Chapter 10
Computational Methods for the Investigations of Heat Transfer Phenomena in Aerospace Applications
Abstract
This chapter introduces computational methods, in particular, computational fluid dynamics (CFD), and also gives a brief description of their development. The basic concepts are presented and then the focus is on the so-called finite volume method. At the end of the chapter, examples related to aerospace-specific topics are given. The important topic of shock wave–boundary layer interaction is also discussed.
Keywords
CFD; Compressible flow; Computer codes; Finite volume method; Reynolds number; Illustrative examples
10.1. Introduction
Computational methods started to have a significant impact on the analysis of aerodynamics and its design in the late 1960s. The so-called panel methods were introduced, and these were based on the distribution of surface singularities on a given configuration. Potential flows (nonviscous flows) around bodies could be solved by these methods. Additional capabilities were added later to the surface panel methods and then it was possible to include higher order more accurate formulations, lifting capability, unsteady flows, and coupling with various boundary layer formulations. However, the panel methods could not offer accurate solutions for high-speed nonlinear flows of current interest, and thus more sophisticated models of the flow field equations had to be developed. Gradually, this development has led to what is now called computational fluid dynamics (CFD). For further details, see Ref. [1].
CFD is an interdisciplinary branch of science and engineering with a broad spectrum of applications. Fluid flow, heat transfer, mass transfer, combustion, and chemical reactions appear in most aspects of modern life and are of significance in automotive, space and aviation, and chemical and process industries, as well as in atmospheric science, energy, medicine, microtechnology, and nanotechnology. The development and applications of CFD have been tremendous, and CFD is used both as a modeling tool and in R&D in many industries nowadays. Besides it continues to be developed for new challenges and is being used in basic research at universities.
In aerospace applications, the fluid flow is compressible and the fluid density varies with its pressure. The flow speed is commonly high and the Mach number is greater than 0.3. Subsonic compressible flows have been found to have a Mach number between 0.3 and 0.8. The relationship between pressure and density is weak, and no shocks will be computed within the flow. Highly compressible flows have a Mach number greater than 0.8. The pressure strongly affects the density, and shocks are possible. Compressible flows can be either transonic (0.8 < Ma < 1.2) or supersonic (1.2 < Ma < 3.0). In supersonic flows, pressure effects are transported only downstream. The upstream flow is not affected by conditions and obstructions downstream.
The total temperature, Tt, is a key parameter and is the sum of the static temperature and the dynamic temperature. There are two ways to calculate the total temperature, see Eq. (10.1):
(10.1)
where V is the velocity and cp is the gas specific heat.
The total pressure, Pt, is another useful quantity for running compressible analyses. It is the sum of the static pressure and the dynamic pressure.
In general, compressible flow analyses are much more sensitive to the applied boundary conditions and material properties than incompressible analyses. If the applied settings do not define a physically real flow situation, then the analysis can be very unstable and may fail to reach a converged solution. Proper specification of the boundary conditions and material properties will greatly improve the chances of a successful analysis.
To include heat transfer in a compressible analysis, it is recommended to apply the total (stagnation) temperature boundary conditions instead of static temperatures at the inlets. Total temperature should also be applied to any solids or walls with known temperature conditions. When there is heat transfer in a compressible analysis, viscous dissipation, pressure work, and kinetic energy terms are calculated. It is very important that the total temperature is specified correctly.
In this chapter a brief summary of the CFD methods, including turbulence modeling; associated problems; and limitations is provided. Examples of CFD applications in aerospace engineering are also provided. Commercially available computer codes and in-house codes are briefly described.
10.2. Governing Equations
All the governing differential equations of mass conservation, transport of momentum, energy, and mass fraction of species can be cast into a general partial differential equation as [2,3],
(10.2)
where ϕ is an arbitrary dependent variable, such as the velocity components and temperature; Γ, the generalized diffusion coefficient; and S, the source term for ϕ. The general differential equation consists of four terms. From left to right in Eq. (10.2), they are referred to as the unsteady term, the convection term, the diffusion term, and the source term.
10.3. Numerical Methods to Solve the Governing Differential Equations
Some numerical methods have been established to solve the governing equations of fluid flow and heat transfer problems. They are the finite difference method (FDM) [4], the finite volume method (FVM) [2,3], the finite element method (FEM) [5,6], the control volume finite element method (CVFEM) [7], and the boundary element method (BEM) [8]. In this chapter, only some details of the FVM will be presented.
