[FMDenaro]

I also suggest to see a chapter in a old book where the multidimensional implementation (while taking into account the presence of the surface integral) is illustrated for the QUICK.

QUICKEST - Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms

B. P. Leonard. Elliptic systems: Finite-difference method IV. In W. J. Minkowycz, E. M. Sparrow, G. E. Schneider, and R. H. Pletcher, editors, Handbook of Numerical Heat Transfer, pages 347–378. Wiley, New York, 1988.

[mihirmakwana6]

Originally Posted by FMDenaro

well, I see the eq.(17) and let me comment a couple of issues:

1) the formula is indeed an upwind for the scalar value on a face, however the upwinding on the non linear flux (uu) would be different;

2) the formula is 1D, that means you do not see in it the surface integral that is conversely in effect in 2D and 3D cases. If you use the Eq.(17) as it is, is likely you get simply a second order accuracy in space in an accuracy testing.

Biased and pure upwinding in 1D were analysed in several papers. When I worked on this issue, we analysed the distribution of the errors in physical and spectral space for FD and FV approaches. If you are interested, you can see Section 4 in https://www.researchgate.net/publica...mes_simulation

you will find some "strange" conclusions ...

1) say i am looking at duu/dx

based on the AVERAGE value of velocity "u" ( first ) i.e whether +ve or -ve
at the face of u-CV, the upwinding for "u" (second) can be applied.


I will look at the paper ( section 4 )

[mihirmakwana6]

Originally Posted by FMDenaro

I also suggest to see a chapter in a old book where the multidimensional implementation (while taking into account the presence of the surface integral) is illustrated for the QUICK.

QUICKEST - Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms

B. P. Leonard. Elliptic systems: Finite-difference method IV. In W. J. Minkowycz, E. M. Sparrow, G. E. Schneider, and R. H. Pletcher, editors, Handbook of Numerical Heat Transfer, pages 347–378. Wiley, New York, 1988.

ok. thankyou.

I will take a look at the book

[FMDenaro]

Originally Posted by mihirmakwana6

1) say i am looking at duu/dx

based on the AVERAGE value of velocity "u" ( first ) i.e whether +ve or -ve
at the face of u-CV, the upwinding for "u" (second) can be applied.


I will look at the paper ( section 4 )

but in a FV method, you do not work on d (uu)/dx ...just think about the term Div (vv) in the differential form that must be integrated to produce the integral form. Therefore, you have to discretize (1/|V|) Int [S] n.(vv) dS.
You see that no derivative appears and the upwind must be applied on the flux function reconstruction.

[mihirmakwana6]

if say its a 1D flow,

after integrating duu/dx w.r.t x i get uu at the faceS of the u-C.V

Now, based on the AVERAGE value of velocity "u" ( first ) i.e whether +ve or -ve
at the face of u-CV, the upwinding for "u" (second) can be applied.

[FMDenaro]

Originally Posted by mihirmakwana6

if say its a 1D flow,

after integrating duu/dx w.r.t x i get uu at the faceS of the u-C.V

Now, based on the AVERAGE value of velocity "u" ( first ) i.e whether +ve or -ve
at the face of u-CV, the upwinding for "u" (second) can be applied.

If you consider ONLY the 1D case, you can work exactly as described in Section 4 in the paper I linked.
The integral will simply produce:

[(uu)est - (uu)west ]/h


Note that, whatever high order reconstruction you use for the flux, the time derivative acts on the volume averaged velocity, therefore you must always expect a second order accuracy in space in the error analys. A proper higher order scaling in the error curve requires special reconstruction bewteen averaged and point-wise velocity as described in Sec.4

[mihirmakwana6]

Originally Posted by mprinkey

That formula will likely work fine. That is just a midpoint interpolation with three upwind and two downwind samples. The difficulty here will be in implementing boundary conditions--either dropping back to lower order or building lopsided stencils that retain 5th order.

Note too that this form is highly accurate, but it is "not as upwind" as QUICK or other more stable upwinding schemes that make more heavy use of upwind data. So, I suspect that solution stability is going to more tenuous.

Near the boundary nodes, as i won't have enough points i cannot use the fifth order scheme. So i will have to use lower order scheme for near boundary points.
This will reduce the overall order of the scheme.

Is there a way ( you did mentioned building lopsided stencils that retain 5th order of which i have no idea ) where overall 5th order is maintained.

[mihirmakwana6]

Originally Posted by FMDenaro

If you consider ONLY the 1D case, you can work exactly as described in Section 4 in the paper I linked.
The integral will simply produce:

[(uu)est - (uu)west ]/h


Note that, whatever high order reconstruction you use for the flux, the time derivative acts on the volume averaged velocity, therefore you must always expect a second order accuracy in space in the error analys. A proper higher order scaling in the error curve requires special reconstruction bewteen averaged and point-wise velocity as described in Sec.4

ok. I will have a look at the paper.

[FMDenaro]

Originally Posted by mihirmakwana6

Near the boundary nodes, as i won't have enough points i cannot use the fifth order scheme. So i will have to use lower order scheme for near boundary points.
This will reduce the overall order of the scheme.

Is there a way ( you did mentioned building lopsided stencils that retain 5th order of which i have no idea ) where overall 5th order is maintained.

to mantain the same order on a boundary you have to use one-sided interpolation...however, let me say that some types of boundary can be treated. For example, on a wall you set just that the normal component of the convective flux integrated on the face of the FV is zero.

