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Third- and fourth-order accurate finite difference schemes for the first derivative of the square of the speed are developed, for both uniform and non-uniform grids, and applied in the study of a two-dimensional viscous fluid flow through an irregular domain. The von Mises transformation is used to transform the governing equations, and map the irregular domain onto a rectangular computational domain. Vorticity on the solid boundary is expressed in terms of the first partial derivative of the square of the speed of the flow in the computational domain, and the schemes are used to calculate the vorticity at the computational boundary grid points using up to five computational domain grid points. In all schemes developed, we study the effect of coordinate clustering on the computed results.

Fluid flow through irregular geometries (that is, geometries that do not conform to known coordinate lines) is encountered in many natural and industrial phenomena. Natural occurrences include flow in channels and in rivers over the curvilinear channel bottom, flow into a dam with a curvilinear free-surface, and groundwater and oil flow through the curvilinear earth layers and reservoirs. Industrial applications of flow through irregular domains include flow of air over wings and streamlined bodies, flow in curvilinear ducts, pipes and conduits, flow through turbines and propeller systems, and flow through artificial human and animal tissues, [

The inherent difficulty in solving the Navier-Stokes equations analytically encouraged the use of the more flexible numerical techniques to obtain approximate, yet accurate, solutions to initial and boundary value problems. The method of finite differences proved to be a flexible approach to a large number of flow problems, and has extensively been used due to its simplicity. This method has become an indispensable tool in the numerical treatment of initial and boundary value problems in Mathematical Physics, and its use in the simulation of two- and three-dimensional fluid flow problems is well-documented and frequently reported in the fluid dynamics literature, (cf. [

In solving boundary value problems that involve viscous fluid flow in two space dimensions, the Navier- Stokes equations are usually cast in vorticity-streamfunction form. The streamfunction possesses Dirichlet conditions if the solid boundary is assumed to be a streamline of the flow. However, no explicit expression exists for the vorticity on a solid boundary. This forces one to impose a Neumann (derivative) boundary condition on the vorticity. Neumann boundary condition is derived using the definition of the vorticity as the Curl of the velocity vector, [

In order to capture the boundary effects more accurately, the use of non-uniform grid with grid clustering at the boundary proved to be a viable tool. Grid refinement has been extensively studied and widely used, and a fairly large number of algebraic and mapping methods have been proposed to produce grid clustering for any given boundary shape, [

In order to overcome the above shortcomings, forward and backward finite difference schemes of various orders of local accuracy continue to be developed to better capture the boundary effects (including better approximation to the vorticiy on a solid boundary), [

The above conclusion has been reached for the case of uniform grid. However, many authors have reached a similar conclusion when a non-uniform grid is used, and some, [

Various authors emphasized that discarding schemes due to their low order of accuracy are “premature” when a non-uniform grid is used, (cf. [

In summary, computation of boundary vorticity accurately is tantamount to finding a most suitable finite difference expression for the first derivative of the tangential velocity component; one that minimizes the unfavourable effects on the coefficient matrix and one that produces the best approximation to the boundary vorticity, whether one employs a uniform or a non-uniform grid.

In the case of fluid flow in curvilinear domains, the domain is first mapped onto a rectangular domain (or a regular domain) where finite differences can be applied. Numerical solution is then sought for transformed governing equations and boundary conditions, [

- The type of grid and spacing used.

- Entry condition (say to a channel).

- The use of flow-field grid points.

In order to shed some light on the above factors, Awartani, Ford and Hamdan, [

The above reported previous work, [

- The type of grid spacing used: Typically, clustered grid near the boundary is expected to produce more accurate results. However, in some coordinate transformation procedures there are factors limiting how “compacted” the grid points can be. An important question to ask is whether a uniform grid in the computational domain can produce a clustered grid in the physical domain.

- Entry conditions to a channel: In case of a uniform entry condition to a channel, there is a correspondence between a grid in the computational domain and a grid in the physical domain, in the sense that clustering in the physical plane produces the same clustering in the computational domain. However, when a more realistic parabolic entry profile is used, the correspondence is lost and the use of clustered grid becomes necessary in the computational domain. We will consider the effects of a parabolic entry profile when the von Mises transformation is used with both uniform and non-uniform grids.

