Details of HOC Formulation

Let us write the steady 2-D incompressible Navier-Stokes equations in non-dimensional form. In streamfunction () and vorticity () formulation, the equations are the following

(1)

(2)

In order to solve these equation, let us use standard three point second order central discretization which are defined as the following

(3)

(4)

where  and  are defined as

(5)

(6)

If we use this second order discretization in equation (1 and 2) then we obtain the following second order accurate finite difference equations

(7)

(8)

In the literature there are many different numerical methods available for the solution of streamfunction and vorticity equations. Using any of these methods if we solve these two equations, the numerical solution we obtain for streamfunction, , and vorticity, , will satisfy the streamfunction and vorticity equations (1 and 2) with second order spatial accuracy, , that is to say our solution will be second order accurate.

Now let us obtain a fourth order compact formulation for the streamfunction and vorticity equations. The following discretizations are spatially fourth order accurate, .

(9)

(10)

In these discretizations, if we can somehow calculate the  and  terms and substitute them, then these discretizations will be fourth order accurate, . If we use these discretizations in streamfunction equation (1), we obtain the following equation

(11)

Now we need to obtain an expression for  and  terms. For these we use the streamfunction equation (1). If we take the x-derivatives of the streamfunction equation we obtain

(12)

(13)

and also if we take the y-derivatives of the streamfunction equation we obtain

(14)

(15)

Now, if we use standard second order central discretizations, defined in Table 1, in these equations then the following expressions are second order accurate

(16)

(17)

(18)

(19)

With the help of these second order accurate expressions, the following expression is fourth order accurate

(20)

similarly the following is also fourth order accurate

(21)

If we substitute these equations (20 and 21) into equation (11) we obtain the following equation

(22)

The solution of this equation (22), is also a solution to streamfunction equation (1) with fourth order spatial accuracy. Therefore if we numerically solve equation (22), the solution we obtain will satisfy the streamfunction equation up to fourth order accuracy .

Now let us obtain a fourth order approximation for vorticity equation. The following expressions are fourth order accurate

(23)

(24)

(25)

(26)

Substituting these (23, 24, 25 and 26) into the vorticity equation (2) we obtain

(27)

For the third order derivatives of the streamfunction appear in this equation (27), we use equations (16 and 17) such that the following terms become

(28)

(29)

In order to find an expression for the third and fourth order derivatives of vorticity, let us take the x- and y-derivatives of the vorticity equation, and here is what we get. The third x-derivative of vorticity is equal to

(30)

with the standard second order central discretization this equation becomes

(31)

Therefore the following term in equation (27) becomes

(32)

The third y-derivative of vorticity is equal to

(33)

with the standard second order central discretization this equation becomes

(34)

Therefore the following term in equation (27) becomes

(35)

The fourth x-derivative of vorticity is equal to

(36)

Again if we use standard second order central derivatives and also substitute for the third order derivatives of streamfunction and vorticity this equation becomes

(37)

Therefore the following term in equation (27) becomes

(38)

The fourth y-derivative of vorticity is equal to

(39)

Again if we use standard second order central derivatives and also substitute for the third order derivatives of streamfunction and vorticity this equation becomes

(40)

Therefore the following term in equation (27) becomes

(41)

If we substitute equations (28, 29, 32, 35, 38 and 41) into equation (27), and rearranging we obtain

(42)

The solution of this long equation (42), is also a solution to vorticity equation (2) with fourth order spatial accuracy. That means if we numerically solve equation (42), the solution we obtain will satisfy the vorticity equation (2) up to fourth order accuracy.

As the final form of the fourth order streamfunction and vorticity equations (22 and 42), we write them as the following

(43)

(44)

where

(45)

We note that the finite difference equations (43 and 44) are fourth order accurate  approximation of the streamfunction and vorticity equations (1 and 2). However, in these equations (43 and 44), if A, B, C, D, E and F are chosen to be equal to zero such that

(46)

then the finite difference equations (43 and 44) simply become

(47)

(48)

These equations (47 and 48) are the standard second order accurate approximation of the streamfunction and vorticity equations (1 and 2). With the way we write the final form of our fourth order equations (43 and 44), we can easily switch between second order and fourth order accuracy just by using homogeneous values for A, B, C, D, E and F or by calculating their values as defined in equation (45).

Now let us rewrite the equations (43, 44 and 45) such that in these equations we replace the finite difference expressions with partial derivatives and then we obtain

(49)

(50)

where

(51)

The equations (49 and 50) are in the same form with the streamfunction and vorticity equations (1 and 2). Therefore any numerical method available for the streamfunction and vorticity equations (1 and 2) can be easily applied to solve equations (49 and 50). We call these two equations (49 and 50) Fourth Order Navier-Stokes (FONS) equations. The solutions of these equations (49 and 50) are fourth order accurate to the streamfunction and vorticity equations (1 and 2) strictly provided that second order accurate central discretizations, tabulated in Table 1, are used and also strictly provided that a uniform grid mesh with and is used. The only difference between the FONS equations and N-S equations are the coefficients A, B, C, D, E and F. Since the FONS equations are in the same form with the N-S equations, any existing code that solve the N-S equations with second order accuracy can easily be modified to provide fourth order accuracy just by adding the coefficients A, B, C, D, E and F into the existing code to obtain the solution of the FONS equations.