EULER SOLUTION USING CARTESIAN GRID WITH LEAST … · 2017-08-23 · Other mesh refinement...

6−9 January 2003 41st

AIAA-2003-1120 Euler Solution Using Cartesian Grid with Least Squares Technique

E. P. C. Koh and H. M. Tsai Temasek Laboratories National University of Singapore Singapore 119260

F. Liu Department of Mechanical and Aerospace Engineering University of California, Irvine Irvine, CA 92697-3975

EULER SOLUTION USING CARTESIAN GRID WITH LEAST SQUARES TECHNIQUE

E. P. C. Koh1 and H. M. Tsai2

Temasek Laboratories National University of Singapore

Singapore 119260

F. Liu3 Department of Mechanical and Aerospace Engineering

University of California, Irvine CA 92697-3975

Abstract subdomains exist. They include methods using unstructured meshes, body-fitted curvilinear meshes, and Cartesian meshes.

This paper discusses an approach that uses “gridless” or “meshless” methods to address the boundary or interface while standard structured grid methods are used everywhere else. The present method uses the Cartesian grid to specify and distribute the computational points on the boundary surface but not to define the geometrical properties. Euler fluxes for the neighbors of cut cells are computed using the gridless method involving a local least-squares curve fit to a “cloud” of grid points. The boundary conditions implemented on the surface points are automatically satisfied in the process of evaluating the surface values in a similar least-squares fashion. No halo points are needed. The overall scheme is robust, stable and converges well for a range of Mach numbers tested. Solutions from the proposed approach are computed for the NACA 0012 and RAE 2822 airfoils and the results are compared with those obtained by a standard Euler solver using body-fitted grids. For grids with equal resolution the method is less accurate for capturing shocks but an improvement in resolution of 60% gives a sharper shock front. The approach offers a number of advantages and its extension to three dimensions is straightforward.

Unstructured meshes are typically constructed from triangles in two-dimensions or tetrahedral cells in three-dimensions (e.g., Refs. 1-4). 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. However, the computational time and cost for unstructured mesh computations are generally higher than the structured mesh approach. The unstructured mesh method tends to be inefficient when applied to large scale three-dimensional problems. In addition, mesh quality is also not easily controlled which will impact convergence and accuracy. Body-fitted hexahedral grids, which belong to the structured grid family, have been widely used. The 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. Highly distorted or skewed cells may occur in some regions of the domain that adversely influence the computation. Moreover, often it is not possible to use a single grid for very complex geometries. More sophisticated methods such as multiblock, patched grids, Chimera or hybrid-type grids are proposed and used in varying degrees of successes (e.g., Refs. 5-7). These methods require transfer of information between the different meshes.

Introduction

A continuing obstacle of Computational Fluid Dynamics (CFD) for configurations with complex geometry is the problem of mesh generation. Several alternatives for dividing the flow field into discrete

Regardless whether an unstructured or a structured method is used, the grid generation involves both surface preparation and griding before the volume grids can begin. In recent years there is a renewed interest in the conceptually simple approach of Cartesian grids8-10. Both structured and unstructured methods of solution can be applied to grids created by this means. It possesses the advantages associated with structured grids and some of its attractions include ease of grid

––––––––––––––––––––––– 1. Associate Scientist, email: [email protected] 2. Principal Research Scientist, email: [email protected] Member

AIAA. 3. Associate Professor, email: [email protected] Senior Member AIAA. Copyright 2003 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc with permission

1 American Institute of Aeronautics and Astronautics

mailto:[email protected]