10.3.1. The Finite Volume Method
In the FVM the domain is subdivided into a number of so-called control volumes. The integral form of the conservation equations is applied to each control volume. At the center of the control volume a node point is placed. At this node the variables are located. The values of the variables at the faces of the control volumes are determined by interpolation. The surface and volume integrals are evaluated by quadrature formulas. Algebraic equations are obtained for each control volume. In these equations, values of the variables for neighboring control volumes appear.
The FVM is very suitable for complex geometries, and the method is conservative as long as surface integrals are the same for control volumes sharing boundary.
The FVM is a popular method particularly for convective flow and heat transfer. It is also applied in several commercial CFD codes. Further details can be found in Ref. [2,3]. A brief illustration is presented in the following section, and an arbitrary control volume is shown in Fig. 10.1.
A formal integration of the general equation across the control volume reads
(10.3)
Then by applying the Gaussian theorem or the divergence theorem,
(10.4)
By summing up all the faces of the control volume, the equation becomes
(10.5)
where the convection flux, Cf; the diffusion flux, Df; and the scalar value of the arbitrary variable ϕ at a face, Φf have to be determined.
10.3.1.1. Convection-Diffusion Schemes
To achieve physically realistic results and stable iterative solutions, a convection-diffusion scheme needs to possess the properties of conservativeness, boundedness, and transportiveness. The upstream, hybrid, and power-law discretization schemes all possess these properties and are generally found to be stable but they suffer from numerical or false diffusion in multidimensional flows if the velocity vector is not parallel with one of the coordinate directions. The central difference scheme lacks transportiveness and is known to give unrealistic solutions at large Peclet numbers. Higher order schemes such as QUICK (quadratic upstream interpolation for convective kinematics) and van Leer may minimize the numerical diffusion but are less numerically stable. Also implementation of boundary conditions can be somewhat problematic with higher order schemes and the computational demand can be extensive because additional grid points are needed, and the expressions for the coefficients in Eq. (10.5) become more complex.
10.3.1.2. Source Term
The source term S may in general depend on the variable ϕ. In the discretized equation, it is desirable to account for such a dependence. Commonly the source term is expressed as a linear function of ϕ.
At the grid point P, S is then written as
(10.6)
In order to prevent divergence, it is required that SP is negative.
The linearization procedure in Eq. (10.6) is commonly used.
10.3.1.3. Solution of the Discretized Equations
The discretized equations have the form of Eq. (10.5) with the ϕ values at the grid points as unknowns. For boundaries not having fixed ϕ values, the boundary values can be eliminated by using given or fixed conditions of the fluxes at such boundaries. Gauss elimination is a direct method to solve algebraic equations. For one-dimensional cases the coefficients form a tridiagonal matrix and an efficient algorithm called the Thomas algorithm or the tridiagonal matrix algorithm (TDMA) is achieved. For two-dimensional and three-dimensional (3D) cases, direct methods require large computer memory and computer time. Iterative methods are, therefore, used to solve the algebraic equations. A popular method is a line-by-line technique combined with a block correction procedure. The equations along the chosen line are solved by the TDMA. Iterative methods are also needed because the equations are nonlinear and sometimes interlinked.
In many situations, e.g., turbulent forced convection, the change in the value of ϕ from one iteration to another is so high that convergence in the iterative process is not achieved. To circumvent this issue and to reduce the magnitude of the changes, underrelaxation factors (between 0 and 1) are introduced.
10.3.1.4. The Pressure in the Momentum Equations
In the momentum equations, a pressure gradient term appears in each coordinate direction (i.e., a source term S). If these gradients are known, the discretized equations for the flow velocities would follow the same procedure as for any scalar. However, in general the pressure gradients are not known but have to be determined as part of the solution. Thus the pressure and velocity fields are coupled and the continuity equation (mass conservation equation) has to be used to develop a strategy.
There are also other related difficulties in solving the momentum and continuity equations. It has been shown that if the velocity components and the pressure are straightforwardly calculated at the same grid points, some physically unrealistic fields, such as checkerboard solutions, may arise in the numerical solution. A remedy to this problem is to use staggered grids. The velocity components are then given staggered or displaced locations. These locations are such that they lie on the control volume faces that are perpendicular to them. All other variables are calculated at the ordinary grid points. Another remedy is to use a nonstaggered or collocated grid where all variables are stored at the ordinary grid points. A special interpolation scheme is then applied to calculate the velocities at the control volume faces. Most commonly, the so-called Rhie–Chow interpolation method is applied [9].