- Use of interior grid points in the approximation: A question that needs to be answered is how many internal grid points should be used in approximating the vorticity on the boundary, and how far from the boundary can grid points be taken for uniform and clustered grids. This work will demonstrate that a natural order of grid points that are closest to the boundary result in more accurate approximations to boundary vorticity.

In order to present our procedures and report the findings of this study, this work is organized as follows. In the following sections, we present a formulation of the problem of viscous fluid flow in a two-dimensional curvilinear configuration and transform the governing equations into von Mises variables. We present a discretization of the computational domain and derive third and fourth order accurate standard finite difference schemes for the first derivative and we test these schemes on a uniform grid and four clustered grids.

We consider a physical problem that gives rise to the need for finite difference expressions for the first derivative. Non-uniform grid arises in a number of ways, including clustering of coordinates and where coordinate transformation is involved. In the current work we consider the von Mises transformation which gives rise to the need for non-uniform grid.

Consider the flow in a two-dimensional, symmetric, dimensionless channel bounded below and above by solid, curvilinear boundaries. The channel is shown in

The flow of a viscous, incompressible fluid through the channel is governed by the equations of continuity and linear momentum, which take the following dimensionless forms, respectively, (cf. [

wherein, for two-dimensional flow,

The dimensionless continuity Equation (1) implies the existence of a dimensionless streamfunction,

and

Elimination of the pressure term from Equation (2) and the introduction of the dimensionless vorticity

facilitate casting the flow equations in the following vorticity-streamfunction form:

Streamfunction equation:

Vorticity equation:

In vorticity-streamfunction, the problem has been reduced from that of solving three equations in three primitive variables to that of solving two equations in the two unknowns

Appropriate boundary and inlet conditions are the no-slip on the upper and lower walls of the channel, and a parallel, parabolic inlet profile at x = a. We assume that the channel is of sufficient length so that inlet and exit conditions are the same, namely:

The absence of explicit vorticity boundary conditions necessitates imposing appropriate inlet, exit and solid boundary conditions. Assuming that Equation (5) is valid on all boundaries then, with

Along the lower and upper boundaries, respectively, we have:

In order to solve Equations (6) and (7), subject to conditions (9a)-(9d), numerically using finite differences, the flow domain is first transformed into a rectangular domain using the von Mises coordinates, and the governing equations and boundary conditions are cast in von Mises variables. The von Mises transformation and the transformed governing equations and boundary conditions are discussed in what follows.

Consider the curvilinear net

The Jacobian of transformation is given by

If

Second partial derivative operators can be obtained by applying operators (12) and (13) onto themselves.

Applying the above transformation to the vorticity-streamfunction Equations (6) and (7), respectively, yields, [

where

The problem has thus been transformed from that of solving Equations (6) and (7) for

on the lower boundary

on the upper boundary

at both inlet and exit

at inlet and exit, and on the lower and upper boundaries

The square of the speed of the flow is given by:

and the velocity components are expressed as

Clearly, vorticity condition on the solid upper and lower channel boundary is given by (17d) in terms of the first derivative of the square of the speed of the flow. In the process of solving Equations (14) and (15) numerically, subject to conditions (17), we need to develop finite difference formulas for the first derivative in (17d).

The rectangular computational domain of _{max} at x = b, and horizontal grid lines ranging from j = 1

at the lower computational boundary _{max} at the upper computational boundary

It has been shown, [

Clustering of grid lines near the computational boundary when the von Mises variables are used can be accomplished in two ways:

1) Selecting a uniform grid in the physical domain and calculating the grid spacing in the computational domain results in a “natural” clustering near the computational boundary. For the given parabolic inlet velocity profile, and the indicated range of streamlines, we can choose y to vary over the interval [−1,1], and calculate the step size

three variable grid spacings in the computational domain, corresponding to a uniform grid in the physical domain with

2) Selecting a grid in the physical domain that is non-uniform and clustered near the physical boundary produces a computational grid that is also clustered near the computational boundary.