generation, lower computational storage requirements, and significantly less operational count per cell compared to body-fitted schemes. The convergence properties of the solver are comparatively better since there are no problems related to skewness or distortion of cells. Furthermore, Cartesian grids offer considerable ease in implementing higher order schemes. 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. The cells of a Cartesian mesh near the body can extend through surfaces of solid components. Accurate means of representations for boundary conditions in cells that intersect surfaces are essential for successful Cartesian schemes. An important feature of the Cartesian approach is the elimination of the need to create a case-specific body-fitted surface mesh which is human labor intensive, and replaces it by a more computer centric solution involving a technique to characterize the intersections between hexahedral mesh cells and body surfaces. The flow grid can be generated independently and the cells of the Cartesian mesh that intersect the surface can be determined without having first to mesh the surface. There is no explicit relation between the surface description and the local flow grid cell structure. The task of surface description, which is unavoidable, is to resolve the geometry, and no longer include tedious surface grid generation process. To ensure adequate representation without the use of excessive grid points, adaptive mesh refinement of the Cartesian grid can be used. A Cartesian based algorithm must first identify cells that intersect solid surfaces and remove or flag cells that are completely enclosed within the solid body. These would not be considered as part of the flow computational domain. Solution using the Cartesian method usually consists of a standard method for the regular cells and a special treatment for the boundary cells. In the literature solvers involving either unstructured or structured grid techniques are reported. Different methods have been proposed in the literature to resolve the boundary conditions. Broadly the methods proposed involve either cut cells for a finite-volume treatment or grid points for a finite-difference construction. A cut-cells method ensures conservation and the same flux computation algorithm is applied. However, 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 neighbouring cells such as in the work of Clarke et al.11 and Ye et al.12. While this approach ensures strict conservation properties, cell formation and merging is not straightforward in three-dimensions. Also it is difficult to ensure that the local geometric properties are in full consistent with the original shape particularly when coarse grids are used as in the implementation of multigrid. The introduction of the finite-difference approach on the other hand avoids all the problems associated with cut cells. For instance, in the work of Epstein et al.13,14,15, cut cells were handled by an extrapolation procedure in which halo points are used at the boundary. Special treatment is needed to ensure that halo points are available at regions with thin surfaces (e.g., a trailing edge) where multiple values are defined at a particular point 13,14. Unlike the cut cells method, this method is not conservative. Another way to improve accuracy is to use higher order representation for the boundary as in the work of Forrer and Jeltsch16 to minimize the lack of conservation. To achieve an accurate solution for the boundary, Cartesian methods require a mesh refinement strategy to resolve the geometry. Wu17 offers an anisotropic refinement strategy that significantly reduces the number of computational cells as compared to traditional isotropic refinement strategy. Other mesh refinement strategies for Cartesian methods can be found in the literature (e.g., Refs. 18-21). In this paper, a different Cartesian method is described for solving the Euler equations. It uses “gridless” or “meshless” methods to address the boundary or interface while standard structured grid methods are used everywhere else. This is different from Batina’s approach22 and other workers who have proposed a completely gridless technique where the governing equations are solved by a local least-squares curve fit using clouds of grid points. A similar approach to the present work was reported by Kirshman and Liu23, who used a finite difference scheme with Van Leer flux splitting technique. In the present work, a finite-volume formulation with central differencing is used with artificial dissipation. Also a different approach is employed in discretizing the surface. The Cartesian grid is used to define and distribute the computational points on the boundary surface. These surface points spacing which are more or less comparable to the Cartesian grid points are better distributed to reduce the possibility of ill-conditioned matrices in the gridless computation with cloud points. Since accuracy is an important issue at the boundary, geometrical properties are defined using the original data of the geometry and not the surface computational points, which may be too coarse. A simplified overall implementation of boundary


condition is adopted. Euler fluxes for the neighbors of cut cells are computed using a ‘gridless’ method and the resulting derivatives are analytically determined for the flow equations. Boundary conditions implemented on the surface points are automatically satisfied in the process of evaluating the surface values by a similar least-squares method. In the next section, the numerical implementation of the Cartesian method for the two-dimensional Euler Equations will be presented. Determination of the gridless cloud points are described followed by the spatial discretization for the flow field and boundary nodes. The discretized Euler equations are solved using the modified four-stage Runge-Kutta scheme developed by Jameson et al24,25.

Numerical Method In this study, the governing equations used are the two-dimensional Euler equations. The Euler equations govern inviscid, incompressible fluid motion and may be written in integral form as

0. =+Ω∂∂

∫∫∫Ω

SdnFUdt

rr (1)

where Ω denotes the volume (area). Fr

is the flux vector: ys lglfF

rrr+= , dS is a surface element and n

r is

the outward normal.

(2)

=

Evu

U

ρρρρ

+

=

uHuv

Puu

f

ρρ

ρρ2

+=

vHPv

vuv

g

ρρρρ

2

Here, P, ρ, u, v, E and H denote the pressure, density, Cartesian velocity components, total energy and total enthalpy. For a perfect gas,

)(21

)1(22 vuPE ++

−=

ργ (3)

ρPEH += (4)

where γ is the ratio of specific heats. Equation (1) represents a system of equations corresponding to the conservation of mass, momentum, and energy. The present method is distinguished from other Cartesian grid approaches in that the ‘gridless’ or ‘meshless’ method is only applied to cells that are in the vicinity of a solid boundary. All other cells will be discretized using the conventional Finite-Volume Method (FVM). With this approach, boundary values

are incorporated into the computation directly, which ensures the success of the method as a whole. Boundary and Surface Nodes Definition Four types of nodes near the solid boundary are identified as shown in Figure 1. Type 1 nodes are nodes whose Euler fluxes can be computed easily using FVM since the values of all four neighbors (left, right, top and bottom) are known and the stencil for FVM is complete. These are termed Fluid nodes. Type 2 nodes are nodes that have cut cells as neighbors and are termed boundary nodes. For these nodes, FVM for flux computation cannot be applied and a gridless method with the use of a least-squares technique will be applied to compute the Euler fluxes. Type 3 nodes are nodes that are being cut by the surface and are termed cut cell nodes and Type 4 nodes are surface nodes that are defined on the body. The boundary points and surface points are managed using an unstructured data approach. Both the fluid nodes and the boundary nodes will be marched in time using the Runge-Kutta multi-step time- stepping scheme while the values of the surface nodes are determined in a least-squares fashion. In the process of computing the surface values, the implemented boundary conditions are satisfied automatically.