10.3.1.5. Procedures for Solution of the Momentum Equations
As mentioned in the preceding section, the velocity and pressure fields are coupled. Thus a strategy has to be developed in the solution procedure of the momentum equations. The oldest algorithm is the SIMPLE (semi-implicit method for pressure-linked equations) algorithm. A pressure field is guessed and then the momentum equations are solved for this pressure field, resulting in a velocity field. Then a pressure correction and velocity corrections are introduced. From the continuity equation an algebraic equation for the pressure correction can be obtained. The velocity corrections are related to the pressure corrections and the coefficients linking the velocity corrections to the pressure corrections depend on the chosen algorithm.
The momentum equations are then solved again but now with the corrected pressure as the guessed pressure. New velocities are obtained and new pressure and velocity corrections are calculated. The whole process is repeated until convergence is obtained.
There are other similar algorithms available today; SIMPLEC (SIMPLE consistent) and SIMPLEX (SIMPLE extended) are common. They differ from SIMPLE mainly in the expression for coefficients linking velocity corrections to the pressure correction [10].
Another algorithm named PISO (pressure implicit splitting operators) [11] has become popular more recently. Originally, it is a pressure–velocity coupling strategy for unsteady compressible flow. Compared to SIMPLE, it involves one predictor step and two corrector steps.
Still another algorithm is SIMPLER (SIMPLE revised). Here the continuity equation is used to derive a discretized equation for the pressure. The pressure correction is then only used to update the velocities through the velocity corrections.
10.3.1.6. Convergence
The solution procedure is in general iterative and then some criterion must be used to decide when a converged solution has been reached. One method is to calculate residuals R as
(10.7)
for all variables. NB means neighboring grid points, e.g., E, W, N, S (East, West, North, South). If the solution is converged, R = 0 everywhere. Practically, it is often stated that the largest value of the residuals [R] should be less than a certain number. If this is achieved the solution is said to be converged.
10.3.1.7. Number of Grid Points and Control Volumes
The widths of the control volumes do not need to be constant or the successive grid points do not have to be equally spaced. Often it is desirable to have a uniform grid spacing. Also it is required that a fine grid is employed where steep gradients appear, whereas a coarse grid spacing may suffice where slow variations occur. The various turbulence models require certain conditions on the grid structure close to solid walls. The so-called high- and low-Reynolds-number versions of these models demand different conditions.
In general, it is recommended that the solution procedure is carried out on several grids with different fineness and varying degrees of nonuniformity. Then it might be possible to estimate the accuracy of the numerical solution procedure.
Adaptive grid techniques might be beneficial to increase the resolution in vital areas, such as resolving the pressure jump after a shockwave. In modeling hypersonic flows, special care must be taken in meshing external flows, especially near-shock conditions, as a fine mesh is required to capture shock effects.
10.3.1.8. Complex Geometries
CFD methods based on the Cartesian, cylindrical, or spherical coordinate systems have limitations in complex or irregular geometries. Using the Cartesian, cylindrical, and/or spherical coordinates means that the boundary surfaces are treated in a stepwise manner. To overcome this problem, methods based on body-fitted or curvilinear orthogonal and nonorthogonal grid systems are needed. Such grid systems may be unstructured, structured, or block structured or composite. Because the grid lines follow the boundaries, boundary conditions can more easily be implemented.
There are also some disadvantages with nonorthogonal grids. The transformed equations contain more terms and the grid nonorthogonality may cause unphysical solutions. Vectors and tensors maybe defined as Cartesian, covariant, contravariant, and physical or nonphysical coordinate oriented. The arrangement of the variables on the grid affects the efficiency and accuracy of the solution algorithm.
Grid generation is an important issue, and today, most commercial CFD packages have their own grid generators, and several grid generation packages, compatible with some CFD codes, are also available. The interaction with various CAD (computer-aided design) packages is also an important issue today.
10.4. The CFD Approach
The FVM described earlier is a popular method particularly for convective flow and heat transfer. It is also applied in several commercial CFD codes. In heat transfer equipment such as heat exchangers, both the laminar and turbulent flows are of interest. Although the laminar convective flow and heat transfer can be simulated, the turbulent flow and heat transfer normally require modeling approaches in addition. By turbulence modeling, the goal is to account for all the relevant physics using as simple a mathematic model as possible. This section gives a brief introduction to the modeling of turbulent flows.
