# Basics of Computational Fluid Dynamics and Numerical Heat Transfer

Computational fluid dynamics (CFD) is one of the branches of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the millions of calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions. Even with high-speed supercomputers only approximate solutions can be achieved in many cases. Ongoing research, however, may yield software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is often performed using a wind tunnel with the final validation coming in flight test.

Even for the cases where analytical solution is available, the interpretation of the solution is often difficult because the solution is expressed as a very complicated series. Numerical solution therefore becomes a desirable approach for heat conduction under non-linear, complex geometric configurations, or complex boundary conditions. Instead of obtaining the analytical expression of the temperature distribution, the results of numerical solutions are given at discrete points. The numerical solution involves three steps: (1) discretization of the computational domain, (2) discretization of the governing equations, and (3) solution of the algebraic equations (Murthy et al., 2006).

## Overview

The most fundamental consideration in CFD is how one treats a continuous fluid in a discretized fashion on a computer. One method is to discretize the spatial domain into small cells to form a volume mesh or grid, and then apply a suitable algorithm to solve the equations of motion (Euler equations for inviscid, and Navier–Stokes equations for viscous flow). In addition, such a mesh can be either irregular (for instance consisting of triangles in 2D, or pyramidal solids in 3D) or regular; the distinguishing characteristic of the former is that each cell must be stored separately in memory. Where shocks or discontinuities are present, high resolution schemes such as Total Variation Diminishing (TVD), Flux Corrected Transport (FCT), Essentially NonOscillatory (ENO), or MUSCL schemes are needed to avoid spurious oscillations (Gibbs phenomenon) in the solution.

If one chooses not to proceed with a mesh-based method, a number of alternatives exist, notably :

• Smoothed particle hydrodynamics (SPH), a Lagrangian method of solving fluid problems,
• Spectral methods, a technique where the equations are projected onto basis functions like the spherical harmonics and Chebyshev polynomials,
• Lattice Boltzmann methods (LBM), which simulate an equivalent mesoscopic system on a Cartesian grid, instead of solving the macroscopic system (or the real microscopic physics).

It is possible to directly solve the Navier–Stokes equations for laminar flows and for turbulent flows when all of the relevant length scales can be resolved by the grid (a Direct numerical simulation). In general however, the range of length scales appropriate to the problem is larger than even today's massively parallel computers can model. In these cases, turbulent flow simulations require the introduction of a turbulence model. Large eddy simulations (LES) and the Reynolds-averaged Navier–Stokes equations (RANS) formulation, with the k-ε model or the Reynolds stress model, are two techniques for dealing with these scales.

In many instances, other equations are solved simultaneously with the Navier–Stokes equations. These other equations can include those describing species concentration (mass transfer), chemical reactions, heat transfer, etc. More advanced codes allow the simulation of more complex cases involving multi-phase flows (e.g. liquid/gas, solid/gas, liquid/solid), non-Newtonian fluids (such as blood), or chemically reacting flows (such as combustion).

## Methodology

In all of these approaches the same basic procedure is followed.

• During preprocessing
• The geometry (physical bounds) of the problem is defined.
• The volume occupied by the fluid is divided into discrete cells (the mesh). The mesh may be uniform or non uniform.
• The physical modeling is defined – for example, the equations of motions + enthalpy + radiation + species conservation
• Boundary conditions are defined. This involves specifying the fluid behaviour and properties at the boundaries of the problem. For transient problems, the initial conditions are also defined.
• The simulation is started and the equations are solved iteratively as a steady-state or transient.
• Finally a postprocessor is used for the analysis and visualization of the resulting solution.

### Discretization methods

The stability of the chosen discretization is generally established numerically rather than analytically as with simple linear problems. Special care must also be taken to ensure that the discretization handles discontinuous solutions gracefully. The Euler equations and Navier–Stokes equations both admit shocks, and contact surfaces.

Some of the discretization methods being used are:

• Finite volume method (FVM). This is the "classical" or standard approach used most often in commercial software and research codes. The governing equations are solved on discrete control volumes. FVM recasts the PDE's (Partial Differential Equations) of the N-S equation in the conservative form and then discretize this equation. This guarantees the conservation of fluxes through a particular control volume. Though the overall solution will be conservative in nature there is no guarantee that it is the actual solution. Moreover this method is sensitive to distorted elements which can prevent convergence if such elements are in critical flow regions. This integration approach yields a method that is inherently conservative (i.e. quantities such as density remain physically meaningful)
$\frac{\partial}{\partial t}\iiint Q\, dV + \iint F\, d\mathbf{A} = 0,$
where Q is the vector of conserved variables, F is the vector of fluxes (see Euler equations or Navier–Stokes equations), V is the cell volume, and $\mathbf{A}$ is the cell surface area.