Determination of Cloud Points

A gridless method requires cloud points that are defined before the start of the computation. Cloud points for a boundary node are set up to compute Euler fluxes whereas cloud points for a surface node are used for boundary condition implementation and surface value computation. For each boundary node, a cloud of 9 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 2a. Only fluid nodes and boundary nodes that fall in the stencil are included. Cut cells and solid cells are omitted and are replaced by surface nodes that are closest to the boundary node. To avoid choosing nodes from the wrong side of the profile at thin surfaces as illustrated in Figure 2b, it is important to ensure that surface nodes selected have outward normal pointing in the direction of the boundary node. In this way, surface nodes on the other surface will not be considered even if it is nearer to the boundary. For each surface node, a one-sided cloud is defined. In each cloud, only fluid nodes and boundary nodes are included. The selection of cloud points is based on their distances to the surface node as well as satisfying the criteria that they fall on the same side as the outward


normal direction. As such, surface values are evaluated using computational nodes on the side of its outward normal as illustrated in Figure 3. The procedure for boundary and surface nodes determination is listed as follow: 1. Cut cells with reference to the Cartesian Mesh and

surface data points are identified. A feasible method for this is to use Alternate Digital Tree (ADT) method whereby the intergrid 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 accomplished using a Non-Uniform Rational B-Splines (NURBS) program. The surface nodes obtained are the nearest point on the surface to the cut cells. Details of the NURBS representation used can be found in Ray and Tsai26. With this latching method, the interval between the surface nodes will be approximately the same as the grid cells interval. Obtaining a set of well-distributed points will help to reduce the possibility of ill-conditioning in the matrices for least-squares computation. A situation arises with the latching method where two surface nodes are too close to each other as shown in Figure 4. Care has to be taken to eliminate one of these points.

3. Normal and tangent vectors for the surface nodes are found using the original data inputs of the object for accuracy rather than using the computational points on the surface. Radius of curvatures at these nodes is also defined. At the tip of airfoil trailing edge, these values are obtained by extrapolation to avoid singularity.

4. Cloud points for the boundary nodes and surface nodes are then defined in the manner described above.

Spatial Discretization Spatial discretization of the Euler equations for all the fluid nodes in the Cartesian grid is performed using the standard Finite-Volume Method. From the control volume for a cell abcd as shown in Figure 5, equation (1) can be written as follows,

0)( =−+∂∂

∫∫∫abcdabcd

gdxfdyUdxdyt

(5)

This expression can be approximated as

( ) 0,,, =+ jijiji QUSdtd (6)

where Si,j is the cell area and Qi,j is the net flux out of the cell. The dependent variables U are known at the

center of each cell. Qi,j can be found by taking the sum over the four sides of the cell as given by,

(7) (∑=

∆−∆4

1kkkkk gxfy )

The fluxes are evaluated by taking the average of the values in the cells on either side of each edge. This formulation is the cell-centered finite-volume scheme. On a Cartesian grid, the quadrilateral abcd is rectangular and the sides coincide with lines of constant x and y. Equation (6) becomes

022

1,1,,1,1, =∆

−+

∆

−+ −+−+

ygg

xff

dtdU jijijijiji (8)

Thus, the scheme reduces to a central difference scheme on a Cartesian grid and is second order accurate in space.

Artificial Dissipation

The Euler equations which are a set of non-dissipative hyperbolic conservation laws require artificial dissipation terms to be added to the discretized Euler equations. This is to suppress the tendency for odd and even point decoupling and to prevent oscillation near shockwaves. An adaptive scheme for adding dissipation by Jameson et al.24 has proven to be effective in practice in numerous calculations of complex steady flows. In this scheme, a third 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 first order dissipative terms is needed to capture the shock. This switch from third to first order is signaled using local pressure gradients. With the addition of dissipative terms Di,j, equation (6) becomes

( ) 0.,,, =−+ jijijiji DQUSdtd (9)

The added diffusive term for a two-dimensional flow is 21,21,,21,21, −+−+ −+−= jijijijiji ddddD (10) Taking the density equation as an example,

)33(

)(

,1,,1,2,21)4(

,21

,,1,21)2(

,21,21

jijijijijiji

jijijijiji

R

Rd

−++++

++++

−+−−

−=

ρρρρε

ρρε (11)