The instantaneous mass conservation, momentum, and energy equations form a closed set of five unknowns u, v, w, p, and T. However, the computing requirements in terms of resolution in space and time for directly solving the time-dependent equations of fully turbulent flows at high Reynolds numbers [the so-called direct numerical simulation (DNS) calculations] are enormous and major developments in computer hardware are needed. Thus DNS is more viewed as a research tool for relatively simple flows at moderate Reynolds number and supercomputer calculations are required. Meanwhile, practicing thermal engineers need computational procedures to provide information on the turbulent processes, but avoiding the need to predict effects of every eddy in the flow. This calls for information about the time-averaged properties of the flow and temperature fields (e.g., mean velocities, mean stresses, mean temperature). Usually, a time-averaging operation, called Reynolds decomposition, is carried out. Every variable is then written as the sum of a time-averaged value and a superimposed fluctuating value. In the governing equations, additional unknowns appear: six for the momentum equations and three for the temperature field equation. The additional terms in the differential equations are called turbulent stresses and turbulent heat fluxes, respectively. The task of turbulence modeling is to provide procedures to predict the additional unknowns, i.e., the turbulent stresses and turbulent heat fluxes, with sufficient generality and accuracy. Methods based on the Reynolds-averaged equations are commonly referred to as the RANS (Reynolds-averaged Navier–Stokes equations) methods. Large eddy simulation (LES) lies between the DNS and RANS approaches in terms of computational demand. Like DNS, 3D simulations are carried out over many time steps but only the larger eddies are resolved. An LES grid can be coarser in space and the time steps can be larger than that for DNS, as the small-scale fluid motions are treated by the so-called subgrid-scale (SGS) model.
10.4.1. Turbulence Models
The most common turbulence models for industrial and aerospace applications are classified as
• zero-equation models
• one-equation models
• two-equation models
• Reynolds stress models
• algebraic stress models (ASMs)
• LESs
The first three models in this list account for the turbulent stresses and heat fluxes by introducing a turbulent viscosity (eddy viscosity) and a turbulent diffusivity (eddy diffusivity). Linear and nonlinear models exist [14–16]. The eddy viscosity is usually obtained from certain parameters representing the fluctuating motion. A popular one-equation model is the Spalart–Allmaras model [17] in which a transport equation is solved for the eddy viscosity. It is mostly used for aerospace and turbomachinery applications but not very common for heat exchangers. In two-equation models, these parameters are determined by solving two additional differential equations. However, one should remember that these equations are not exact but approximate and involve several adjustable constants. Models using the eddy viscosity and eddy diffusivity approach are isotropic in nature and cannot evaluate nonisotropic effects. Various modifications and alternate modeling concepts have been proposed. Examples of models of this category are the k-ε and k-ω models in high- or low-Reynolds-number versions as well as in linear and nonlinear versions. A lately popular model is the so-called V2F model introduced by Durbin [18]. It extends the use of the k-ε model by incorporating near-wall turbulence anisotropy and nonlocal pressure-strain effects, while retaining a linear eddy viscosity assumption. Two additional transport equations are solved: one for the velocity fluctuation normal to walls and another for a global relaxation factor. More recently the shear stress transport (SST) k-ω model by Menter [19] has become popular, as it uses a blending function of gradual transition from the standard k-ω model near solid surfaces to a high-Reynolds-number version of the k-ε model far away from solid surfaces. Accordingly, it accurately predicts the onset and the size of separation under adverse pressure gradients.
In the Reynolds stress equation models (RSMs), differential equations for the turbulent stresses (the Reynolds stresses) are solved and directional effects are naturally accounted for. Six modeled equations (i.e., not exact equations) for the turbulent stress transport are solved together with a model equation for the turbulent scalar dissipation rate ε. The RSM models are quite complex and require large computing efforts and are therefore not widely used for industrial flow and heat transfer applications, such as in heat exchangers.
The ASM and explicit ASM (EASM) present an economic way to account for the anisotropy of the turbulent stresses without solving the Reynolds stress transport equations. An idea is that the convective and diffusive terms are modeled or even neglected and then the Reynolds stress equations are reduced to a set of algebraic equations.
