparable forced and natural convection simultaneously are rather limited. However, the recent advent of high heat flux computing and LASER equipment and the need for their cooling has made mixed convection more relevant. The present review traces the development of studies in mixed convection over the last half a century. The most tricky and complex question in this respect may be that of the onset of turbulent flow in mixed convection. A clear and acceptable criterion for the transition of laminar flow to turbulent in this regime is still evasive. Hence, the review has culminated into a relook into the studies dedicated to these transition characteristics.

The deeper insights of relationships between large and small scales lead to the development of large eddy simulation (LES), where large scales are explicitly resolved and small scales being universal are modeled. With the advent of high computing power, it is feasible now to successfully simulate the complex turbulent flows of engineering interest using LES. The paper starts with a brief discussion on features of turbulence leading to LES and subgrid-scale models. The evaluation of LES to resolve the physics of transitional and turbulent flows are made based on illustrations, where the few being previous studies of the author and his research group. Although results demonstrate an immense potential of LES to simulate the transitional and turbulent flows as an alternative to DNS with moderate computational cost, there exist several bottlenecks even today. The requirement of very fine meshes near walls is one of such bottlenecks in using LES at high Reynolds number flows. The hybrid LES-RANS, which was invented to eliminate the limitations, is also discussed here in brief. As a concluding remark, it can be stated that the method is particularly suitable and superior to RANS for situations, where unsteadiness and large-scale structures dominate the flow.

We follow Brian Spalding’s contributions to the field of combustion modelling and simulation, from his early theoretical works and ‘analogue models’ to his more recent thinking on population-based turbulence and combustion models. We revisit the genesis of his popular Eddy-Breakup Model, and highlight how the same notion, i.e. the central role of the fragmentariness of turbulent flow, permeated his thinking throughout the years. We highlight the connections between Spalding’s ideas and other prevailing turbulent combustion models.

The empirical/analytical combustion design methodology practiced since middle 1970s comprises continuously evolving conventional design practice with significantly increased roles played by hypotheses formulation, its direct or indirect verification, semi-analytical models and multidimensional computational tools. Rapid advances in “applicable CFD” and turbulent combustion modeling during the early 1970s with remarkable contributions made by the team led by Prof. Spalding combined with resources (dollars, people, and facilities) provided by industry (Garrett, Allison, GE Aviation, Goodrich, Parker, and Woodward), government (NASA, the US Air Force, Army and Navy) and numerous universities led to formulation and successful applications of empirical/analytical design methodology in several gas turbine combustion technology and design programs. These programs included NASA staged combustion Concept 3 (in 1977), two Army combustor concepts for small engines (1978), 2100, 2400, and 2900 °F temperature rise combustors (1981–1983), the two first product combustors (1986), two near-stoichiometric temperature high-performance combustors (1993), entitlements for ultralow NO_x premix/pre-vaporized and partially premixed mixers (1993, 2008), an RQL combustor for the largest turbofan engine (1996), the second-generation lean-dome combustion technology TAPS demonstration (2003) and its product introduction in GEnx (2009), NASA LDI-2 and LDI-3 technology demonstrations in 2014 and 2018, respectively. An overview of these activities along with the most recent CFD simulation and diagnostics activities are described in this chapter as a recognition of 50 Years of CFD in Engineering Sciences that has provided useful insight for advancing combustion technologies and products while simultaneously improving design process efficiency.

A review is presented of the evolution of heat and mass transfer in modern fuel cells, and the role of computational fluid dynamics in the prediction of their performance. Both polymer electrolyte and solid oxide fuel cells are considered. The mathematical details of the mass transfer driving force and the transferred substance state, as well as the distributed resistance analogy, are derived. It is shown how the transferred substance state may be used to prescribe generalised convection–diffusion boundary conditions (inlet/outlet/wall). The combination of the mass transfer driving force and the application of the distributed resistance analogy concept to fuel cell stack models are explained in detail. In addition to the governing equations for thermofluids, the mathematical modelling of fuel cells requires additional thermodynamic, electrochemical kinetic and electric considerations to be taken into account (physicochemical hydrodynamics). Moreover, the results of original research conducted over two decades and culminating in very recent results are presented and explained. This work is research-in-motion and some future possibilities are outlined in the conclusion.

