Computation of compressible fluid flow

The first type of cloud points is set up for each boundary node to compute the Euler fluxes whereas the second type of cloud points is set up for each surface node, for boundary conditio

Acknowledgements

The author would like to, first and foremost, express gratitude and appreciation to her supervisor, Associate Professor, Dr Yeo Khoon Seng His guidance, support and assistance have, without doubts, contributed significantly to the completion of the author’s research as well as to this thesis Professor Yeo has given the author the space and freedom of exploration and self-development while providing direction and invaluable feedback in times of need

The author would also like to thank Dr Tsai Her Mann and Dr Lum Kai Yew, Principal Investigators at Temasek Laboratories, National University of Singapore (NUS) Dr Tsai has offered the author numerous ideas and suggestions to approach problems and have assisted the author in overcoming difficulties in her work Dr Lum has motivated the author in the formulation of the boundary implementation using the least-squares with constraint approach Lively discussions with Dr Lum have also provided the author with much insight and inspiration

Sincere gratitude also goes out to the author’s colleagues at Temasek Laboratories, in particular, Dr Zhang Zhengke for his patience in imparting his CFD knowledge Useful and engaging discussions with PhD student, Chew Choon Seng, have also helped the author significantly

Table of Contents

Acknowledgements i Nomenclature ii

Summary v

1 Introduction

2 Literature Review

2.1 Development of structured grid methods

2.2 Development of unstructured grid methods

2.3 Development of Cartesian grid methods

2.3.1 Boundary condition using finite volume

2.3.2 Boundary condition using finite difference

ghost cells method 2.3.3 Boundary condition using immersed boundary

method 2.4 Development of meshless methods

3.3.4 Implementation of surface boundary conditions

4.1 Solution of flow over single airfoil

4.1.1 Symmetric NACA 0012 airfoil Case 1: M = 0.5, α = 3.0° ∞

Case 2: M = 0.85, α = 0.0°

1 5

57911131415

18

181920202224252627283031

34

34363639

4.1.2 Asymmetric RAE 2822 airfoil Case 1: M = 0.5, α = 3.0° ∞

Case 2: M = 0.75, α = 3.0° ∞

4.2 Solutions of flows over multi-component objects

Case 1: Flow over dual NACA 0012 airfoils Case 2: Flow over NLR with flap airfoil Case 3: Flow over 3-element airfoil 4.3 Additional remarks

5.1 Boundary implementation using least-squares with constraint

5.1.1 Formulation

5.1.2 Constraint equations for ρ, ρE & P

5.1.3 Constraint equations for velocity 5.2 2-D solution of flow over circular cylinder

5.3 3-D solution of flow over uniform wing

5.4 Additional remarks

6 Conclusion

4747505253565962

63

63646567697377

79

Summary

Mesh generation for complex geometries is a continuing obstacle in Computational Fluid Dynamics (CFD) Thus, there is a strong need for a fast and efficient numerical method that can handle these complex configurations In this work, a Cartesian method for the computation of steady-state solution for compressible Euler equations is presented

The proposed method attempts to combine the advantages of the conventional Cartesian grid method and the gridless method while avoiding their shortcomings The Cartesian method is used over the bulk of the computational domain for its efficiencies while the gridless method is only employed in handling solid boundaries In this way,

we arrive at a general solution method that is flexible and efficient for problems with complex geometries

In this boundary implementation, two types of cloud points are used The first type of cloud points is set up for each boundary node to compute the Euler fluxes whereas the second type of cloud points is set up for each surface node, for boundary condition implementation and surface value determination The spatial discretization for the Euler equations is based on the cell-centered finite-volume approach The discretized equations are then solved using the modified four-stage Runge-Kutta scheme Numerous 2-D test cases involving flow over a single airfoil and flow over multi-component objects are computed The results compare well with referenced body-fitted curvilinear grid solutions and converge well for the wide range of Mach numbers tested

An alternative gridless implementation is suggested as an improvement to the above gridless approach This involves using least-squares with constraint via the Lagrange Multiplier principle to implement the boundary conditions The formulation is general and is easily extensible to three dimensions Although there is no significant improvement in terms of accuracy and conservation over the previous scheme, the least-squares with constraint approach is better in terms of implementation Some preliminary three-dimensional results using the new approach are presented

List of Figures

Figure 3: Cloud points for (a) boundary nodes; (b) boundary nodes at thin

Figure 4: Cloud points for surface nodes (Closest 8 points to surface node) 30

Figure 5: Boundary implementation for surface nodes 33

Figure 6: Close-up view of a stretched grid for the NACA 0012 airfoil 35

Figure 7: Close up view of body-fitted grid with NACA 0012 35

Figure 8: Solution for NACA 0012 with M∞ = 0.5, α = 3° (a) Cp plot (b) Mach

contour plot (Cartesian grid) (c) Mach contour plot (Body-fitted grid) (d) Convergence plot [(b) and (c) are plotted with contours Mmin=0.1,

Figure 9: Solution for NACA 0012 with M∞ = 0.85, α = 0° (a) Cp plot (b)

Mach contour plot (Cartesian Grid) (c) Mach contour plot fitted grid) (d) Convergence plot [(b) and (c) are plotted with contours Mmin=0.1, Mmax=1.1, interval=0.034.] 41

(Body-Figure 10: Solution for NACA 0012 with M∞ = 0.8, α = 1.25° (a) Cp plot (b)

Mach contour plot (Cartesian grid) (c) Mach contour plot fitted grid) (d) Convergence plot [(b) and (c) are plotted with contours Mmin=0.1, Mmax=1.1, interval=0.026.] 43

(Body-Figure 11: Solution for NACA 0012 with M∞ = 2.0, α = 0° (a) Cp plot (b)