For calculation of the turbulent heat fluxes, most commonly, a simple eddy diffusivity (SED) concept is applied. The turbulent diffusivity for heat transport is then obtained by dividing the turbulent viscosity by a turbulent Prandtl number. Such a model cannot account for nonisotropic effects in the thermal field but still this model is frequently used in engineering applications. There are some models presented in the literature to account for nonisotropic heat transport, e.g., the generalized gradient diffusion hypothesis (GGDH) and the WET (wealth = earnings × time) method. These higher order models require that the Reynolds stresses are calculated accurately by taking nonisotropic effects into account. If not, the performance may not be improved. In addition, partial differential equations can be formulated for the three turbulent heat fluxes but numerical solutions of these modeled equations are rarely found. Further details can be found in, e.g., Ref. [20].
In the LES model the time-dependent flow equations are solved for the mean flow and the largest eddies, while the effects of the smaller eddies are modeled. The LES model has been expected to emerge as the future model for industrial applications but it is still limited to relatively low Reynolds number and simple geometries. Handling wall-bounded flows with focus on the near-wall phenomena such as heat and mass transfer and shear at high Reynolds number presents a problem because of the near-wall resolution requirements. Complex wall topologies also present problems for LES.
Nowadays, models obtained by combining the LES and RANS-based methods have been suggested. Such models are called hybrid models and the detached eddy simulation (DES) [21] is an example.
In an article [22], it was stated that the prospects are good for steady problems with RANS turbulence modeling to be solved accurately even for very complex geometries if technologies for solution adaptation become mature for large 3D problems. It was also conjectured that the future rate of growth for supercomputers will only be half of the rate in the past 20 years. This is expected to slow down the pure LES reliability for aerospace applications at high Reynolds numbers but the hybrid RANS–LES approaches were judged to have a great potential. Also a breakthrough in turbulence modeling to predict separation and laminar–turbulent transition was not foreseen to appear in the near future.
10.4.2. Wall Effects
There are two standard procedures to account for wall effects in numerical calculations of turbulent flow and heat transfer: one is to employ low-Reynolds-number modeling procedures and the other is to apply the wall function method. The wall functions approach includes empirical formulas and functions linking the dependent variables at the near-wall cells to the corresponding parameters on the wall. The functions are composed of laws of the wall for the mean velocity and temperature, and formulas for the near-wall turbulent quantities. The accuracy of the wall function approach is increasing with increasing Reynolds number. In general the wall function approach is efficient and requires less CPU time and memory size but it becomes inaccurate at low Reynolds numbers. When low-Reynolds-number effects are important in the flow domain, the wall function approach ceases to be valid. The so-called low-Reynolds-number versions of the turbulence models are introduced and the molecular viscosity appears in the diffusion terms. In addition, damping functions are introduced. Also the so-called two-layer models have been suggested to enhance the wall treatment. The transport equation for the turbulent kinetic energy is solved, whereas an algebraic equation is used for, e.g., the turbulent dissipation rate.
10.4.3. CFD Codes
Several industries and engineering and consulting companies worldwide are nowadays using the commercially available general-purpose so-called CFD codes for simulation of fluid flow, heat and mass transfer, and combustion in aerospace applications. Among these codes are the ANSYS FLUENT, ANSYS CFX, CFD++, and STAR-CCM+. Also many universities and research institutes worldwide apply commercial codes, besides using their in-house developed codes. Nowadays, open-source codes such as OpenFOAM are also freely available. There are also specialized codes such as DPLR (data-parallel line relaxation) and VULCAN (viscous upwind algorithm for complex flow analysis) for hypersonic flows. DPLR is designed for supersonic and hypersonic flows under nonequilibrium conditions, whereas VULCAN is designed for internal flows in scramjet engines. Further details and brief descriptions of the codes can be found in Ref. [23].
However, to successfully apply such codes and to interpret the computed results, it is necessary to understand the fundamental concepts of computational methods. Important issues are also the description of complex geometries and the generation of suitable grids. The commercial codes commonly have their own grid generation tool, e.g., ANSYS ICEM, but stand-alone software such as Pointwise are also popular. The codes are generally also compatible with various CAD tools.