Proton Exchange Membrane Fuel Cell (also called Polymer Electrolyte Membrane Fuel Cell) PEMFC is an electrochemical device that converts the chemical energy in the Hydrogen–Oxygen reaction directly into electrical energy. The reaction takes place at low temperatures, with water and heat as products. The conversion efficiency could be as high as 70%. These make it very attractive as a power source for many applications in electronics, automotive and back-up generator. Much research has been performed on PEMFC over the past 30 years to improve performance and reliability. This paper reviews such works, with an emphasis on computational methods as a supplement to experimental studies. It starts with a review of the fundamentals of PEMFC, illustrates principles of operation, and finally discusses computational studies which are largely based on standard computational fluid dynamics (CFD) methods. These range from one-dimensional, isothermal, single-phase to three-dimensional, non-isothermal, two-phase flow through porous media. The CFD methods are supplemented with electrical charge equations. Although the state-of-the-art is very advanced and can simulate accurately a single cell, or a small stack with a few cells, large stacks containing tens or hundreds of cells, typical of many practical applications, cannot still be resolved with existing computer resources.

In this chapter, a comprehensive review of numerical studies for falling liquid films over plain flat surfaces and horizontal tubes have been presented in terms of flow hydrodynamics and coupled heat and mass transfer. The early studies on the falling film transport models were based on simplified assumptions, and the reduced equations yielded approximate solutions. Developments in Computational Fluid Dynamics (CFD) led by Professor Spalding at Imperial College since the 1960s have enabled the solution of the full set of equations, and space- and time-accurate solutions. The present review primarily discusses recent studies that are based on the solution of the full set of coupled liquid–gas flow equations and highlights some key observations based on these studies. For liquid film flow over plain flat surfaces, the review highlights the important role of interfacial waves and the associated enhancement in the sensible heat transfer rates. However, the impact of these interfacial waves on coupled heat and mass transfer and the potential interaction of these waves with the gas medium is not fully understood. For film flow over horizontal tubes, the recent literature has made significant progress through full-scale models and employing sharp interface capturing techniques. Time- and space-resolved calculations for falling film evaporation over horizontal tubes are currently limited in the literature, but could reveal key underlying mechanisms and/or assist in developing underlying models related to dry-out conditions.

In the past, various numerical approaches have been followed and utilized to analyze the bubble growth and heat transfer during boiling of liquids over a heated substrate. The present chapter focuses to review significant works associated with the simulations of nucleate and film boiling regimes. Most of the numerical approaches differ in the interface capturing techniques or the microlayer modeling in case of nucleate boiling. In the present chapter, the overall development in the research related to boiling studies has been overviewed and the advancement in the studies particularly related to the pool boiling through numerical simulation has been discussed.

This paper deals with the prediction of airflow and temperature distributions in data centers with the goal of achieving proper cooling of the computer equipment. The focus is on raised-floor data centers, but the material is equally applicable to other designs. First, the concept of a raised-floor data center is introduced and the cooling challenge is described. In this arrangement, cooling air is supplied through perforated tiles. The flow rates of the cooling air must meet the cooling requirements of the computer racks placed next to the tiles. These airflow rates are governed primarily by the pressure distribution under the raised floor. Thus, the key to modifying the flow rates is to influence the flow field in the under-floor plenum. Computational Fluid Dynamics (CFD) studies are presented to provide insight into various factors affecting the airflow distribution and the corresponding cooling and to explore various methods for controlling the airflow distribution. Then attention is turned to the above-floor space, where the focus is on preventing the hot air from entering the inlets of computer servers. Different strategies for achieving this prevention are considered. CFD modeling is ideal for understanding the behavior of these strategies and for determining their effectiveness. Some recent studies in these areas are summarized.