The discretization of the computational domain is a process that divides the computational domain into many control volumes as shown in the figure below (Patankar, 1980). Each subdomain is bounded by faces represented by the dashed lines. The physical properties of the cell are defined at the grid points. In Practice A, the faces are always located midway between the grid points. On the contrary, the grid points are always located in the center of the control volume in Practice B. Obviously, if a uniform grid is used, the two practices result in identical grid size and therefore, any discussion on the differences between two practices are meaningful only if the grid size is not uniform. Since the faces in Practice A are located in the midway between grid points, the heat flux across the faces can be more accurately calculated. The disadvantage of Practice A is that the properties of the entire control volume are represented by a point not located in the center of the control volume, which will result in inaccuracy. On the other hand, the properties of the entire control volume in Practice B are represented by the grid point at the center which is a better representation. In addition, Practice

Discritization of the computational domain.

B can easily handle discontinuity of thermophysical properties or boundary conditions. The discretization of governing equations can be done by local or point-wise representation of the partial differential equations (finite difference method; FDM), or weighted integral of the partial differential equations (finite element method; FEM). Patankar (1980) proposed a finite volume method (FVM) involving obtaining the discretized equation by performing integration on the governing equation over the small region. While the resultant algebraic equations for FVM and FDM are often similar, the FVM can guarantee conservation of the mass, momentum and energy on each cell, regardless of the size of the cell. The FVM can also be very easily extended to convective heat transfer because mature numerical methods have been developed in the last three decades.

• Finite element method (FEM). This method is popular for structural analysis of solids, but is also applicable to fluids. The FEM formulation requires, however, special care to ensure a conservative solution. The FEM formulation has been adapted for use with the Navier–Stokes equations. Although in FEM conservation has to be taken care of, it is much more stable than the FVM. approach (Interscience). Consequently it is the new direction in which CFD is moving. Generally stability/robustness of the solution is better in FEM though for some cases it might take more memory than FVM methods.(Huebner, 1995)
In this method, a weighted residual equation is formed:
$R_i = \iiint W_iQ\,dV^e$
where Ri is the equation residual at an element vertex i , Q is the conservation equation expressed on an element basis, Wi is the weight factor and Ve is the volume of the element.
• Finite difference method. This method has historical importance and is simple to program. It is currently only used in few specialized codes. Modern finite difference codes make use of an embedded boundary for handling complex geometries making these codes highly efficient and accurate. Other ways to handle geometries are using overlapping-grids, where the solution is interpolated across each grid.
$\frac{\partial Q}{\partial t}+ \frac{\partial F}{\partial x}+ \frac{\partial G}{\partial y}+ \frac{\partial H}{\partial z}=0$
Where Q is the vector of conserved variables, and F, G, and H are the fluxes in the x, y, and z directions respectively.
• Boundary element method. The boundary occupied by the fluid is divided into surface mesh.
• High-resolution schemes are used where shocks or discontinuities are present. To capture sharp changes in the solution requires the use of second or higher order numerical schemes that do not introduce spurious oscillations. This usually necessitates the application of flux limiters to ensure that the solution is total variation diminishing.

### Turbulence models

Turbulent flow produces fluid interaction at a large range of length scales. This problem means that it is required that for a turbulent flow regime calculations must attempt to take this into account by modifying the Navier–Stokes equations. Failure to do so may result in an unsteady simulation. When solving the turbulence model there exists a trade-off between accuracy and speed of computation.

### Direct numerical simulation

Direct numerical simulation (DNS) captures all of the relevant scales of turbulent motion, so no model is needed for the smallest scales. This approach is extremely expensive, if not intractable, for complex problems on modern computing machines, hence the need for models to represent the smallest scales of fluid motion.

### Reynolds-averaged Navier–Stokes

Reynolds-averaged Navier–Stokes (RANS) equations are the oldest approach to turbulence modeling. An ensemble version of the governing equations is solved, which introduces new apparent stresses known as Reynolds stresses. This adds a second order tensor of unknowns for which various models can provide different levels of closure. It is a common misconception that the RANS equations do not apply to flows with a time-varying mean flow because these equations are 'time-averaged'. In fact, statistically unsteady (or non-stationary) flows can equally be treated. This is sometimes referred to as URANS. There is nothing inherent in Reynolds averaging to preclude this, but the turbulence models used to close the equations are valid only as long as the time over which these changes in the mean occur is large compared to the time scales of the turbulent motion containing most of the energy.

RANS models can be divided into two broad approaches:

Boussinesq hypothesis

This method involves using an algebraic equation for the Reynolds stresses which include determining the turbulent viscosity, and depending on the level of sophistication of the model, solving transport equations for determining the turbulent kinetic energy and dissipation. Models include k-ε (Spalding), Mixing Length Model (Prandtl) and Zero Equation (Chen). The models available in this approach are often referred to by the number of transport equations they include, for example the Mixing Length model is a "Zero Equation" model because no transport equations are solved, and the k-ε on the other hand is a "Two Equation" model because two transport equations are solved.

Reynolds stress model (RSM)

This approach attempts to actually solve transport equations for the Reynolds stresses. This means introduction of several transport equations for all the Reynolds stresses and hence this approach is much more costly in CPU effort.

### Large eddy simulation

Volume rendering of a non-premixed swirl flame as simulated by LES.