Convergence plot (c) Close up Mach contour plot (Cartesian grid) (d) Close up Mach contour plot (Body-fitted grid) [(c) and (d) are plotted with contours Mmin=0.25, Mmax=1.9, interval=0.12.] 46

Figure 12: Solution for RAE 2822 with M∞ = 0.5, α = 3° (a) Cp plot (b) Mach

contour plot (Cartesian grid) (c) Mach contour plot (Body-fitted grid) (d) Convergence plot [(b) and (c) are plotted with contours

Mmin=0.05, Mmax=0.8, interval=0.019.] 49

Figure 13: Solution for RAE 2822 with M∞ = 0.75, α = 3° (a) Cp plot (b) Mach

contour plot (Cartesian grid) (c) Mach contour plot (Body-fitted grid) (d) Convergence plot [(b) and (c) are plotted with contours Mmin=0.1,

Figure 14: Solution for dual NACA0012 airfoil with M∞ = 0.5, α = 0° (a) Close

up view of stretched grid (b) Cp plot (c) Mach contour plot with contours Mmin=0.02, Mmax=1.0, interval=0.034 (d) Convergence plot 55

Figure 15: Solution for NLR airfoil with flap with M∞ = 0.2, α = 0° (a) Close up

view of stretched grid (b) Cp plot (c) Mach contour plot with contours Mmin=0.025, Mmax=0.375, interval=0.012 (d) Convergence

Figure 16: Solution for three-element airfoil with M∞ = 0.2, α = 20° (a) Close

up view of stretched grid (b) Cp plot (c) Mach contour plot with contours Mmin=0.048, Mmax=0.72, interval=0.052 (d) Convergence

Figure 17: Solution for circular cylinder using least-squares with constraints

approach with M∞ = 0.38 (a) Close up view of stretched grid (b) Mach contour plot (Least-squares with constraint) (c) Mach contour plot (Previous method) (d) Mach contour plot (Body-fitted grid) (e) Comparison of Cp plots (f) Comparison of convergence plots [(b), (c) and (d) are plotted with contours Mmin=0.052, Mmax=0.714,

Figure 18: Cartesian grid for NACA 0012 uniform wing surface 74

Figure 19: Close up view of cross-sectional X-Z plane with surface nodes

Figure 20: Solution for uniform NACA 0012 wing using least-squares with

constraints approach with M∞ = 0.85, α = 0° (a) Cp contour plot with contours Cpmin=-0.78, Cpmax=0.66, interval=0.103 (b) Mach contour plot with contours Mmin=0.1, Mmax=1.1 interval=0.034 (c)

As technology becomes more widely available in industry and academia, CFD is used

to provide insights into many aspects of fluid motion Nowadays, CFD methodologies are routinely employed in the fields of aircraft, turbomachinery, car and ship designs

It is also applied in meteorology, oceanography, astrophysics, in oil recovery and also

in architecture Hence, CFD is becoming an increasingly important design tool in engineering and also a substantial research tool in certain physical sciences The future advancement of fluid dynamics will depend on a proper balance of all three approaches, with CFD helping to interpret and understand the results of theory and experiment, and vice versa

Chapter 1 Introduction

Over the course of last decade, significant progress has been made on developing numerical methods for the solution of the compressible Euler and Navier-Stokes equation In general, these numerical methods can be classified by the mesh they use, which falls under the category of structured grid or unstructured grid methods Structured grids and unstructured grids each have their own specific advantages and shortcomings Examples of structured grids are body-fitted hexahedral grids and Cartesian grids Body-fitted grid has the advantage of ease in boundary implementation due to the body-aligned nature of the grid However, the major drawback of this is the difficulty of mesh generation for complex geometry Cartesian grid, on the other hand, does not encounter any problems with mesh generation However, as the grid is not body-aligned the cells near the boundary are cut by the surface which makes accurate implementation of boundary conditions complicated The success of Cartesian method depends greatly on having an accurate means of representation for the boundary

Unstructured meshes are typically constructed from triangles in two-dimensional or tetrahedral cells in three-dimensional The main advantage of unstructured grids is the ease of grid generation about complex configuration since the cells may be oriented in any arbitrary way to conform to the geometry However, the computational time and cost for unstructured mesh computations are generally higher which makes it inefficient when applying to large scale three-dimensional problems In general, structured grids are favored for its simpler data structure, which leads to smaller computing times since no indirect addressing is required while unstructured grids are favored for its flexibility in mesh generation when handling arbitrarily complex geometries

Chapter 1 Introduction

Most practical fluid dynamics problems, e.g., flows around aircraft, in turbines, and physiological flows such as in hearts and arteries, entails flow that take place in domains with complex configuration With the increasing demands on the complexity

of flow simulations, grid generation has to become more and more sophisticated Initially, simple structured meshes constructed either by algebraic methods or by using partial differential equations are used As geometrical configurations become more complex, this led to the use of multi-block and Chimera approach where grids are broken down into a number of topologically simpler blocks that are either matched exactly at interfaces or are allowed to overlap However, the generation of a structured, multi-block grid for a complicated geometry may still take weeks to accomplish

Up till now, the problem of mesh generation is still a continuing obstacle of CFD for configurations with complex geometry There is a strong need for a fast and efficient numerical method that can handle complex configurations In this work, a Cartesian method for the computation of compressible Euler equations is developed Cartesian grid is chosen because of the numerous advantages associated with its use such as ease

of grid generation and lower computational storage requirement The boundary conditions will be implemented using a gridless approach which comprises of only clouds of points The points in each cloud do not need to have any specific connectivity with one another This work attempts to combine the advantages of the conventional Cartesian grid method and the gridless method while avoiding their shortcomings The Cartesian method is used over the bulk of the computational domain for its efficiencies while the gridless method is adopted for its flexibility in

