For the numerical method for compressible flow, the time-marching technique is one of the most important methods. The principle of a time-marching method is to consider the solution of a stationary problem as the solution after a sufficiently long calculation time of the time-dependent equations describing the particular problem.
The attraction of the time-marching method is the ability to compute mixed subsonic and supersonic flows. The equations may be solved by either finite-difference or finite-volume form. In the former, it is usual to transform the computational domain into a uniform rectangular grid and to express the derivatives of the flow variables in terms of values at the nodes of this grid. In the finite-volume method the equations are regarded as these for the conservation of mass, energy, and momentum applied to a set of interlocking control volumes formed by a grid in the physical plan. When equations are solved in this way, it is easier to ensure conservation of mass and momentum than in the differential approach, but numerical schemes are necessary to ensure stability.
The advantages of the time-marching methods are as follows: (a) the same code can be used for the solution of all flow regimes: subsonic to supersonic, viscous or inviscid equations; (b) the technique is easy to implement and vectorize to take an advantage of supercomputing capabilities. The equations are solved as a coupled system. The scalar variables (pressure, velocities, density, enthalpy, entropy, and temperature) are solved simultaneously; (c) the code is flexible so that it can employ
both external and internal flow calculations; (d) for complex flow geometry and flow fields, the technique is comparable in other methods. The major disadvantages are as follows: (a) since the technique is time iterative and often requires very small time steps, the computational efficiency is low. This shortage stimulates the author to develop a new kind of economical time-marching algorithm; (b) the technique is not suitable for incompressible flows. This also stimulates the author to solve the eigenvalue stiffness problem.
Since the flows in turbomachinery components contains both incompressible and compressible parts, the development of the methods that can be used for both incompressible and compressible flows is very important. The pressure-based meth-ods solve a Poisson equation for pressure, and the momentum equations in an uncoupled manner by iterating until a divergence-free flow field is satisfied. On the other hand, it is evident from the material presented in the earlier section that very efficient and accurate algorithms have been developed for solving hyperbolic systems governing compressible flow using time-iterative methods. Many attempts have been made to combine these two methods and hope to form a kind of new method that can be used in both impressible and compressible flows.
Two methods have been proposed. One involves recasting the incompressible flow equation as hyperbolic systems, which is called pseudo-compressibility tech-nique. The first method is to premultiply the time derivative by a suitable matrix and the other is to use a perturbed form of the equations in which specific terms are dropped such that the physical acoustic waves are replaced by pseudo-acoustic modes. This technique is called preconditioning method [6].
Pseudo-compressibility technique is based on making the governing equations behave like a hyperbolic system through adding the time derivative of a pressure term. In the steady state, these equations and solutions are exactly those of steady incompressible flow, even though the temporal solutions are not accurate. The conti-nuity equation after modification introduces spurious pressure waves of finite speed into the flowfield through the pseudo-compressibility parameter. Most researchers linearized the equations using a truncated Taylor series and discretized them by using a two-point central difference in the spatial directions. The most popular solution techniques are the standard implicit approximate factorization schemes.
The pseudo-compressibility technique has been used in the turbine blade flow cal-culation. However, the pseudo-compressibility parameter is difficult to determine.
It is strongly influenced by the flow structure.
The preconditioning method, on the other hand, is based on modifying the time-dependent Navier–Stokes equations. The preconditioning matrix was added in the time derivative term. The basic idea of the preconditioning procedure is to start with the time derivatives expressed in terms of the viscous dependent variables that appear in the diffusion terms. The procedures modify the eigen structure of the invis-cid components of the set of the Navier–Stokes equations such that the condition number remains bounded as the freestream Mach number becomes small. Provided that the numerical discretization is modified to reflect the new eigen system, the use of time-derivative preconditioning can allow a single algorithm to provide accurate, efficient solutions to the Navier–Stokes equations at all flow speeds.
For obtaining the stable scheme, most of the researchers used the implicit scheme to yield a promising method for rapid calculations of fully coupled flows at all speeds. However, the selection of the preconditioning matrix is according to expe-rience and the flow characteristics. The limitations of current methods encourage us to investigate the new methods. Authors [2 and 3] proposed a timing matching method through modifying the governing equations to prevent eigenvalue stiffness when it uses in incompressible flow range.