The other terms in equation (10) are defined in a similar way. The coefficients of and 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

)2(ε )4(ε


oscillations near discontinuities. The second difference of pressure is defined as

The spatial derivatives can be found by differentiating equation (17) and is given as follow,

jijiji

jijijiji PPP

PPP

,1,,1

,1,,1, 2

2

−+

−+

+++−

=ν (12) =∂

xaayhyaa

xh

3231 +=∂∂

+∂ (19)

Then ),,,max( ,1,,1,2

)2()2(,21 jijijijiji −+++ = ννννκε (13)

),0max( )2(,21

)4()4(,21 jiji ++ −= εκε (14)

Hence by determining the values of a0, a1 and a2, the spatial derivatives are found and the Euler fluxes evaluated. Conservation of mass, momentum, and energy a concern for gridless method as pointed out by Batina22. This can be controlled by sufficient resolution and compromising this only at points where this is applied. As noted the bulk of the computational domain uses a fully conservative finite volume formulation.

with 1)2( =κ , 321)4( =κ . By this definition, is

significant near shock regions and first order dissipation is used. is significant where the flow is smooth and third order dissipation is used.

)2(ε

)4(ε

ji

R,2

1+ is chosen to give the dissipative terms the proper

scale and is approximated by

∆+

∆=

+

+

+ji

ji

ji

ji

ji tS

tS

R,

,

,1

,1

,21 2

1 (15)

The cloud points of each boundary node are selected such that it encompasses fluid nodes, boundary nodes and surface nodes. A mixture of different types of nodes in each cloud will ensure the transfer of information between these nodes.

The time step limit ∆t is given by

jiji

jiji

St

,,

,, µλ +=∆ (16)

Artificial dissipative terms for boundary nodes are computed using the same equations, as described above. Cut cell values are approximated by the surface node values in order to compute the 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.

where λ and µ are the average spectral radii of the Jacobian matrices in i and j directions.

Temporal Discretization Boundary Node Treatment

As a boundary node does not have a complete stencil for FVM, a gridless method is used to compute the Euler fluxes for these nodes. This is a very simple method for boundary treatment as it uses only clouds of points and does not require the points to have any pre-specified connectivity amongst one another nor the formation of meshes or cells. The approach is simple and requires a separate treatment which makes it easy to implement into any scheme.

The Runge-Kutta multi-stage time integration24 is applied for both the fluid and boundary nodes in the same routine. No separate treatment is required for the boundary nodes. The modified four-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 follow,

For each boundary node, a cloud of points is defined. The governing equation is solved through a local least-squares curve fit using the cloud of grid points. In each cloud, each term of the fluxes as defined in equation (2) is assumed to vary linearly according to a function (17) xyayaxaayxh 3210),( +++=

4,

1,

3

,

0,

4,

2

,

0,

3,

1

,

0,

2,

0

,

0,

1,

,0,

213141

jin

ji

jijiji

jijiji

jijiji

jijiji

njiji

uu

RS

tuu

RS

tuu

RS

tuu

RS

tuu

uu

=

∆−=

∆−=

∆−=

∆−=

=

+

(20)

Performing a least-squares curve fit in a given cloud results in four equations

=

∑∑∑∑

∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑

iii

ii

ii

i

iiiiiiii

iiiiii

iiiiii

iiii

hyxhyhx

h

aaaa

yxyxyxyxyxyyxyyxyxxxyxyxn

3

2

1

0

2222

22

22

(18) with the residual R given by jijiji DQR ,,, −= (21) 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 following stages.

RV

nP

nE

n

nVV

t

tn

2

00

00

ρ

ρ

=∂∂

=∂∂

=∂∂

=∂∂

=

(24) The scheme is explicit with a CFL condition of

22≤CFL and has second-order accuracy in time for a non-linear equation. In order to increase the allowable stability bound, implicit residual smoothing is employed. The residuals from neighboring points are smoothed using a Laplacian operator. For a residual, Ri, at mesh point I, and smoothed residual, iR ,

where R is the local surface radius of curvature. Taking the curvature of the body into consideration will result in better accuracy.

iii RRR 2∇+= ε (22) with constant ε usually taken as 0.5 and the discrete form of is an undivided difference, which becomes, 2∇ 11 )21( −+ −++−= iiii RRRR εεε (23)

A one-sided cloud of points for each surface node is set up as shown in Figure 6. Each cloud contains only one surface node and the values of flow variables at the other cloud points is resolved to the normal and tangential direction of that surface node for easy implementation of the boundary conditions. More surface nodes may be incorporated in each cloud, which means more conditions will have to be satisfied in each cloud. This may result in more accurate solution but at present, this has not been incorporated.

Equation (23) is solved directly. With the use of implicit residual smoothing, the Courant number in practice can be increased to 9.0. 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.