Chapter 1 Introduction

arrive at a general solution method that is flexible and efficient for problems with complex geometries With an appropriate means of representation for the boundary, there is certainly great potential for Cartesian methods to handle the computation of arbitrary complex configurations in two-dimensions as well as in three-dimensions

In the next section, a literature review of the development of numerical methods and schemes for computation of Euler and Navier-Stokes equation will be presented Following that, a methodology of the proposed scheme will be described Steady-state solutions of various two-dimensional test cases will be presented in Section 4 to demonstrate the potential of the hybrid Cartesian gridless method proposed Further development of the method and preliminary three-dimensional results are also presented in Section 5

Chapter 2 Literature Review

Chapter 2

Literature Review

The developments of accurate and efficient methods that can deal with arbitrarily complex geometry represent a significant advance in CFD A grid that is not well suited to the problem can lead to unsatisfactory results, instability, or lack of convergence Accurate representation of multi-scale, time-dependent physical phenomena is required Although a variety of mesh generation techniques is now available, the generation of meshes around complicated, multi-component geometries

is still a tedious and difficult task

Currently, the two most widely used methods for dealing with complex geometries involve the use of either structured or unstructured meshes A type of structured mesh that is widely used for flow computation is the body-fitted, curvilinear grid1-3 Its main advantage is the ease in implementing boundary conditions due to the body-aligned nature of the mesh However, a major drawback for this is the difficulty of mesh generation for complex geometry This may give rise to occurrence of highly distorted

or skewed cells in some regions of the domain that adversely influence the computation Very often, it is not possible to use a single type of grid for highly complex geometries This led to more sophisticated methods such as multiblock,

Chapter 2 Literature Review

patched grids or Chimera grids and are used in varying degrees of success (e.g., Refs 4-6).

Multiblock approach involves dividing the physical space into a number of topologically simpler blocks which can be easily meshed These blocks are matched at boundary such that they fill up the entire domain There is an exchange of information between the blocks at interface where they meet To allow for greater flexibility, the grid points on both sides of an interface need not match exactly With this, finer mesh blocks can be used at regions that require flow or geometry refinement However, there

is an increased overhead for the conservative treatment for the boundary due to the enhanced flexibility7, 8 Although the multiblock methodology offers implementation

of the flow solver on a parallel computer by means of domain decomposition, the grid generation process in the case of complex configurations still takes weeks or months to complete

Another methodology developed is the Chimera technique9-12 The basic idea here is to generate first the grids separately around each geometrical entity in the domain After that, the grids are combined together in such a way that they overlap each other where they meet These methods require transfer of information between the different meshes, the identification of mesh intersection points and interpolation of data on all the overlapping meshes It is also important to ensure an accurate transfer of quantities between the different grids at the overlapping region As such, the extension of the overlap is adjusted accordingly to the required interpolation order The advantage of the Chimera technique over the multiblock approach is that the grids can be generated completely independent of each other This allows for greater flexibility in terms of

Chapter 2 Literature Review

grid generation and also avoids the trouble of having to take care of the interface between the grids However, the problem with Chimera technique is that the conservation properties of the governing equations are not satisfied through the overlapping region This may affect the accuracy of the solution Moreover, if an individual body geometry is very complex, the problem of mesh generation remain unresolved unless multiple embedded meshes are used on a single component, as in the flexible mesh embedding techniques (FAME) approach13, 14

The second type of grids used for flow computation is the unstructured grids Unstructured meshes are typically constructed from triangles in two-dimensions or tetrahedral cells in three-dimensions (e.g., Refs 15-19) The main advantage of traditional unstructured grids is the ease of grid generation about complex configurations since the cells may be oriented in any arbitrary way to conform to the geometry It is popular because triangular and tetrahedral grids can be generated automatically, independent of the complexity of the domain Significant progress has been made in the area of automatic tetrahedral mesh generation which sees many commercial CFD software using unstructured grids for their computation Another advantage associated with the use of unstructured grids is the relative ease of local refinement, i.e refining the grid in the regions where it is necessary Local or adaptive refinement for unstructured grid can be handled in a relatively seamless manner They provide a natural setting for adaptation with no major changes to the flow solver20, 21

However, the computational time and cost for unstructured mesh computations are

Chapter 2 Literature Review

grid points are suitably ordered and neighboring points can be easily identified using their grid index, unstructured mesh computing requires sophisticated data structure to store the relation of the faces to the cells or the nodes to the elements in order to identify their neighboring cells Hence, unstructured grid methods require much higher memory storage as compared to structured grid approaches This is especially so when applied to large-scale three-dimensional problems, where unstructured mesh method tends to be less efficient

Another problem with unstructured mesh is that the ease with which complex configurations can be meshed is often offset by the difficulties associated with the generation of highly stretched tetrahedral meshes Stretched meshes are necessary for the efficient computation of certain inviscid flows, such as flows over wings, and for all flow containing thin viscous layers Different approaches such as the use of hybrid grid or mixed grid have been adopted in an attempt to address this problem Hybrid grid is composed of structured and unstructured zone, while mixed grid is made up of different elements (prisms, tetrahedral, pyramids, etc.) The problem with hybrid grid

is that two different flow solvers are needed – one structured and one unstructured, which has to be coupled Mixed grid is favored over hybrid grid, but its challenge is to develop data structures and numerical schemes for the flow solver, which can handle varying element types in a seamless way The generation of mixed grids is also non-trivial for geometrically demanding cases Although some degree of success has been achieved22, 23, these two approaches still suffer from disadvantages, such as the requirement for excessive user intervention and the generation of low quality meshes