Problems of validating computational models for naturally convective flows are discussed. It is shown that even in the case of laminar flow when all property variations are included in simulations, a good agreement with experimental data can be problematic due to inaccuracy in boundary conditions. For turbulent flows, the validation becomes even more challenging as the difference between experimental and numerical results could be the effect of an inadequate turbulence or subgrid model or the use of inappropriate boundary conditions. As a result, the better agreement with experimental data might be obtained by “improving” the turbulence or subgrid model, or by obtaining “more accurate” boundary conditions, or a combination of the two. This issue is explored using the example of LES in simulating buoyancy-driven flow in a tall rectangular cavity. The challenge of unknown boundary conditions has also been explored for a reduced laboratory model of an open-ended channel heated from one side. In this case, in order to obtain agreement between the numerical and the experimental data, the disturbances needed to be introduced at the inlet of the channel. By extending a computational domain to include the laboratory it was shown that the large structures meandered around the laboratory and disrupted the flow in the channel. Similar large disturbances were also detected experimentally. Therefore, it is shown that for a reduced model only tuning is possible and validation in a classical sense cannot be achieved.

The Generalized Integral Transform Technique (GITT) is reviewed as a hybrid numerical–analytical approach for linear or nonlinear diffusive and convective–diffusive partial differential formulations, including an important class of conjugated problems in heat transfer and fluid flow analyses. This chapter focus is on the handling of irregular regions and heterogeneous domains, as a tribute to Prof. D. B. Spalding, who stimulated this research direction in a private communication with the first author, back in 1994. First, formal solutions for nonlinear diffusion and convection–diffusion formulations are reviewed, including the alternatives of adopting nonlinear and/or convective eigenvalue problems, either on total or partial transformation schemes. Next, the GITT itself is formalized in the solution of linear and nonlinear eigenvalue problems, including the direct integral transformation of problems defined in irregular domains, based on simpler auxiliary eigenvalue problems written for the same geometry. Then, a single domain reformulation strategy is discussed, which accounts for heterogeneities on either physical properties or geometrical forms, by rewriting the different media transitions as space variable equation coefficients and source terms. The two complementary strategies are then illustrated through representative examples in convection and conjugated conduction–convection problems, confirming the excellent convergence characteristics of the proposed eigenfunction expansions, toward the establishment of sets of benchmark reference results. The present hybrid solutions are also co-verified against results from purely numerical general-purpose CFD codes.

The Method of Weighted Residuals (MWR) is used to compare the finite element method (FEM) with the finite volume method (FVM) through nodal recursion relations. Both methods reside under the general MWR structure, with the underlying switch between the two methods established through the weighting function. Both methods yield comparable spatial accuracy for steady-state conditions. However, the flexibility of the FEM permits additional options that can increase accuracy, but generally at the expense of additional time and resource constraints.

Computational Fluid Dynamics appears to be poised on the threshold of rapid advances powered by the recent developments in deep machine learning. Deep machine learning will be used to improve the speed, accuracy and, the user-friendliness of CFD software. The applications of CFD will expand beyond the usual aerospace and mechanical/thermal areas to include areas such as biomedical, sport, food processing, environmental, fire safety, buildings ventilation and energy efficiency, and a host of other areas of social relevance. Deep machine learning will be routinely used to generate digital twins/reduced order models which will have a profound impact on the way that CFD is utilized. Standardized interfaces will be developed to embed the digital twins into CAD/PLM software and even spreadsheets. This will enable engineers to rapidly assimilate these models into the product development process and thereby create optimal designs, without needing the services of a CFD expert. These models will also be used for optimal control. In addition, these models can be combined with experimental and field data using Internet-of-Things (IoT) to provide for real-time monitoring of the device and assessing the need for preventive maintenance, etc. This has profound implications for product safety in the field. The social benefits are obvious. In short, CFD will become ubiquitous but will be buried inside digital twins/reduced order models so that it is usable by engineers, whereas CFD experts will be more engaged in creating them using high fidelity computations and of course, in extending the application of CFD into diverse areas of human activity.