Physical grid clustering can be accomplished in various ways, including the use of elementary functions. For the current work, we employ the square root, the cubic root and the square functions to define the clustering in the physical y-direction. These are illustrated in Tables 1-5, wherein we provide a comparison with the case of uniform grid in the physical domain (

At the inlet to the channel, the computational velocity conditions are:

j | ||||
---|---|---|---|---|

1 | 0.01 | |||

2 | 0.01 | |||

3 | 0.01 | |||

4 | 0.01 | |||

5 | 0.01 |

j | ||||
---|---|---|---|---|

1 | 0.005012567 | |||

2 | 0.005037944 | |||

3 | 0.005063713 | |||

4 | 0.005089883 | |||

5 |

j | ||||
---|---|---|---|---|

1 | 0.003344507 | |||

2 | 0.003367105 | |||

3 | 0.003390089 | |||

4 | 0.003413469 | |||

5 |

j | ||||
---|---|---|---|---|

1 | 0.0199 | |||

2 | 0.0197 | |||

3 | 0.0195 | |||

4 | 0.0193 | |||

5 |

j | ||
---|---|---|

1 | ||

2 | ||

3 | ||

4 | ||

5 |

Using (20), we compute

j | |||||
---|---|---|---|---|---|

1 | 0 | 0 | 0 | 0 | 0 |

2 | 0.00039601 | 0.0001 | 0.000044593 | 0.001552674 | 0.0491659667 |

3 | 0.00156816 | 0.0004 | 0.000178975 | 0.006026702 | 0.0948030258 |

4 | 0.00349281 | 0.0009 | 0.000404064 | 0.013157739 | 0.1380939921 |

5 | 0.00614656 | 0.0016 | 0.000720798 | 0.022696458 | 0.1794589879 |

The definition of vorticity is expressed at the lower boundary of the computational domain, in von Mises coordinates as the following derivative of the square of the speed of the flow:

In order to obtain finite difference expressions for

In order to derive forward differencing schemes of local accuracy n for the first derivative along the grid line (i,1) using

where E is the local truncation error. The weights,

where

The coefficients,

where

For uniform grid,

Using (23) in (22), and equating to zero the coefficients of the first n partial derivatives (including the zero’th derivative) of

The order of the leading error term, E, will be the coefficient of

Based on the above description, Siyyam, Ford and Hamdan, [

1) The relationship between the order of the scheme, n, and the minimum number, k, of grid points used is that

2) The number of standard schemes that use k = n + 1 grid points when

given by the combination:

3) If

4) If

The above observations are used to construct

We employ the procedure above to derive a standard fourth-order accurate scheme (that is, n = 4) using five grid points. We thus take

Order of standard scheme | Minimum number of grid points required (k = n + 1) | Number of possible standard schemes when | Number of possible standard schemes when |
---|---|---|---|

First-order (n = 1) | 2 | C(5,2) = 10 | C(1,1).C(4,1) = 4 |

Second-order (n = 2) | 3 | C(5,3) = 10 | C(1,1).C(4,2) = 6 |

Third-order (n = 3) | 4 | C(5,4) = 5 | C(1,1).C(4,3) = 4 |

Fourth order (n = 4) | 5 | C(5,5) = 1 | C(1,1).C(4,4) = 1 |

or equivalently

coefficients

The leading local truncation error has the following leading term that is the coefficient of

Equations (32) translate into the following five equations in the five coefficients:

Solution to this system of 5 equations in 5 unknowns is given by

where

For the four types of non-uniform grids given in Tables 1-4, the values of

Using (40)-(43) in (35)-(39), coefficients

For uniform grid with step size h, coefficients

and the fourth-order expression for the first derivative takes the form:

Grid # | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Values of the coefficients

In order to arrive at a third-order (n = 3) locally accurate forward differencing scheme for the first derivative, Awartani, Ford and Hamdan, [

Letting

then the coefficients satisfy the following conditions:

Equations (52) represent a determinate system consisting of 4 equations in 4 unknowns that has the following unique solution:

where

Grid | |||||
---|---|---|---|---|---|

Grid 1 | 14,306.07323 | −16,095.95974 | 2032.465715 | −261.1816983 | 18.60248878 |

Grid 2 | 56,687.98222 | −63,619.31368 | 7871.374257 | −987.5683431 | 86.5377706 |

Grid 3 | 12,783.0685 | −19,849,012,084 | 17,520.61174 | −2180.649425 | 150.0190806 |

Grid 4 | 3647.395614 | −142,573.05 | 543.2005178 | −72.92902426 | 5.475543678 |

Grid 5 uniform | 156.2500391 | −300.000075 | 225.0000563 | −100.000025 | 18.75000469 |

and

The leading error term,

The four standard, third-order accurate schemes that use the four grid points j = 1, 2, 3 and 4, are as follows:

(1,2,3,4) Scheme:

Solution (53), above, provides the following expression that was initially derived by Awartani, Ford and Hamdan, [

where

For uniform grid, (57) yields the following scheme:

Values of the coefficients

Other standard third-order accurate schemes obtained in this work employ grid points

Grid | ||||
---|---|---|---|---|

Grid 1 | 13,672.62702 | −15,079.7727 | 1520.916705 | −113.7705973 |

Grid 2 | 54,200.53239 | −59,683.49924 | 5916.931052 | −433.9642054 |

Grid 3 | 121,583.6377 | −133,812.6289 | 13,190.00854 | −961.00854 |

Grid 4 | 3480.328141 | −3850.98425 | 401.9421534 | −31.26164889 |

Grid 5 uniform | 137.5000344 | −225.0000563 | 112.5000281 | −25.00000625 |

the form given by (52), with coefficients

(1,2,3,5) Scheme:

where

For uniform grid, (59) yields the following scheme:

Values of the coefficients

(1,2,4,5) Scheme:

where

Grid | ||||
---|---|---|---|---|

Grid 1 | 13,183.73878 | −14,295.48845 | 1126.106905 | −14.35723799 |

Grid 2 | 52,267.16645 | −56,598.30806 | 4384.864483 | −53.72287539 |

Grid 3 | 117,250.3235 | −126,909.795 | 9777.680061 | −118.2085627 |

Grid 4 | 3355.33057 | −3648.30425 | 297.0706287 | −4.09648027 |

Grid 5 uniorm | 131.2500328 | −200.00005 | 75.00001875 | −6.250001563 |

For uniform grid, (62) yields the following scheme:

Values of the coefficients

(1,3,4,5) Scheme:

where

For uniform grid, (65) yields the following scheme:

Grid | ||||
---|---|---|---|---|

Grid 1 | 11789.2926 | −12058.4901 | 324.5056047 | −55.30810146 |

Grid 2 | 46733.75961 | −47768.30974 | 1242.031037 | −207.4808993 |

Grid 3 | 104833.6526 | −107130.427 | 2753.697604 | −456.9231722 |

Grid 4 | 3001.166302 | −3074.036981 | 88.57579336 | −15.70511441 |

Grid 5 uniform | 118.7500297 | −150.0000375 | 50.0000125 | −18.75000469 |

Values of the coefficients

Six standard, second-order accurate schemes have been developed by Awartani, Ford and Hamdan, [

(1,2,3) Scheme:

where

For uniform grid, (3.49) reduces to the following second-order accurate formula:

Combinations of standard first-order schemes that use the two grid points

For the sake of comparison of results, we list below the natural order (1,2)-scheme that produced the most accurate results in [

(1,2)-Scheme:

For uniform grid, (72) reduces to:

Grid | ||||
---|---|---|---|---|

Grid 1 | 4272.561304 | −6070.25437 | 2073.745892 | −276.0528255 |

Grid 2 | 16,799.27007 | −23,721.13304 | 7961.140282 | −1039.277311 |

Grid 3 | 37,581.59177 | −52,958.56974 | 17,668.47142 | −2291.493454 |

Grid 4 | 1105.334666 | −1590.210946 | 562.7195912 | −77.84331141 |

Grid 5 uniform | 81.25002031 | −225.0000563 | 200.00005 | −56.25001406 |

An expression for the vorticity on the lower computational channel boundary is given by

In order to approximate

the schemes developed in this work. These values are computed on the uniform and non-uniform grids given in Tables 1-5, using the square of the speed values of

In the following Tables, we list the computed values of

Comparison of the schemes that use the natural order of grid points in conjunction with the five grids used, is given in

Standard, first-order accurate scheme of Equation (72) produces the results in

Standard, second-order accurate scheme that uses the natural order of grid points produces results illustrated in

1) The Percentage Relative Error decreases with increasing accuracy of the scheme employed, for both variable and uniform grids.

2) Grids with lowest Percentage Relative Errors for all standard schemes are, respectively: Grid 3, Grid 2, Grid 1, Gird 4, Grid 5.

3) Schemes that employ the natural order of grid points produce smaller Percentage Relative Errors, for all grids employed.