Large eddy simulations (LES) is a technique in which the smaller eddies are filtered and are modeled using a sub-grid scale model, while the larger energy carrying eddies are simulated. This method generally requires a more refined mesh than a RANS model, but a far coarser mesh than a DNS solution.

### Detached eddy simulation

Detached eddy simulations (DES) is a modification of a RANS model in which the model switches to a subgrid scale formulation in regions fine enough for LES calculations. Regions near solid boundaries and where the turbulent length scale is less than the maximum grid dimension are assigned the RANS mode of solution. As the turbulent length scale exceeds the grid dimension, the regions are solved using the LES mode. Therefore the grid resolution for DES is not as demanding as pure LES, thereby considerably cutting down the cost of the computation. Though DES was initially formulated for the Spalart-Allmaras model (Spalart et al., 1997), it can be implemented with other RANS models (Strelets, 2001), by appropriately modifying the length scale which is explicitly or implicitly involved in the RANS model. So while Spalart-Allmaras model based DES acts as LES with a wall model, DES based on other models (like two equation models) behave as a hybrid RANS-LES model. Grid generation is more complicated than for a simple RANS or LES case due to the RANS-LES switch. DES is a non-zonal approach and provides a single smooth velocity field across the RANS and the LES regions of the solutions.

### Vortex method

The Vortex method is a grid-free technique for the simulation of turbulent flows. It uses vortices as the computational elements, mimicking the physical structures in turbulence. Vortex methods were developed as a grid-free methodology that would not be limited by the fundamental smoothing effects associated with grid-based methods. To be practical, however, vortex methods require means for rapidly computing velocities from the vortex elements – in other words they require the solution to a particular form of the N-body problem (in which the motion of N objects is tied to their mutual influences). A long-sought breakthrough came in the late 1980’s with the development of the Fast Multipole Method (FMM), an algorithm by V. Rokhlin (Yale) and L. Greengard (Courant Institute) that has been heralded as one of the top ten advances in numerical science of the 20th century. This breakthrough paved the way to practical computation of the velocities from the vortex elements and is the basis of successful algorithms.

Software based on the Vortex method offer the engineer a new means for solving tough fluid dynamics problems with minimal user intervention. All that is required is specification of problem geometry and setting of boundary and initial conditions. Among the significant advantages of this modern technology;

• It is practically grid-free, thus eliminating numerous iterations associated with RANS and LES.
• All problems are treated identically. No modeling or calibration inputs are required.
• Time-series simulations, which are crucial for correct analysis of acoustics, are possible.
• The small scale and large scale are accurately simulated at the same time.

### Vorticity Confinement method

The Vorticity Confinement method (VC) is an Eulerian technique, well known for the simulation of turbulent wakes. It uses a solitary-wave like approach to produce stable solution with no numerical spreading. VC can capture the small scale features to over as few as 2 grid cells. Within these features, a nonlinear difference equation is solved as opposed to finite difference equation. VC is similar to shock capturing methods, where conservation laws are satisfied, so that the essential integral quantities are accurately computed.

### Two phase flow

The modeling of two-phase flow is still under development. Different methods have been proposed. The Volume of fluid method gets a lot of attention lately, but the Level set method and front tracking are also valuable approaches. Most of these methods are either good in maintaining a sharp interface or at conserving mass. This is crucial since the evaluation of the density, viscosity and surface tension in based on the values averaged over the interface.

### Solution algorithms

Discretization in space produces a system of ordinary differential equations for unsteady problems and algebraic equations for steady problems. Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of (usually) nonlinear algebraic equations. Applying a Newton or Picard iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefinite in the presence of incompressibility. Such systems, particularly in 3D, are frequently too large for direct solvers, so iterative methods are used, either stationary methods such as successive overrelaxation or Krylov subspace methods. Krylov methods such as GMRES, typically used with preconditioning, operate by minimizing the residual over successive subspaces generated by the preconditioned operator.

Multigrid is especially popular, both as a solver and as a preconditioner, due to its asymptotically optimal performance on many problems. Traditional solvers and preconditioners are effective at reducing high-frequency components of the residual, but low-frequency components typically require many iterations to reduce. By operating on multiple scales, multigrid reduces all components of the residual by similar factors, leading to a mesh-independent number of iterations.

For indefinite systems, preconditioners such as incomplete LU factorization, additive Schwarz, and multigrid perform poorly or fail entirely, so the problem structure must be used for effective preconditioning.[6] The traditional methods commonly used in CFD are the SIMPLE and Uzawa algorithms which exhibit mesh-dependent convergence rates, but recent advances based on block LU factorization combined with multigrid for the resulting definite systems, have led to preconditioners which deliver mesh-independent convergence rates.[7]

## References

Murthy, J.Y., Minkowycz, W.J., Sparrow, E.M., and Mathur, S.R., “Survey of Numerical Methods,” Handbook of Numerical Heat Transfer, 2nd ed., eds., Minkowycz, W.J., Sparrow, E.M., and Murthy, J.Y., pp. 3-51, Wiley, New York.

Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC.

Huebner, K. H., Thornton, E. A., and Byron, T. D., The Finite Element Method for Engineers, 3rd ed., Wiley Interscience(1995).

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