10.5. Topics Not Treated
There are several additional topics that are of importance in CFD modeling and simulations of aerospace problems. Among them, topics that are not being treated in this chapter are
• implementation of boundary conditions
• adaptive grid methods
• local grid refinements
• solution of algebraic equations
• convergence and accuracy
• parallel computing
• animation
10.6. Examples
10.6.1. Chemical Nonequilibrium Turbulent Flow in a Scramjet Nozzle
In this section, details and results from a numerical investigation of nonequilibrium flow in a scramjet single expansion ramp nozzle are presented with a chemical reaction model including seven species and eight finite rate steps [24]. The generic geometry of the scramjet nozzle is depicted in Fig. 10.2. The flow is internal at high temperature and pressure and external at high velocity and low pressure.
The internal fluid goes through a diverging nozzle before being mixed with the external flow. The top wall of the nozzle is designed for maximum trust according to the characteristic line method, and it is longer than the bottom wall making the nozzle to be a single expansion ramp nozzle. The total length of the nozzle is L = 18.54 H1, the length of the bottom wall is Ls = 3.12 H1, and the diffusive angle of the bottom wall θ = 6°, where H1 is the height of the nozzle inlet.
The FVM with a fully implicit scheme was used to solve the conservative unsteady RANS equations, including variations in density, pressure, velocities, mass fractions, and total and thermal energy per unit mass, with appropriate source terms. The heat fluxes are contributed by the species transport. Turbulence is handled by RNG (Re-Normalisation Group) k-ε model. The relation between the thermal energy per unit mass and pressure and density is handled as for nonequilibrium. A second-order upwind scheme is adopted in general but for the k-ε equations first-order upwind scheme was used. Nonequilibrium wall functions were employed to link the viscous region near the wall with the fully turbulent developed region. To consider the effect of the external flow on the nozzle performance, the domain of the external flow was extended. To describe the complicated flow field a multiblock technique was adopted for the structured grid generation, including local refinements. The grid is shown in Fig. 10.3.
The boundary conditions were for external flow the far field pressure was prescribed, no-slip and adiabatic walls were assumed, and total pressure, temperature, Ma number, and mass fractions were given at the nozzle inlet. Finite rate chemistry reaction models for hydrogen combustion in air were used. Seven species were considered, i.e., H2O, OH, O2, H, H2, O, and N2. Eight reactions were considered.
10.6.1.1. Some Results
Fig. 10.4 shows the total temperature contours of the chemical nonequilibrium flow inside the nozzle. Because of combustion the total temperature sharply increases at the inlet of the nozzle. Then the increasing trend is diminished.
A typical picture of the mass fraction distribution of OH is shown in Fig. 10.5. At first it increases and then it is reduced in the main flow direction. The variation is nonmonotonic because chemical reactions occur close to the wall.
Fig. 10.6 presents an example of the mass fraction distributions of H2O, O2, H2, and N2 along the nozzle walls. The distribution of H2 and O2 decreases gradually, whereas the distribution of H2O increases gradually and of N2 does not change. The largest variations are close to the inlet of the nozzle. This means that the chemical reactions mainly appear close to the nozzle inlet.
The major conclusions from the study can be summarized as (1) the chemical reactions occurred mainly close to the nozzle inlet, (2) the inlet temperature and pressure affect the strength of the chemical reactions, and (3) close to the nozzle inlet, the mass fractions of some species varied significantly.
10.6.2. Shock Wave–Boundary Layer Interactions
Shock wave–boundary layer interactions (SBLIs) have significant interest in the aerospace community, as they occur in both external and internal aerodynamic problems at transonic, supersonic, and hypersonic speeds. There is a big interest in accurate numerical prediction methods using CFD. A review [25] presented the advances in understanding low-frequency unsteadiness, heat transfer prediction capability, phenomena in complex multishock boundary layer interactions, and flow control techniques. Several workshops have also been held through the years. Important conclusions are that in RANS predictions of SBLI the choice of turbulence model and the grid density are very important in enabling more accurate solutions.
10.7. Conclusions
Computational approaches for analysis of transport phenomena based on CFD were briefly summarized and reviews of recent works were provided. However, many challenges need to be overcome before CFD can take over as a primary tool in aerospace applications. The following issues need to be addressed: code and performance improvements; introduction of multiprocessor technology; improvements in grid technique including unstructured grids, adaptive grids, and deformable boundaries; validation and improvement of turbulence modeling; handling of boundary layer transition; and improvement of multiphase flow modeling. Besides, well-documented validation data bases are needed.
Results have revealed that the CFD approach, in many cases, can not only demonstrate important physical effects but also provide satisfactory results in decent agreement with corresponding experiments.
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