A detailed review of various methodologies for spatial and temporal discretization on unstructured grids is discussed in Ref 24

Chapter 2 Literature Review

In recent years there is a renewed interest in the conceptually simple approach of Cartesian grids25-27 Both structured and unstructured methods of solution can be applied to grids created by this means Since the geometrical complexity of flow simulations is rapidly increasing, the ability to generate grids fast and with minimum user interaction becomes more and more important For simple geometry, it may not be economical to employ a Cartesian method since the number of grid points required to ensure sufficient resolution is higher than body-fitted grid method However, to be used in an embedded grid manner there is great potential for a Cartesian method in dealing with complex configurations when grid generation becomes the limiting factor for body-fitted grid method

Cartesian grid possesses all the advantages associated with structured grids which include ease of grid generation, lower computational storage requirements, and significantly less operational count per cell compared to body-fitted schemes The numerical solution of the equations of fluid dynamics is greatly simplified if the discretization is performed in a Cartesian coordinate system In addition, Cartesian solver has better convergence properties since there are no problems related to skewness or distortion of cells Embedded grids, adaptive local grid refinement, and other grid refinement strategies could be used to provide better resolution of geometry and flow features (e.g., Refs 28-34) Wu30 offers an anisotropic refinement strategy that significantly reduces the number of computational cells as compared to traditional isotropic refinement strategy Computational efficiency can also be greatly improved

Chapter 2 Literature Review

Furthermore, Cartesian grids offer considerable ease in implementing higher order schemes

Cartesian grid methods have been used with significant success for computing flows over complicated geometries In 1993, De Zeeuw and Powell35 presented an adaptively

refined Cartesian mesh solver for the Euler equations In 1994, both Pember et al.36

and Quirk37 presented adaptive Cartesian mesh approaches for the solution of the Euler equations In 1995, Coirier and Powell38 investigated the accuracy of Cartesian mesh approaches for the Euler equations and in 1996 they39 presented a solution adaptive approach for both viscous and inviscid flows in two-dimensions

Yang and co-workers have also focused on Cartesian mesh methods for the Euler equations in compressible flow In 1997 they presented a method for compressible flows for static and moving body problems40, 41, which they extended to three-dimensions in 200042 A similar approach was used by Causon et al.43 in 2000 for the

solution of the shallow-water equations In 1997, Almgren et al.44 presented a Cartesian grid projection method for the incompressible Euler equations in complex geometries In 1998, Johansen and Colella45 presented a second-order accurate method for solving Poisson’s equation on irregular two-dimensional domains This approach

was extended in 2001 by McCorquadale et al.46 to the solution of the time dependent heat equation An assessment of the accuracy of Cartesian mesh approaches has been made by Coirier and Powell38 These results have demonstrated the great potential of Cartesian methods in handling flow with complex geometries

Chapter 2 Literature Review

However, the main challenges in using a Cartesian method are the problems in dealing with the arbitrary boundaries As the grids are not body-aligned, Cartesian cells near the body can extend through surfaces of solid components The boundaries of the complex geometries are generally approximated as a series of staircase steps Hence, accurate means of representations for surface boundary conditions are essential for the success of Cartesian schemes Different methods have been proposed in the literature

to resolve the boundary conditions either using unstructured or structured grid techniques Broadly the methods proposed involve either cut cells for a finite-volume treatment, grid points for a finite-difference construction or immersed boundary approaches

2.3.1 Boundary condition using finite volume cut-cell method

A Cartesian cut cell mesh is generated by “cutting” solid bodies out of a background Cartesian mesh This result in the formation of fluid, solid and cut cells and the boundaries is represented by different types of cut cell Using proper interpolation strategies the flow variables on the resulting irregular cells can be computed according

to the boundary conditions on the body This method allows a clear distinction between the solid and the fluid by practically generating a boundary-fitted grid around the body In this way, the difficult and case-specific problem of generating a structured

or unstructured mesh is replaced by the more general problem of finding the intersection points between a surface geometry and a background Cartesian mesh Recent successful applications of the cut cell method to two-dimensional inviscid and viscous flow problems can be found in Refs 35, 37, 47 and 48

Chapter 2 Literature Review

boundary This lead to the formation of various irregular “interface-cells” as described

in Quirk37 Although a cut-cell method ensures conservation, but the task of computing the volume and fluxes for all the irregularly shaped cut cells entails a considerable increase in complexity More crucially, the method can at times lead to the creation of very small cells at the boundary which poses problems of numerical stability

To alleviate this problem, rule-based cell merging techniques were used to combine cut

cells that are too small with neighboring cells as in the work of Clarke et al.49 and Ye

et al.50 Udaykumar et al.51 also extended the formulation to treat moving boundaries with good results for a variety of two-dimensional problems Another cell merging

approach proposed by Kirskpatrick et al.52 in three-dimensional uses the concept of

“master/slave” pair to link two cells and treat each cell as a distinct entity instead of merging two cells to form a single cell Other three-dimensional problems which uses cut-cell formulations are also presented in Refs 44, 45 and 53

While these approaches ensure strict conservation properties, cell formation and merging is not straightforward in three-dimensions and will result in high computational overhead Furthermore it is difficult to ensure that the local geometric properties are in full consistency with the original shape particularly when coarse grids are being used for multigrid implementation However, the cut cell approaches in principle still possess the potential to greatly simplify and automate the difficult task of mesh generation It can also provide a method that can deal efficiently with steady or unsteady compressible flows around arbitrarily complex geometries, which are either stationary or in relative motion without having to remesh the grid