The variables are assumed to vary according to the function: f = b0+b1ξ+b2η+b3ξη (25) Boundary Conditions Taking Vt as example, applying the Neumann condition in equation (24) on the surface node will result in the following:

The main challenge in using Cartesian grids is in the boundary condition implementation as the grids are not body aligned. A satisfactory boundary condition implementation is an important part of the algorithm. As noted above in the present work, gridless method with least-squares technique is used to implement the surface boundary condition such that surface boundary nodes are included in the solution. The far field boundary condition is constructed using Riemann invariants.

032 =+=∂

∂bc

t bbV

bc ξη

b2 = -b3ξbc (26) Substituting equation (26) back into equation (25), the number of unknowns is reduced from four to three and the new function is given as: Vt = b0+b1ξ+b3η(ξ-ξbc) (27)

Surface Boundary Condition The equation is solved using the least-squares method

as described earlier for the three unknowns b0, b1 and b3. This solution automatically satisfies the boundary condition and the flow variable at the surface node is defined.

Surface node values are found in a similar least-squares fashion using a one-sided cloud of points as described above. The boundary conditions are implemented such that in the process of solving the surface values, the boundary conditions are satisfied.

Far Field Boundary Condition

For inviscid condition the flow slips over the surface and hence it is tangent to the surface. The slip condition is thus imposed on the boundary and the other flow variables on the surface nodes are approximated as follow:

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. ∞+== RRnqq en

rr (28)

))(1(25.0 ∞+−= RRc eγ (29)


where )1(2. −−= ∞∞∞ γcnqR

rr (30)

C )1(2. −+= γeee cnqR

rr (31)

and subscript e signifies extrapolated values, while subscript ∞ refers to free stream values. At the outflow boundary, the velocity vector is defined as, nnqqqq ene

rrrrr).( −+= (32)

and at the inflow boundary, nnqqqq n

rrrrr).( ∞∞ −+= (33)

The set is completed using the enthropy by setting, (34) ∞== ssorss e

respectively for outflow or inflow boundary.

Results and Discussions Results are presented for two-dimensional airfoils using the method described above. To show the robustness of the method, solution for a symmetric NACA 0012 airfoil and an asymmetric RAE 2822 airfoil are computed. All the results are computed using stretched grids such as shown in Figure 7a. A close-up view of the stretched grid is provided in Figure 7b. Solution is first obtained on a coarser stretched grid with 144x130 cells. The cells within the chord length are then refined to obtain a 200x180 grid, which is used to improve the accuracy of the solution. The coarser grid has 80 cells within the length of the airfoil, resulting in approximately 160 surface nodes defined on the airfoil boundary. This resolution is deliberately chosen to be comparable to the reference body-fitted grid (160x32) solution obtained using Jameson’s FLO52 code. A close-up view of the body-fitted grid is shown in Figure 8. The finer stretched grid has 130 cells in the x-direction, resulting in approximately 260 surface nodes. Both the Cartesian grid and body-fitted grid extend to a distance of about 25 chords from the airfoil profile. NACA0012 Airfoil For the symmetric airfoil, four flow cases with various combinations of freestream Mach number and angle of attack α are computed: (1) = 0.85, α = 0°; (2) = 0.5, α = 3.0°; (3) = 0.8, α = 1.25°; (4) = 2.0, α = 0°. All four cases were computed using CFL number of 2.0 with local time-stepping to accelerate convergence to steady state. For the present work, implicit residual smoothing is not implemented. For each of the cases, a comparison of the pressure distribution followed by the convergence history is

presented. The pressure distributions of the pressure

coefficient

∞M

∞M

∞M

∞M∞M

221

∞∞

∞−=

qpp

p ρ

∞M

are plotted with the negative

pressure upward and the convergence histories show the mean rate of change of the density. All solutions are obtained using single precision operations. Figure 9a shows the Cp plot for case 1 with = 0.85, α = 0°. The solutions for the three grids (coarse Cartesian grid, fine Cartesian grid, and body-fitted grid) are plotted on the same graph. Despite the fact that in the Cartesian method the edges are approximated with jagged cells, the coarse grid solution matches well with the body-fitted grid solution along the entire surface except that it is not able to capture the shock as sharply. This is due to smearing near the shock region when the least-squares method is applied on the boundary and surface nodes. Using the finer Cartesian grid, the result obtained is better than the body-fitted grid. A sharper shock front is captured as seen from the Figure 9a. However, four surface points are still needed to capture the shock. The convergence plots for both the finer and coarser Cartesian grids are shown in Figure 9b.