[mprinkey]

Originally Posted by FMDenaro

to mantain the same order on a boundary you have to use one-sided interpolation...however, let me say that some types of boundary can be treated. For example, on a wall you set just that the normal component of the convective flux integrated on the face of the FV is zero.

That problem is also for faces NEAR a boundary, not just on a boundary. Consider the face on the opposite side of a cell on a wall. The wall face can have zero flux, but the opposite face will want either 2 or 3 data points in the wall direction...depending on the upwinding direction. There is only one cell data point between it and the wall. So, at minimum, this will require interpolation formula with 4 points on one side and 1 point on the other and the direction of that stencil is dictated by the WALL location, not the direction of the flux.

[FMDenaro]

Originally Posted by mprinkey

That problem is also for faces NEAR a boundary, not just on a boundary. Consider the face on the opposite side of a cell on a wall. The wall face can have zero flux, but the opposite face will want either 2 or 3 data points in the wall direction...depending on the upwinding direction. There is only one cell data point between it and the wall. So, at minimum, this will require interpolation formula with 4 points on one side and 1 point on the other and the direction of that stencil is dictated by the WALL location, not the direction of the flux.

yes, you are right
As I wrote above, the solution is in the one-sided interpolation at high degree that does not depend on the upwinding. However, near a wall the normal velocity component will be very small and the diffusive flux will be predominant over the convective (in turbulence, we have at least 3-4 nodes within y+=1, i.e. the viscous sub-region). That justifies the fact one forces the stencil according to the wall location

[FMDenaro]

just consider that on a wall du/dx=dw/dz=0 (x streamwise, z spanwise), therefore from the continuity equation dv/dy=0 with v=0 at the wall...

[mprinkey]

Originally Posted by FMDenaro

yes, you are right
As I wrote above, the solution is in the one-sided interpolation at high degree that does not depend on the upwinding. However, near a wall the normal velocity component will be very small and the diffusive flux will be predominant over the convective (in turbulence, we have at least 3-4 nodes within y+=1, i.e. the viscous sub-region). That justifies the fact one forces the stencil according to the wall location

Of course. There are lots of ways to "cheat." 8) Lots of codes use high-order convective interpolate and just 2nd FV quadrature and get better results because the lower-order solutions were swamped by diffusion errors in the convective terms. Maybe still formally 2nd order, but much better resolution. That is why I mentioned the LES papers initially--all of these issues were discussed in detail in the early days of LES over "real" geometries.

Chasing uniform high-order accuracy is a difficult task, as I'm sure you well know.

[mihirmakwana6]

The paper
THREE-DIMENSIONAL INCOMPRESSIBLE FLOW CALCULATIONS WITH ALTERNATIVE DISCRETIZATION SCHEMES

by

Panos Tamamidis & Dennis N. Assanis


to which i referred for the 5th order upwind scheme given by eqn. 17 is

http://i.imgur.com/GlFRysZ.jpg?1


The paper to which they refer i.e Rai[13] is

http://i.imgur.com/Z8cEyLf.jpg?1

to which i don't have access

Now when i fit a 4th order polynomial with 3 upstream points and 2 downstream points
then i get different co-efficients

http://i.imgur.com/EgNDdw9.jpg?1

the way i fitted the polynomial is

phi(x=2.5dx) = L0*phi_d2 + L1*phi_d1 + L2*phi_u1 + L3*phi_u2 + L4*phi_u3

where Li are lagrange polynomial at i ( = 0 to 4) with x_i = i*dx

I used the same analogy for quick scheme and it gave me the correct co-efficients.

So, why a discrepancy with the formula ( eqn. 17 ) ??

[mprinkey]

Originally Posted by mihirmakwana6

The paper
THREE-DIMENSIONAL INCOMPRESSIBLE FLOW CALCULATIONS WITH ALTERNATIVE DISCRETIZATION SCHEMES

by

Panos Tamamidis & Dennis N. Assanis


to which i referred for the 5th order upwind scheme given by eqn. 17 is

http://i.imgur.com/GlFRysZ.jpg?1


The paper to which they refer i.e Rai[13] is

http://i.imgur.com/Z8cEyLf.jpg?1

to which i don't have access

Now when i fit a 4th order polynomial with 3 upstream points and 2 downstream points
then i get different co-efficients

http://i.imgur.com/EgNDdw9.jpg?1

the way i fitted the polynomial is

phi(x=2.5dx) = L0*phi_d2 + L1*phi_d1 + L2*phi_u1 + L3*phi_u2 + L4*phi_u3

where Li are lagrange polynomial at i ( = 0 to 4) with x_i = i*dx

I used the same analogy for quick scheme and it gave me the correct co-efficients.

So, why a discrepancy with the formula ( eqn. 17 ) ??

I can't speak to the veracity of your work. I didn't review it. But the Eqn 17 that you quoted is clearly wrong. Just by simple inspection, if ALL of the upwind and downwind samples are, say, 1, the interpolant should be 1 as well. And that is obviously not true for Eq 17. More generally, all of the coefficients on the interpolation scheme need to sum to unity. My guess is that the 540 coefficient should be 54. That, all by itself, would call the veracity of that reference into question. But I don't have either paper to review...nor really the time or motivation. There are plenty of well touted references about for simple finite differences interpolation schemes...they do date back all the way to Newton!

[Alisha]

Did you tried MUSCL?