Chapter 2 Literature Review

2.3.2 Boundary condition using finite difference ghost cells method

In a finite-difference approach, boundary conditions are enforced by an interpolation procedure28, 29, 54 through the use of ghost cells or halo points Ghost cells are defined

as cells in the solid that have at least one neighbor in the fluid For each ghost cell, an interpolation scheme that implicitly incorporates the boundary condition on the boundary is then devised A number of options are available for constructing the interpolation scheme55 For instance, in the work of Epstein et al.28, cut cells were handled by an extrapolation procedure in which halo points are used at the boundary

Tidd et al.29 also used a finite difference formulation to treat the boundary conditions They introduced the concept of a “multi-value dummy point”, where some nodes may contain more than one value defined These are nodes that belong to the finite difference stencil of another node, but are hidden from the other node by the surface geometry These nodes usually occur at thin surfaces (e.g., a trailing edge) and require special treatments Another technique using reflected ghost cells is that of Dadone and Grossman56, 57 In their work, flow properties at the ghost cells are determined based on

an assumed flow field model in the vicinity of the wall consisting of an isentropic vortex of constant total enthalpy, which satisfy the normal momentum equation such that normal velocity at the wall vanishes A wide variety of flow simulation including compressible flow past a circular cylinder and an airfoil using the ghost cell method is

also demonstrated in Ghias et al.58

However, unlike the cut cells method, this method is not conservative To minimize the lack of conservation, higher order representation for the boundary is used to

Chapter 2 Literature Review

need for special treatment at nodes where multiple values are defined at a particular point28, 29 is still problematic when considering thin geometry Because of the complexity involved, the method is unpopular among current researchers

2.3.3 Boundary condition using immersed boundary method

For the immersed boundary methods, the effect of a stationary or moving boundary is accounted for by introducing an external force field in the equations of motion such that the fluid satisfy the boundary conditions on the solid boundaries The method was first used by Peskin and McQueen60-63 to study blood flow in the human heart To avoid numerical instabilities due to the stiffness of the problem, a delta function was used to distribute the forcing over 3−4 grid nodes in the vicinity of the boundary The spreading of the forcing function results in smearing of the interface and increases the spatial resolution requirements This appears to be a major obstacle in extending the method to viscous computation To overcome this limitation, LeVeque and Li64, 65

proposed the immersed interface method (IIM) This method modifies the governing

equations at grid nodes only in the immediate vicinity of the interface by adding forcing functions constructed to enforce a set of appropriate jump conditions at the interface It maintains sharp-interfaces between different phases and is second-order

accurate The IIM was initially applied to solve 2D elliptic equations and Stokes flow

problems with flexible boundaries and is recently extended to solve the 2D, incompressible Navier–Stokes equations66

Another approach recently proposed by Mohd-Yusof and Fadlun et al.67, 68 does not require the explicit addition of discrete forces to the governing equations Instead, it applies boundary conditions at the grid nodes closest to the solid boundary The specific values of various flow variables at such near-boundary nodes are calculated by

Chapter 2 Literature Review

interpolating linearly along an appropriate grid line between the nearest interior node, where flow variables are available from the solution of the governing equations, and the point where the grid line intersects the solid boundary, where physical boundary conditions are known This method can be thought of as accounting for the presence of

a solid boundary at the grid nodes nearest to the boundary In this sense, it is

conceptually related to the IIM

This approach is very promising for 3-D flows and has been successfully applied to a variety of problems The success of the implementation depends greatly on the choice

of the solution reconstruction scheme near the interface In Fadlun et al.68, a simple one-dimensional scheme was suggested where the solution is reconstructed along an arbitrarily selected grid line The method is straightforward and works well for bodies that are largely aligned with the grid lines As for complex bodies, there is ambiguity since several points will be produced This gives rise to the need for multidimensional

scheme Kim et al.69 suggested a bilinear reconstruction procedure Recently,

Gilmanov et al.70 introduced a more general scheme, which is applicable to complex three-dimensional boundaries without special treatments They discretized the surface with triangular mesh and interpolated the solution along the normal to the body

direction Gilmanov et al.70 reported nearly second-order spatial accuracy and excellent agreement with benchmark computations on body-fitted meshes

A group of ‘meshless’ or ‘gridless’ methods are being increasingly proposed in the past two decades as an alternative to conventional grid-based methods for the solution

Chapter 2 Literature Review

opposed to mesh-based methods, do not require pre-specified connectivity between grid points The main requirement from the meshless points is their individual locations, which is why these methods are able to negotiate irregular boundaries with ease and flexibility There is no need to determine any cell edges, sides, faces, surfaces

or vertices

The numerical analysis process for meshless schemes consists first a generation of a distribution of points, termed global cloud of points, within the analysis domain This distribution of points in the domain can be obtained from either one or more of the available grid generations Hence, a meshless solver can be used on all types of grids

or point distributions71 For each of these points, a local cloud of neighboring points is then selected A local approximation is chosen for the unknowns to be found in terms

of the point values using typically least-squares procedures Finally, the derivation of a discrete set of algebraic equations is obtained by substituting the point approximations into the governing partial differential equations of the problem

A number of meshless methods have been proposed so far These include the smooth particle hydrodynamics (SPH) method72, the generalized finite difference (GFD) method73, 74, the diffuse element (DEM) method75, the element free Galerkin (EFGM) method76, the reproducing kernel particle method (RKPM)77, the moving least-squares reproducing kernel method (MLSRK)78, the partition of unity method (PUM)79, the hp-clouds method80, the finite point method (FP)81 and the meshless local Petrov-Galerkin (MLPG) method82 The meshless methods can be grouped into integral and non-integral types The integral type involves performing the numerical integration to solve the weak form of the partial differential equations Examples of this are the DEM,