∞M

For case 2 with = 0.5, α = 3.0°, the Cp plot for the subsonic flow is presented in Figure 10a and the convergence plot in Figure 10b. From the Cp plot, the coarse grid solution is almost identical to the body-fitted grid solution. This shows that the Cartesian method presented above is able to provide an equally good solution for the subsonic case using the same number of cells as that in the body-fitted method. With the finer grid, the Cartesian method is able to resolve the peak at the leading edge better than the body-fitted grid. The Mach number contour plot of the fine Cartesian grid for the subsonic case is presented in Figure 10c. In case 3, the transonic flow of = 0.8 with α = 1.25° is computed and the C

∞M

p and convergence plot are shown in Figure 11a and 11b, respectively. A similar trend is observed for the Cp plot between case 1 and case 3 except that for case 3, a weak shock is developed on the lower surface of the airfoil. The weak shock captured by the fine grid solution appears to be sharper than that by the body-fitted grid method. Figure 11c shows the Mach number contour of the fine Cartesian grid for this case. Supersonic flow of = 2.0 with α = 0° is tested in case 4 and the results are presented in Figure 12a and Figure 12b. The C

∞M

p plots for both the coarse and fine Cartesian grids match the body-fitted grid solution. No distinguishable difference is observed from the graph.


However, a comparison of the contour plots for the fine Cartesian grid with the body-fitted grid reveals some differences. Close-up view of the Mach contour plots for both grids are shown in Figures 12c and 12d. Note that in the construction of the fine Cartesian grid, refinement of cells was only done for cells within the airfoil and the interval between the cells away from the airfoil remains the same as that of the body-fitted grid. Thus, this shows that more cells are required in the streamwise-direction in order to capture the shock more sharply. A further test is included to find out the stability of the proposed Cartesian method. A solution for high Mach number, = 5.0 with α = 0° is computed. The numerical scheme used in this work is not particularly suited for high Mach number computation. The purpose of this test is purely to test the stability of the method and not to be concern about the accuracy of the solution. The C

∞M

p plot, convergence plot, and the pressure contour plot are shown in Figures 13a, 13b, and 13c, respectively. The solution is obtained without changing the artificial dissipation coefficient 1)2( =κ

and 321

=)4(κ . However, the CFL number has to be

lowered to 0.9 for solution convergence. From this, we can see that higher stability can be achieved using a Cartesian method. It can compute without increasing the damping coefficient for such high-speed computation. RAE 2822 Airfoil Using the same Cartesian grid used for the NACA0012 airfoil, the two-dimensional flows over RAE2822 airfoil are computed. Figures 14a and 14b show the Cp plot and convergence plot for a subsonic flow over the RAE 2822 airfoil with

= 0.5 and α = 3°. The solution obtained using the coarse Cartesian grid matches very closely to the body-fitted grid solution except at the peak location on the upper surface of the airfoil where it has a slightly lower value. Using the finer Cartesian grid, the C

∞M

p plot obtained matches exactly with the body-fitted solution. At the peak location, it is able to predict a higher value than the body-fitted solution. The Mach number contour plot of the fine grid solution for the subsonic case is presented in Figure 14c. Results for transonic flow with = 0.75 and α = 3° are presented in Figures 15a and 15b. From the C

∞Mp plot,

coarse grid solution for the airfoil lower surface matches better with the body-fitted solution than its upper surface solution. Two areas of mismatch were

spotted. Firstly, a gentler slope is obtained for the shock on the upper surface. Secondly, the solution near the leading edge with high curvature is not well resolved. The fine grid solution on the other hand matches the body-fitted grid solution for both upper and lower surfaces of the airfoil. Hence for the transonic case, more points are needed on the upper surface of the airfoil in order for the coarse grid to obtain the same accuracy as the body-fitted grid. The Mach number contour plot of the fine grid solution for this case is presented in Figure 15c.

Conclusions and Future Work A method has been presented for the treatment of boundary conditions for the solution of the Euler Equations on a Cartesian grid. The approach involves using “gridless” or “meshless” methods to address the boundary or interface while standard structured grid methods are used everywhere else. Boundary conditions are implemented on the geometry surface and are automatically satisfied as the surface values are solved in a least-squares fashion. This method simplifies the implementation of boundary conditions for a Cartesian method and exploits the numerous advantages of a Cartesian mesh at the same time. Moreover the Cartesian grid forms a natural means to distribute computational points for the meshless scheme. For a simple geometry, such as airfoils, it is not economical to employ a Cartesian method since number of grid points required to ensure sufficient resolution is higher than body-fitted grid method. However, 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. Embedded grid could be used for resolution and multigrid computations13,14. The method shows great potential for extension to three-dimensional problems where its advantages will be significant. It can also be exploited for other applications such as adaptive shock, local refinement, and interpolation between grids in overset methods. These issues are currently being investigated. The overall scheme is robust and stable and it converges well for a range of Mach numbers tested. Solutions from the proposed approach are computed for symmetrical and asymmetrical airfoils and the results are compared with standard body-fitted grids solutions. The method is less accurate for capturing the shocks but an improvement in resolution of 60% gives sharper shock fronts compared to a body-fitted grid method with equal number of grid points.