There are two classes of methods for solving the time-dependent equations:
explicit and implicit. Explicit methods, where spatial derivatives are evaluated using the known conditions at the old time level, are simpler and more easily vectorizable. The implementation of the boundary conditions is straightforward.
The coding can be easily extended to include a time-dependent calculation. Their major disadvantages lie in the conditional stability dictated by courant-Friedrichs-lewy limit. The convergence is usually slower and requires more computational time than an implicit method. For implicit methods, on the other hand, the unknown variables are derived from a simultaneous solution of a set of equations. Implicit methods usually allow larger time steps and faster convergence, and are attractive for both steady and unsteady flows. Both explicit and implicit schemes are widely employed in the computations of turbomachinery flows. However, both of these methods are still under development.
The most widely used explicit methods in turbomachinery flow are Lax–
Wendroff scheme, MacCormack scheme, and Runge–Kutta scheme [6]. The Runge–Kutta scheme allows larger time step than that with other schemes. Artificial dissipation must be purposefully added to these schemes, usually after each stage in the updating procedure. All of these schemes provide inherent artificial dissipation due to truncation error. The most widely used implicit techniques in turbomachin-ery flow are approximate factorization and upwind schemes. The literature shows that operational CFL number in the order of 10 for viscous turbomachinery flow calculations provided the best error damping properties although linear stability analysis provided that the schemes are unconditionally stable in two dimensions [1–3]. The implicit scheme enables large time steps but may introduce factorization error. The results for a turbine cascade indicate that all techniques provide nearly identical pressure distribution. The major differences lie in the CFL number and CPU time for convergence. Both explicit and implicit techniques are widely used, with explicit techniques more commonly used in industry. Both techniques can incorporate local time stepping. The maximum time step for a four-stage Runge–
Kutta method is CFL= 2.8, and typical CFL numbers for the implicit lower upper scheme and alternating direction implicit scheme are between 5 to 10, [6]. There-fore, the choice of explicit versus implicit technique and the time accuracy of a given algorithm may have less to do with the accuracy of the predicted results than the nature of a numerical damping and an application grid.
In the turbomachinery flow field simulation, most researchers [20–32] focused on the flow field analysis. In the flow and heat transfer analysis, the governing equations may be solved in either finite-difference or finite-volume form. The finite-difference scheme usually provides a good convergent rate. The finite-volume
scheme provides a very simple and automatic conservation of mass and momentum scheme, and it also has good physical properties because the scheme is obtained from working in a physical grid. The argument as to whether finite-difference or finite-volume schemes are preferable is not resolved, and both types are still widely used. Either scheme is written in the form of difference scheme. The finite-difference equations use to simulate flow and heat transfer must be numerically stable, spatially and temporally accurate and efficient, conservative so that flow discontinuities can be removed from every grid without any distortion, and easily applicable in generalized coordinates.
Aerodynamic designers have been using the CFD approach as one of the design tools to generate engineering design data. The potential of CFD approaches to revolutionize a turbomachinery analysis is exemplified by impressive calculations such as those of flows around turbine stators and rotors. A general method for CFD in the design of turbomachinery, which would include turbulence viscous effects with minimum approximations and high efficiencies, could provide a major tool for both researchers and design engineers to improve turbine performance and durability. One of the motivations of this part has led to a major effort in developing cascade flow analysis of various degrees of sophistication. Despite significant gains in the computational speed of modern computers, most numerical schemes still require prohibitive amounts of computational time to obtain solutions of the Navier–
Stokes equations. Despite the challenges of the three-dimensional complicated flow simulation, a two-dimensional flow simulation is still an interesting topic of a research scientist [1,4].
Although considerable progress has been made in the past 20 years in the numer-ical simulation of steady two-dimensional turbine cascade flows, these numernumer-ical results were still in general not reliable enough [1]. Further improvement is still needed in the turbulence modeling, grid generation, and efficiency of the numerical scheme. The fact is that turbulence models themselves have not always performed well. The choice of the numerical method might also be a crucial factor for the successful implementation of a numerical scheme. The present study demonstrates the efficiency of the numerical scheme and potential usage of this scheme to solve cascade flow and heat transfer problems.