Chapter 2 Literature Review

EFGM, hp-clouds, etc The non-integral type involves transforming the strong form of partial differential equations into a system of linear or non-linear algebraic equations and solving the resultant algebraic equations FP and GDF are examples of this type Such formulations have been implemented successfully within the field of fracture mechanics and crack propagation76

Meshless methods for compressible flow where the governing equations are hyperbolic

have been developed by Deshpande et al.83, 84, Batina85, Morinishi86, and Sridar and Balakrishnan87, all of which belong to the non-integral type and make use of a least-squares procedure to solve the resulting over-determined system of equations They differ from one another in the way they introduce upwinding The work of Deshpande

et al.83 deals with an upwind implementation based on the Kinetics Theory of Gases, resulting in a least-squares kinetic upwind method (LSKUM)84, while that of Batina85

is based on a centered scheme using artificial dissipation

Batina85 pioneered the use of a pure gridless method for compressible fluid dynamics However, there are issues associated with global conservation of mass, momentum and energy in the main flow field where shock waves may occur In addition, such methods tend to be less efficient compared to conventional methods using structured meshes due to extra work in the construction of the least-squares fitting over clusters

of grid points It is also not straightforward to implement acceleration techniques such

as the multigrid method Consequently, such pure gridless methods have not found wide use in the computational fluid dynamics community despite its purported flexibility

_ Chapter 3 Numerical Schemes and Methodology

Chapter 3

Numerical Schemes and Methodology

An important feature of Cartesian approaches is the elimination of the need to create a case-specific, body-fitted surface mesh which is human intensive In place of this, a more computer centric solution is used, involving a technique to handle the intersections between hexahedral mesh cells and body surfaces This is significant when handling complex configurations especially in three dimensions, when mesh generation becomes a limiting factor for body-fitted grids

3.1 Introduction

In this work, a different Cartesian method is proposed An attempt is made to combine the advantages of the conventional Cartesian grid method and the gridless method while avoiding their shortcomings Unlike Batina’s approach85, where the gridless technique is used for the whole flow field, we use a Cartesian method to solve the flow equations for the grid points in the interior of the computational domain where the finite-difference or finite-volume stencils are complete The gridless method, however,

is used to implement boundary conditions and treat grid points for which the difference or finite-volume stencil is not complete The Cartesian method is used over the bulk of the computational domain for its efficiencies while the gridless method is adopted for its flexibility in handling the complex arbitrary distribution of grid points near solid boundaries The gridless method is only applied to a layer of cells around

_ Chapter 3 Numerical Schemes and Methodology

the object instead of the entire flow field In this way, a general solution method that is flexible, efficient, and accurate for problems with complex geometries is achieved A similar approach to the present work was reported by Kirshman and Liu88, who used a finite difference scheme with Van Leer flux splitting technique

In this work, a finite-volume formulation with central differencing for the Euler equations is used and a simple gridless approach is employed in discretizing the surface In the next sections, the numerical method that combines the Cartesian method for the two-dimensional Euler Equations and gridless method for treating boundary conditions are presented Determination of the gridless cloud points for the boundary and the spatial discretization for the flow field and boundary nodes are also detailed

Euler equations are obtained from the Navier-Stokes equations by neglecting all shear stresses and heat conduction terms This is a valid approximation for flows at high Reynolds numbers outside the viscous region developing in the vicinity of solid surfaces The set of Euler equations allows discontinuous solutions in cases of shock waves occurring in supersonic flows Since the gradients of the fluxes are not defined

at discontinuity surfaces, the solutions across discontinuities can be represented via an integral approach 89

_ Chapter 3 Numerical Schemes and Methodology

computer program to solve the above system of equations is based on a version of

FLO52 by Jameson

3.2.1 Governing Equations

The two-dimensional Euler equations consisting of the mass, momentum, and energy

conservation laws that govern the motion of an inviscid fluid may be written in integral

form as

0

∫+

, (1)

where Ω denotes volume (area in two dimensions); Fr

is the flux vector; dS is a

surface element and nr is the outward normal; and

u U

ρρρρ

+ρ

ρ

=

uH uv

P u

u f

ρ

=

vH

P v vu

v

where f and g are the x and y components of the flux vector Fr

Here, p, P, u, v, E and

H denote the pressure, density, Cartesian velocity components, total energy and total

enthalpy For a perfect gas, ( u v )

) (

ρ+

=E P

H , where γ is the

ratio of specific heats

3.2.2 Spatial discretization

Spatial discretization of the Euler equations for an interior node in the Cartesian grid is

performed using a second-order cell-centered finite-volume method (FVM)

Considering the control volume for a cell abcd as shown in Figure 1, Eq (1) can be

written as follows,

0)

gdx fdy Udxdy

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This expression can be approximated as

(S i,j U i,j)+Q i,j =0

dt

d

, (4)

where Si,j is the cell area and Qi,j is the net flux out of the cell Let (i, j) denote the cell

center of such a grid with grid spacing x∆ and ∆y , in the x and y directions,

respectively In a cell-centered finite-volume scheme, the dependent variables U are

known at the center of each cell and the fluxes are evaluated by taking the average of

the values in the cells on either side of each edge

On a Cartesian, uniform grid, this reduces to the following central-difference scheme

which is second-order accurate in space,

02

2

1 , 1 , , 1 ,

1

∆

−+

x

f f

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3.2.3 Artificial dissipation

The Euler equations constitute a set of non-dissipative hyperbolic conservation laws