[6] Alonso, J. J. and Jameson, A. “Fully-Implicit Time-Marching Aeroelastic Solutions” AIAA Paper 94-0056, Jan 1994.

The method described is not restricted to a particular numerical scheme. In the current work block-structured grids are used as it requires fewer words of storage per grid point. But one can also incorporate it in other schemes, e.g., factored and directional solvers by Jameson and Caughy27 for increased computational speed. The implementation of such solvers within the present framework for boundary condition is an interesting possibility.

[7] Meakin, R. L. “Object X-Rays for Cutting Holes in Composite Overset Structured Grids” AIAA Paper A01-31051, Jun 2001.

[8] Coirier, W. J. and Powell, K. G. “An Accuracy Assessment of Cartesian Mesh Approaches for the Euler Equations” Journal of Computational Physics, Vol. 117, pp.121-131, 1995

[9] Berger, M. J. and LeVeque, R. J. “An Adaptive Cartesian Mesh Algorithm for the Euler Equations in Arbitrary Geometries” AIAA Paper 89-1930, June 1989.

Future work involves refining the mesh near the boundary to improve the accuracy of the solution and to examine the issue of number of points to capture shocks. Potentially for better shock capturing, a weighted-least squares and/or step function (for regions near shocks) approximation may be better than the simple least squares used in the present work. Acceleration techniques such as multigrid and implicit residual smoothing could be implemented for faster convergence.

[10] Clarke, D. K., Salas, M. D. and Hassan, H. A. “ Euler Calculations for Multielement Airfoils Using Cartesian Grids” AIAA Journal, Vol. 24, pp.353-358, 1986.

[11] Clarke, D. K., Salas, M. D. and Hassan, H.A. “Euler Calculations for Multi-Element Airfoils using Cartesian Grids” AIAA Journal, Vol. 24, pp.353-358, 1986.

The authors would like to thank Assoc. Professor Yeo Khoon Seng of Mechanical and Production Engineering Department, NUS, and Dr. Cai Jinsheng of Temasek Laboratories for their suggestions and helpful discussion. The support of the Directorate of Research & Development, Defence Science and Technology Agency, Singapore (under POD 0001820, program monitors, Mr. Pek Beng Tit and Ms. Ng Keng Bee) is also gratefully acknowledged.

[12] Ye, T., Mittal, R., Udaykumar, H. S. and Shyy, W. “An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries” Journal of Computational Physics, Vol. 156, pp.209-240, 1999.

[13] Epstein, B., Luntz, A. and Nachson, A. “Multigrid Euler Solver about Arbitrary Aircraft Configurations, with Cartesian Grids and Local Refinement” AIAA Paper 89-1960, Jun 1989.

[14] Tidd, D. M., Strash, D. J., Epstein, B., Luntz, A., Nachshon, A. and Rubin, T. “Multigrid Euler Calculations over Complete Aircraft” Journal of aircraft, Vol 29, No.6, 1992.

References [1] Batina, J. T. “Unsteady Euler Algorithm with

Unstructured Dynamic Mesh for Complex Aircraft Aerodynamic Analysis” AIAA Journal, Vol. 29, pp.327-333, 1991.

[15] MGAERO User’s Manual “A Cartesian Multigrid Euler Code for Flow Around Arbitrary configurations.” Analytical Methods, Inc.

[2] Weatherill, N. P., Hassan, O., Marchant, M. J., and Marcum, D. L. “Adaptive Inviscid Flow Solutions for Aerospace Geometries Using Efficiently Generated Unstructured Tetrahedral Meshes” AIAA Paper 93-3390, 1993.

[16] Forrer, H. and Jeltsch, R. “A Higher-order Boundary Treatment for Cartesian-Grid Methods” Journal of Computational Physics, Vol.140, pp.259-277, 1998.

[17] Wu, Z. N. “Anisotropic Cartesian Grid Method for Viscous Flow Computation” Computational Fluid Dynamics Review 1998, M. Hafez and K. Oshima, eds., World Scientific, New Jersey, USA.

[3] Jameson, A. and Mavriplis, D. “Finite Volume Solution of the Two-Dimensional Euler Equations on a Regular Triangular Mesh” AIAA Journal, Vol. 24, pp.611-618, 1986.

[4] Frink, N.T. and Pirzadeh, S.Z. “Tetrahedral Finite-Volume Solutions to the Navier-Stokes Equations on Complex Configurations” 10th International Conference on Finite Elements in Fluids, Tucson, USA, 1998.