Time-marching algorithms [2,14] have been used primarily in gas turbine cas-cade flow and heat transfer analysis. The basic principle of a time-marching method is to consider the solution of a stationary problem as the solution after sufficiently long calculation time of the time-dependant equations. The computation starts with a rough perturbation that develops under certain boundary conditions until it reaches a convergent state. In this approach, the governing equations are replaced by a robust time-difference approximation with which steady or time-dependent flows of interest can be solved. At each time level, a system of algebraic equations is solved. The time-marching algorithms can be categorized into two types: explicit and implicit schemes. An explicit method, in general, is easier to program and to vectorize, allowing incorporation of boundary conditions and turbulence models in a simpler way than an implicit method. A main shortcoming of the explicit method is its susceptibility to numerical stability, which constrains the time-step size rather
severely. The implicit scheme, on the other hand, is generally unconditionally stable, so that a relatively large time step can be adapted. In addition to these algorithms, there exists a family of hybrid algorithms, which is widely used in the cascade-flow simulations. The hybrid algorithms possess the positive features of the explicit and implicit algorithms, providing a relatively rapid convergence process and having a less restricted stability constraint. The essence of this algorithm is the use of explicit and implicit finite-difference formulae at alternative computational mesh points. However, the nature of the basic formulae for the hybrid algorithm remains the same as for the traditional implicit and explicit schemes.
In viscous calculations, dissipating properties always present due to the exis-tence of diffusive terms. Away from the shear layer regions, however, the physical diffusion is generally not sufficient to prevent the numerical oscillation of the schemes. Thus, the artificial dissipation terms are usually used in order to sta-bilize and permit a higher CFL number in most of the interactive schemes for the compressible flow computation. As is commonly recognized, the artificial dissipa-tion will influence the accuracy of the numerical results. It has been shown that essentially grid-converged solutions for the high-speed viscous flows over aerody-namic shapes can be obtained if sufficiently fine meshes for the computation are provided. However, the computation based on fine meshes is very time consuming, making it difficult to assess the numerical accuracy of grid-converged solutions.
The techniques to properly treat dissipation terms are still an important topic of the computational fluid. An attempt was made by Turkel and Vatsa [15] to improve the accuracy of the solutions on a given grid in order to reduce the required number of grid points for obtaining a specified level of accuracy. The essential mechanism used in their method [15] is to reduce the level of the artificial viscosity by replacing the scalar form of the artificial viscosity with a matrix form. It has been shown that the numerical accuracy of the Navier–Stokes solutions is improved through the use of the matrix-valued dissipation model. In this chapter, the dissipation terms were not treated through modifying convective fluxes as in traditional methods. Instead, the dissipation terms were incorporated into the time-derivative terms to form a new time discretization scheme to reduce the level of artificial viscosity. And also in this way, the eigenvalue-stiffness problem associated with the time-derivative term can be prevented when the calculations are performed in a low Mach number flow. This characteristic can partly control the effect of eigenvalue stiffness, which is well known on the convergence of both explicit and implicit schemes. Furthermore, the present scheme is a hybrid scheme that combines the advantages of the implicit and explicit schemes, and is a second-order time and spatial derivative scheme that allows more accurate calculation. Because the higher CFL number can be used with the present method, it is more economic than the ordinary implicit scheme.
The effect of eigenvalue stiffness on the convergence of both explicit and implicit schemes is well known, and normally two distinct methods have been suggested for controlling the eigenvalues to enhance convergence. There are two kinds of methods that can solve this problem. One of the methods to overcome this problem is to premultiply the time derivative by a suitable matrix, and the other is to use a perturbed form of the equations in which specific terms are dropped such that
the physical acoustic waves are replaced by pseudo-acoustic modes. The present method follows the first idea and incorporates the matrix-valued dissipation terms in time-derivative terms. These terms prevent the eigenvalues of the time-derivative terms from becoming stiff when the calculations are performed in the flow where the Mach number is relatively small. This characteristic can partly control the effect of eigenvalue stiffness that is well known on the convergence of both explicit and implicit schemes. Furthermore, the present scheme employs a hybrid algorithm that combines the advantages of the implicit and explicit schemes. The present scheme also processes a spectral radio technique to simplify the calculation and avoid the approximate factorization, thus increasing the stability. This new method is a second-order time and spatial derivative scheme permitting a high CFL number.