Discritizing the Euler equations using central finite volume method lead to odd and

even point decoupling An addition of artificial dissipation terms serves to suppress a

tendency for odd and even point decoupling as well as to prevent oscillations near

shockwaves An adaptive scheme using 2nd and 4th order differences for adding

dissipation is used

The adaptive scheme for adding dissipation by Jameson et al.90 has proven to be

effective in practice in numerous calculations of complex steady flows In this scheme,

a fourth order dissipative term is added throughout the domain to provide a base level

of dissipation sufficient to prevent nonlinear instability However, this is not enough to

prevent oscillations in the neighborhood of shock waves Thus near a shock wave

region a switch to second-order dissipative terms is needed to capture the shock This

switch from third to first order is signaled using local pressure gradients

* ,

* , 1

* , 2

4 , ,

* ,

* , 1

2 , , , , ,

33

2

2 1 2

2

j i j j

i j i j i

j i j i j i j i j i

U U U

U

U U

d

− +

+ +

+ + +

+

−+

−

−

−

=ξ

ξ ξ

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(9)

H v u

U* = ρ,ρ ,ρ ,ρ )

Similar expressions can be written for the other directions and are omitted here The

dissipation terms are added to all four equations, but in the energy equation the fourth

component, ρE, is replaced by ρH This ensures that the steady state satisfies H = H∞ =

constant

The term λ in Eq (8) is a measure for the inviscid fluxes and is defined as

j i j i j

,

2 2

ξ,i ,j λ ,i ,j λ ,i ,j

2 2

i j i j i

j i j

i j i j i

S a S

V

S a S

V

, , ,

, , ,

,

, , ,

, , ,

,

2 1 2

1 2

1 2

1

2 2

2 2

+ +

+ +

+ +

+ +

∆+

∆

⋅

=

∆+

∆

⋅

=

η η

η

ξ ξ

ξ

λ

λ

(12) are the spectral radii of Jacobian matrices associated with the ξ and η directions For

the Cartesian grid, ξ and η directions coincide with x and y directions respectively In

Eq (12), V denotes the velocity vector, a is the speed of sound, and ∆S is the cell face

vector

The purpose of using the scaling factor defined in Eq (11) is to avoid excessive

dissipation for cells with high aspect ratios but to maintain effective damping The

parameter ω can take values between 0 and 1 A value of ω = 1.0 is used for a grid of

low aspect ratio (inviscid flows)

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The coefficients ε(2)and ε(4) in Eq (8), control the amount of second-order and

fourth-order diffusion They are chosen in a self-adaptive way so as to produce a low level of

diffusion in regions where the solution is smooth, but prevent oscillations near

discontinuities Their local values are determined by an appropriate pressure sensor for

the presence of shock waves:

j i j i j i

j i j i j i j i

P P P

P P P

, 1 , , 1

, 1 , , 1 ,

2

2

− +

− +

++

) 2 ( )

2 ( ,

),

0max( ( 2 )

, , ) 4 ( )

4 ( ,

κ By this definition, ε(2)is significant near shock regions and

first order dissipation is used ε(4)is significant where the flow is smooth and third

order dissipation is used

3.2.4 Temporal discretization

The Runge-Kutta multi-stage time integration90 is applied for both the fluid and

boundary nodes in the same routine No separate treatment is required for the boundary

nodes The modified multi-stage Runge-Kutta scheme is used as it gives the best ratio

of allowable time step to computational work per time step The integration is given as

follows,

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4 ,

1 ,

3 ,

0 ,

4 ,

2 ,

0 ,

3 ,

1 ,

0 ,

2 ,

0 ,

0 ,

1 ,

,

0 ,

213141

j

n j i

j j i j i

j i j i j i

j i j i j i

j i j i j i

n j i j i

u u

R S

t u

u

R S

t u

u

R S

t u

u

R S

t u

u

u u

where Di,j contains all the dissipative terms To reduce the computational cost of

evaluating dissipative terms at each stage of the Runge-Kutta time integration, Di,j is

calculated at the first two stages and is frozen for the remaining stages

The scheme is explicit with a CFL condition of CFL≤2 2 and has fourth-order

accuracy in time for a non-linear equation For steady state solutions, local time

stepping is used to accelerate convergence With this, the time step for each cell varies

and is based on applying the CFL criterion to the local flow condition

3.2.5 Far field boundary condition

In the far field, the usual characteristic analysis based on one-dimensional Riemann

invariants is used to determine the values of the inviscid flow variables The normal (to

the cell face) velocity component and the speed of sound are defined as

)(

5.0

_ Chapter 3 Numerical Schemes and Methodology

)1(2

and at the inflow boundary,

n n q q q

The gridless method involves the use of only clouds of points and does not require the

points to have any pre-specified connectivity with one another nor the formation of

meshes or cells This eases the boundary implementation for the Cartesian grid since

the grid is not body-aligned and the cells are cut by the object surface These randomly

cut cells are handled in an unstructured manner separately by the gridless method In

this way, there is greater flexibility in dealing with objects with complex geometries

As with any gridless scheme, it is important to first classify the nodes near the surface

before setting up the cloud points This information is provided as preprocessor work

before running the code The gridless strategy adopted here involves treating fluid

nodes along the boundary of the object and surface nodes on the object separately The

boundary nodes require the Euler fluxes to be determined while the surface nodes need

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to satisfy the boundary conditions These will be explained in details in the following sections

3.3.1 Nodal definition

Four types of nodes near the solid boundary are identified as shown in Figure 2 Type

1 nodes are nodes whose Euler fluxes can be computed easily in the finite-volume method since the values of all four neighbors (left, right, top and bottom) are defined, i.e., the computational stencil for the finite volume is complete in the 2nd-order scheme

These are termed fluid nodes Type 2 nodes are nodes that have cut cells as neighbors and are termed boundary nodes For these nodes, the finite-volume method cannot be

applied and a gridless method with the use of a least-squares technique will be applied

to compute the Euler fluxes The fluid and boundary nodes made up the computational nodes Type 3 nodes are nodes whose cells are cut by solid wall surfaces and are

termed cut-cell nodes and Type 4 nodes are surface nodes that are defined on the body