Scheme Grid | 1^{st} order (1,2) scheme | 2^{nd} order (1,2,3) scheme | 3^{rd} order (1,2,3,4) scheme | 4^{th} order (1,2,3,4,5) scheme |
---|---|---|---|---|

Grid 1 | −1.9867357 (0.666%) | −1.991132154 (0.443%) | −1.9920387 (0.398%) | −1.992431358 (0.378%) |

Grid 2 | −1.993381972 (0.331%) | −1.995622482 (0.218%) | −1.996072644 (0.196%) | −1.996257356 (0.187%) |

Grid 3 | −1.995569677 (0.221%) | −1.997066733 (0.146%) | −1.997366868 (0.131%) | −1.997498503 (0.125%) |

Grid 4 | −1.973488941 (1.325%) | −1.982419735 (0.879%) | −1.984135078 (0.793%) | −1.98505092 (0.747%) |

Grid 5 (uniform) | −1.843726 (7.813%) | −1.909891241 (4.509%) | −1.9246767643 3.766% | −1.931826679 (3.408%) |

4) Using clustered Grid 3 produces the lowest Percentage Relative Error, for all standard schemes used.

Forward-difference schemes that approximate the derivative at a typical grid point (i,j) in the computational flow domain, using up-to 5 grid points, employ the gridlines j, j + 1, j + 2, j + 3, j + 4, shown in

The standard forward-difference uniform and non-uniform schemes that are based on the natural order of grid lines are listed below for the sake of completeness and ease of reference.

(j, j + 1)-Scheme: first order

For non-uniform grid:

For uniform grid:

(j, j + 1, j + 2) Scheme: second order

For non-uniform grid:

For uniform grid:

(j, j + 1, j + 2, j + 3) Scheme: third order

Scheme Grid | (1,2,3,4) scheme | (1,2,3,5) scheme | (1,2,4,5) scheme | (1,3,4,5) scheme |
---|---|---|---|---|

Grid 1 | −1.9920387 (0.398%) | −1.9917441 (0.412%) | −1.990900404 (0.454%) | −1.986352479 (0.682%) |

Grid 2 | −1.996072644 (0.196%) | −1.995920807 (0.203%) | −1.99548624 (0.225%) | −1.99313533 (0.343%) |

Grid 3 | −1.997366868 (0.131%) | −1.997266348 (0.136%) | −1.996973186 (0.151%) | −1.995385341 (0.231%) |

Grid 4 | −1.984135078 (0.793%) | −1.983628605 (0.818%) | −1.981979149 (0.901%) | −1.973188713 (1.341%) |

Grid 5 uniform | −1.92467676435 (3.766%) | −1.92229302 (3.885%) | −1.917526191 (4.124%) | −1.903225703 (4.838%) |

For non-uniform grid:

For uniform grid:

(j, j + 1, j + 2, j + 3, j + 4) Scheme: fourth order

For non-uniform grid:

where

For uniform grid:

For uniform grid with step size h, coefficients

and the fourth-order expression for the first derivative takes the form:

The main theme of this work has been the development and testing of numerical schemes for the first derivative using uniform and non-uniform grids. The schemes have been applied to the computation of corner vorticity in the case of viscous fluid flow through a two-dimensional curvilinear channel that has been mapped onto a rectangular computational domain using the von Mises transformation.

The following has been accomplished in this work.

1) A fourth-order accurate scheme that uses 5 grid points has been developed and tested.

2) Third-order accurate schemes not previously reported have been developed and tested in this work.

Results have clearly demonstrated that:

1) Non-uniform grids produce more accurate values of corner vorticity than uniform grids.

2) The clustered grid that consistently produced the best approximation to corner vorticity is Grid 3, which is based on a stretching of physical coordinates using the cubic root function. This produced the smallest grid- spacing closest to the boundary in both the physical and computational domains.

3) The schemes that produced the best approximation to corner vorticity are the standard schemes which employ the natural order of grid points.

S. O. Alharbi, gratefully acknowledges support for this work through his PhD scholarship from Majmaah University, Kingdom of Saudi Arabia.

S. O. Alharbi,M. H. Hamdan, (2016) High-Order Finite Difference Schemes for the First Derivative in Von Mises Coordinates. Journal of Applied Mathematics and Physics,04,524-545. doi: 10.4236/jamp.2016.43059