[18] Aftosmis, M. J., Berger, M. J. and Adomavicius, G. “A Parallel Multilevel Method for Adaptively Refined Cartesian Grids with Embedded Boundaries” AIAA Paper 2000-0808, Jan 2000.

[19] Zeeuw, D. D. and Powell, K. G. “An Adaptively Refined Cartesian Mesh Solver for the Euler Equations” Journal of Computational Physics, Vol. 104, pp.56-68, 1993.

[5] Meakin, R. L. “Automatic Off-Body Grid Generation for Domains of Arbitrary Size” AIAA Paper A01-31050, Jun 2001.


[20] Durbin, P. A. and Iaccarino, G. “An Approach to Local Refinement of Structured Grids” Journal of Computational Physics, Vol. 181, pp.639-653, 2002.

[21] Ham, F. E., Lien, F. S. and Strong, A. B. “A Cartesian Grid Method with Transient Anisotropic Adaptation” Journal of Computational Physics, Vol. 179, pp.469-494, 2002.

[22] Batina, J. T. “A Gridless Euler/Navier-Stokes

Solution Algorithm for Complex-aircraft Applications” AIAA Paper 93-0333, Jan 1993.

[23] Kirshman, D. J. and Liu, F. “Gridless Boundary

Condition Treatment for a Non-Body-Conforming Mesh” AIAA 2002-3285, June 2002.

[24] Jameson, A., Schmidt, W. and Turkel, E.

“Numerical Solutions of the Euler Equations By Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes.” AIAA Paper 81-1259, 1981.

[25] Jameson, A. “Transonic Flow Calculations for

Aircraft” Numerical Methods in Fluid Dynamics, Lecture Notes in Mathematics 1127.

[26] Ray, T., and Tsai, H.M. “Some Issues in NURBS Representation of Airfoil Shapes for Optimization” Proceedings of the 8th International Conference on Numerical Grid Generation in Computational Field Simulations, 2002.

[27] Jameson, A. and Caughey, D. A. “How Many Steps are Required to Solve the Euler Equations of Steady Compressible Flow: In Search of a Fast Solution Algorithm”, AIAA 2001-2673, June 2001.


Figure 1: Classification of grid nodes.

Figure 2a: Cloud points for boundary nodes.

Figure 2b: Cloud points for boundary nodes at thin surfaces.

Figure 3: Cloud points for surface nodes

(Closest 8 points to surface node in shaded region).

Figure 4: Duplicate surface nodes

Figure 5: Control Volume of a single cell.

Figure 6: Boundary implementation for surface nodes.

nr

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

tr

i i+1/2 i+1 i-1/2 i-1

j-1

j-1/2

j

j+1/2

j+1

a b

c d

η ξ nr

nr

nr n

r


Figure 7a: Stretched Cartesian Grid.

Figure 9a: Cp plot for NACA 0012 with M∞ = 0.85, α = 0°.

Figure 7b: Close up of stretched Cartesian Mesh with NACA 0012.

Figure 9b: Convergence plot for NACA 0012 with

M∞ = 0.85, α = 0°.

Figure 8: Close up of Body-fitted Grid with NACA 0012.




M∞ = 0.5, α = 3°.


M∞ = 0.8, α = 1.25°.

Figure 10c: Mach contour plot(fine grid) for NACA 0012 with

M∞=0.5, α=3°. Mach Contours Mmin=0.1, Mmax=0.67, interval=0.019


M∞=0.8, α=1.25°. Mach Contours Mmin=0.1, Mmax=1.1, interval=0.026

Figure 11a: Cp plot for NACA 0012 with M∞ = 0.8, α = 1.25°.




M∞ = 2.0, α = 0°.


Figure 12c: Close up Mach contour plot for NACA 0012 with M∞=2.0, α=0° using fine Cartesian grid. Mach Contours Mmin=0.25, Mmax=1.9,

interval=0.12


M∞ = 5.0, α = 0°.

Figure 12d: Close up Mach contour plot for NACA 0012 with M∞=2.0, α=0° using Body-Fitted Grid. Mach Contours Mmin=0.25, Mmax=1.9,

interval=0.12


M∞=5.0, α=0°. Mach Contours Mmin=0.5, Mmax=3, interval=0.13


Figure 14a: Cp plot for RAE 2822 with M∞ = 0.5, α = 3°.

Figure 15a: Cp plot for RAE 2822 with M∞ = 0.75, α = 3°.

Figure 14b: Convergence plot for RAE 2822 with

M∞ = 0.5, α = 3°.

Figure 15b: Convergence plot for RAE 2822 with

M∞ = 0.75, α = 3°.

Figure 14c: Mach contour plot(fine grid) for RAE 2822 with


Figure 15c: Mach contour plot(fine grid) for RAE 2822 with



EULER SOLUTION USING CARTESIAN GRID WITH LEAST … · 2017-08-23 · Other mesh refinement...

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