Boundary and surface nodes are managed using an unstructured data approach

The procedure for boundary and surface nodes determination is listed as follows:

1 Cut cells with reference to the Cartesian grid and surface data points are identified

A feasible method for this is to use Alternate Digital Tree (ADT) approach92whereby the inter-grid information is classified in a tree structure for quick identification With the cut cells found, solid cells that fall completely inside the object are marked out and boundary nodes are identified

2 A set of well-distributed points on the surface is obtained by `latching’ the center

of each cut cell on to the surface of the object to obtain the surface nodes This is

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cells Details of the NURBS representation used can be found in Ray and Tsai93 With this latching method, the interval between the surface nodes will be approximately the same as the grid cells interval The use of a set of well-distributed points will help to reduce the possibility of ill-conditioned matrices in the least-squares computation

3 Normal and tangent vectors for the surface nodes are found using the original data inputs of the object instead of the computational points on the surface for better accuracy The radius of curvature at these nodes is also defined for boundary implementation At the tip of the airfoil trailing edge, these values are obtained by extrapolation to avoid singularity

Type 1: Fluid nodes Type 2: Boundary nodes Type 3: Cut cell nodes Type 4: Surface nodes

Figure 2: Classification of grid nodes

3.3.2 Selection of cloud points

In this boundary implementation, two types of cloud points are used The first type of cloud points is set up for each boundary node to compute Euler fluxes whereas the second type of cloud points is set up for each surface node, for boundary condition implementation and surface value determination For each boundary node, a cloud of 9

_ Chapter 3 Numerical Schemes and Methodology

points with a mixture of fluid nodes, boundary nodes and surface nodes are selected

from its vicinity This is determined using a 3x3 stencil over the boundary node as

shown in Figure 3a Fluid nodes and boundary nodes that fall in the stencil are

included while cut cells and solid cells are omitted These are replaced by surface

nodes that are closest to the boundary node and have outward normals pointing in the

direction of the selected boundary nodes This criterion is set to avoid choosing nodes

from the wrong side of the profile at thin surfaces as illustrated in Figure 3b

a) b) Figure 3: Cloud points for (a) boundary nodes; (b) boundary nodes at thin surfaces

For each surface node, a one-sided cloud comprising only fluid nodes and boundary

nodes is defined The selected cloud points must fall on the same side as the outward

normal direction of the surface node and the selection is based on the shortest distance

to the node As such, surface values are evaluated using computational nodes on the

side of its outward normal as illustrated in Figure 4 A mixture of different types of

nodes in each cloud ensures the transfer of information between these nodes

nr

nr nr nr

_ Chapter 3 Numerical Schemes and Methodology

nr

t

r

Figure 4: Cloud points for surface nodes (Closest 8 points to surface node)

3.3.3 Boundary node treatment

As boundary nodes do not have a complete stencil for FVM, a gridless method is used

to compute the Euler fluxes for these nodes The boundary nodes are treated separately

This makes the method easy to be implemented into any scheme Using the cloud

points defined for each boundary node as given above, the governing equation is

solved through a local least-squares curve fit using these points In each cloud, each

term of the fluxes as defined in Eq (2) is assumed to vary according to a function

xy a y a x a a y x

i i

i i i

i i i i i i i i

i i i i i i

i i i i i i

i i i i

h y x

h y

h x h

a a a a

y x y x y x y x

y x y y x y

y x y x x x

y x y x

N

3 2 1 0

2 2 2 2

2 2

2 2

where N is the number of points in the cloud

The spatial derivatives can be found by differentiating Eq (25) and are given as

follows,

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x a a y

h y

a a x

h

3 2 3

∂

∂+

=

∂

∂

(27)

Hence by determining the values of a1, a2 and a3, the spatial derivatives are found and

the Euler fluxes evaluated Although simple to implement, the gridless method is not a

conservative method as pointed out by Batina85 However, by applying the gridless

method only on boundary nodes, non-conservation is thus restricted to a layer of cells

surrounding the object The rest of the computational domain is computed using a fully

conservative finite-volume formulation The lack of conservation can be minimized

with the use of sufficient resolution near the surface

The artificial dissipative terms for boundary nodes are computed in the same manner

as the fluid nodes Special treatment is required to approximate the cut-cell values

using the surface node values for the computation of local pressure gradient Tests

show that setting the cut cell values to zero will result in large oscillations near the

surface boundary due to an underestimation of the local pressure gradient

Both the fluid nodes and the boundary nodes are marched in time using the

Runge-Kutta multi-stage time-stepping scheme while the values of the surface nodes are

determined in a least-squares fashion Surface boundary conditions are embedded in

the computation of the flow variables on the surface of the body as described below

3.3.4 Implementation of surface boundary conditions

The main challenge in using Cartesian grids is in the implementation of surface

boundary conditions as the grids are not body aligned A gridless method with a

least-squares technique is used to implement the surface boundary conditions Surface node

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The boundary conditions are implemented such that in the process of solving the

surface values, the boundary conditions are satisfied

Under inviscid conditions the flow slips over the surface and is hence tangential to it

The slip condition is thus imposed on the boundary and the other flow variables on the

surface nodes are approximated as follows:

R

V n

where R is the local surface radius of curvature; the subscripts n and t stand for the

normal and tangential directions, respectively, along the surface By taking the

curvature of the body into consideration, better accuracy is attained

As shown in Figure 5, the cloud for each surface node contains only one surface node

and the values of flow variables at the other cloud points are resolved in the tangential

(ξ) and normal (η) directions of that surface node (here ξ and η are used in place of t

and n.) This is for easy implementation of the boundary conditions since they are given

in the ξ and η directions