While finite volume methodologies (FVM) have predominated in fluid flow computations, many flow problems, including groundwater models, would benefit from the use of boundary methods, such as the Complex Variable Boundary Element Method (CVBEM). However, to date, there has been no reporting of a comparison of computational results between the FVM and the CVBEM in the assessment of flow field characteristics. In this work, the CVBEM is used to develop a flow field vector outcome of ideal fluid flow in a 90-degree bend which is then compared to the computational results from a finite volume model of the same situation. The focus of the modelling comparison in the current work is flow field trajectory vectors of the fluid flow, with respect to vector magnitude and direction. Such a comparison is necessary to validate the development of flow field vectors from the CVBEM and is of interest to many engineering flow problems, specifically groundwater modelling. Comparison of the CVBEM and FVM flow field trajectory vectors for the target problem of ideal flow in a 90-degree bend shows good agreement between the considered methodologies.
Finite volume methodologies have traditionally been used to analyse fluid flow problems, including groundwater models, through the use of computational fluid dynamics (CFD) software packages such as Fluent and OpenFOAM. And while various other domain modelling approaches are also common in fluid flow analysis, such as the finite element and finite difference numerical approaches, they are all limited by the necessity of a volume mesh, the characteristics of which can have a significant effect on both the accuracy and solution time of the computations [
While the advantages of the CVBEM over domain modelling methods, such as FVM, are specifically described by Johnson et al. [
Such a CVBEM vector flow field is a direct result of the CVBEM approximation function for the conjugate component. In addition, the vector flow field can be developed using vector gradients of the CVBEM potential function outcome, which has application to three-dimensional flow problems. What is particularly new, as presented in this paper, is the development of a procedure to develop stream function flow trajectory vectors based upon vector calculus gradients of the CVBEM potential function (that is, the real part of the CVBEM complex variable function outcome), as opposed to being based upon the CVBEM stream function (the imaginary part). Because the CVBEM solution solves the boundary value problem (BVP), the CVBEM flow trajectory vectors should properly represent the ideal fluid flow direction and magnitude of the flow regime. In other words, for the considered important ideal fluid flow application problem, the CVBEM solution should be the exact solution to the BVP and the produced fluid flow trajectory vectors should be correctly determined. The flow field vector outcome from EasyCFD is thus used to verify and validate the development of flow trajectory vectors by the CVBEM for ideal flow problems.
The CVBEM originates from the real variable Boundary Element Method (BEM) that was developed by Carlos Brebbia [
To develop the finite volume solution, the CFD software EasyCFD was used to setup and solve the 90-degree bend fluid flow model. EasyCFD is a CFD software tool for the numerical simulation of fluid flow in a boundary fitted mesh. The Navier-Stokes equations: mass, momentum, and energy, are solved via a finite volume methodology. Specific details and validation regarding the EasyCFD program can be found in several publications [
The selected test problem is of two-dimensional ideal fluid flow in a 90-degree horizontal bend. This test problem has been the subject of several computational modelling assessments and is considered in the current work due to the availability of the analytic solution, and the challenge of developing the flow field vector trajectories for a highly spatially variable flow field problem. The CFD model used to simulate flow in a 90-degree horizontal bend is shown in
The modelling domain in which the results are compared is a square defined within the total modelling space as shown in
In order to directly compare the FVM results to the CVBEM application and the available analytic solution, it was necessary to approximate ideal flow in the FVM model. This was accomplished in EasyCFD by defining the fluid as water (ρ = 1000 kg/m3, μ = 0.001 N*s/m2), the flow type as laminar (to ignore turbulent effects), and the boundary walls as symmetry boundary conditions (i.e. slip-walls). The FVM solution was considered converged when all residuals (u, w, mass) were less than 5e−6.
For the analytic solution, the mathematical description of the streamline function is directly available from the conjugate function of the complex variable monomial w(z) = z2 [
CVBEM basis function specification includes complex variable monomials as well as the usual sums of products of complex polynomial and complex logarithm basis functions. Thus, three outcomes are available for comparison; namely, the FVM computational outcome of a set of highly discretized point estimates; the CVBEM approximation function outcomes; and the analytic solution.
For the CFD application, the flow field is developed by a finite volume computation that is made in addition to the usual post-processing interpolation of point estimates of fluid flow properties for the subject problem. For the analytic solution and the CVBEM outcome approximation function, flow field vectors are determined by direct use of the conjugates function or by application of the vector gradient operator upon the modelling outcome of the CVBEM approximation potential function. The CVBEM test problem solution can be seen in
Comparisons of the vector trajectories between modelling approaches are displayed in
as to vector direction for the considered FVM and CVBEM applications.
A quantitative comparison of the error between the 2 methods with respect to vector magnitude and direction can be found in
between the vector magnitude (
defined as a velocity vector pointing in the positive x direction with the angle increasing counter-clockwise.
The comparison of vector magnitudes shows that the maximum absolute error is less than 0.1 m/s, with an average absolute error of 0.03 m/s, or average relative error of 1.1%. Similar agreement is found when comparing vector direction, which shows that the maximum absolute error is 10.1 degrees, with an average absolute error of 0.4 degrees. Even better agreement is found when comparing points not located along the x- or y-axis (0.7 degrees maximum absolute error, 0.15% average relative error).
Comparison of the CVBEM and FVM flow field trajectory vectors for the target problem of flow in a 90-degree bend shows good agreement between the considered methodologies, achieving an average relative error of 1.1% in velocity magnitude and 0.15% in velocity direction. This is the first such work in which velocity vectors developed by the CVBEM are compared to the results from an FVM model and the results indicate that the flow trajectory vectors developed from the complex variable boundary element method are correctly determined and properly represent the ideal fluid flow velocity and direction.
The authors would like to thank the cadets and faculty of the United States Military Academy, Department of Mathematical Sciences, who provided invaluable feedback and support in development of this paper.
Bloor, C., Hromadka II, T.V., Wilkins, B. and McInvale, H. (2016) CVBEM and FVM Computational Model Comparison for Solving Ideal Fluid Flow in a 90-Degree Bend. Open Journal of Fluid Dynamics, 6, 430-437. http://dx.doi.org/10.4236/ojfd.2016.64031